On the 3D Navier-Stokes Equations with Stochastic Lie Transport

Abstract

We prove the existence and uniqueness of maximal solutions to the 3D SALT (Stochastic Advection by Lie Transport) Navier-Stokes Equation in velocity and vorticity form, on the torus and the bounded domain respectively. In particular we demonstrate the efficacy of Goodair et al. (Existence and Uniqueness of Maximal Solutions to SPDEs with Applications to Viscous Fluid Equations, 2023. Stochastics and Partial Differential Equations: Analysis and Computations, pp.1-64) in showing the well-posedness for both the velocity and vorticity form of the equation, as well as obtaining the first analytically strong existence result for a fluid equation perturbed by Lie transport noise on a bounded domain.

1 Introduction

The theoretical analysis of Stochastic Navier-Stokes Equations dates back to the work of Bensoussan and Temam [4] in 1973, where the problem of existence of solutions is addressed in the presence of a random forcing term. The well-posedness question for additive and multiplicative noise has since seen significant developments, for example through the works [1, 7, 24, 29, 39, 42, 43] and references therein. The interest in this problem has expanded into analytical properties of these solutions, particularly along the lines of ergodicity, which can be seen in [17, 18, 22, 26, 27, 34]. In the present work our concern is the Navier-Stokes Equations with Stochastic Lie Transport, derived through the principle of Stochastic Advection by Lie Transport (SALT) introduced in [35]. We consider the equation

$$\displaystyle \begin{aligned} {} u_t - u_0 + \int_0^t\mathcal{L}_{u_s}u_s\,ds - \nu\int_0^t \Delta u_s\, ds + \int_0^t Bu_s \circ d\mathcal{W}_s + \nabla \rho_t= 0 \end{aligned} $$

(1)

where u represents the fluid velocity, $\rho$ the pressure,¹ $\mathcal {W}$ is a Cylindrical Brownian Motion, $\mathcal {L}$ represents the nonlinear term and B is a first order differential operator (the SALT Operator) formally addressed in Sect. 2.3. Intrinsic to this stochastic methodology is that B is defined relative to a collection of functions $(\xi _i)$ which physically represent spatial correlations. These $(\xi _i)$ can be determined at coarse-grain resolutions from finely resolved numerical simulations, and mathematically are derived as eigenvectors of a velocity-velocity correlation matrix (see [10, 11, 12]). We pose the equation (1) in 3 dimensions and impose the divergence free constraint on u. We shall consider the problem both over the torus $\mathbb {T}^3$ and a smooth bounded domain $\mathscr {O} \subset \mathbb {R}^3$. In the case of the torus we supplement the equation with the zero-average condition (as is classical), whilst for the bounded domain we impose the boundary condition

$$\displaystyle \begin{aligned} {}u \cdot n = 0, \qquad w=0\end{aligned} $$

(2)

where n represents the outwards unit normal at the boundary, and w the fluid vorticity. These are the so called Lions boundary conditions, considered in [38] and shown to be a particular case of the Navier boundary conditions in [36] (note that this is done in 2D, whilst a treatment of the Navier boundary conditions in 3D can be found in [28]). The significance of such a boundary condition is well documented in that work by Kelliher, and can be seen in other works such as [28] where the boundary layer is explicitly addressed. The precise mathematical interpretation of these conditions, and the operators of (1), are explicated in Sect. 2.2. A complete derivation of this equation can be found in [45].

This work continues the theoretical development of fluid models perturbed by a transport type noise, the significance of which was posed as early as 1992 in the paper [6]. The area has garnered substantial attention in more recent years with the seminal works [35, 41], in which the authors establish a new class of stochastic equations driven by transport type noise which serve as fluid dynamics models by adding uncertainty in the transport of fluid parcels to reflect the unresolved scales. This paper partners that of [32] where we showed the existence and uniqueness of maximal solutions to Stochastic Partial Differential Equations satisfying an abstract framework, built to cope with a general transport type noise as we see in (1). The importance of such equations in modelling, numerical schemes and data assimilation is reviewed there, along with the theoretical developments of these equations: let us briefly mention some interesting results [3, 5, 21, 23]. We only draw particular attention here to the Navier-Stokes Equations, and results on a bounded domain. The Navier-Stokes Equations have been studied with transport type noise, for example in the works [14, 19, 25, 43], though typically solutions are analytically weak and where strong solutions are considered major concessions in the noise are made. In these cases a cancellation property is evident in the noise term, so the resulting energy balance is formally the same as for the deterministic equation without noise. These difficulties have been addressed on the torus, in the likes of the papers [13, 37] and those further addressed in [32], but extending a control of this noise term to a bounded domain remains open. Furthermore, whilst the presence of viscosity does improve the solution theory, it invites additional challenges in controlling the noise. Energy methods require non-standard Sobolev inner products to conduct the required integration by parts for the viscous term in the bounded domain, so we must provide novel estimates on the transport type noise in these inner products. The problem of analytically strong solutions to fluid equations perturbed by a transport type noise in the bounded domain has been considered in [8], though the authors assume that the gradient dependency is of a small enough size to be directly controlled and that the noise terms are traceless under Leray Projection; such assumptions are designed to circumvent the technical difficulties of a first order noise operator on a bounded domain, and is a luxury that the SALT equations do not have.

The goal of this paper is to apply the abstract framework established in [32], providing a rigorous justification of the results first announced in [31] and extending them to the vorticity form of Eq. (1) on a bounded domain. The purpose of this is twofold:

• 1. To demonstrate the efficacy of the criteria from [32]. This is most pertinent in our treatment of the velocity form on the torus; whilst the well-posedness results that we present are new, one would expect them to hold. This belief is compounded by the well-posedness results for both viscous and inviscid fluid equations perturbed by Lie transport noise on the torus [2, 13, 15, 16, 37] for which the Navier-Stokes equations do not present any intrinsic additional difficulty, though still contributing a necessary piece to the somewhat sparse literature. We hope to convey that verifying the assumptions of [32] is now the simplest and most efficient way to prove the well-posedness of this equation and, by extension, associated stochastic viscous fluid equations.

• 2. To obtain the well-posedness of the equation in the presence of a boundary, which does present intrinsic additional difficulty beyond the existing results.

In the interests of brevity we provide a shorter manuscript here, though greater detail can be found in the former arXiv version [33]. In Sect. 2 we establish the stochastic and functional framework necessary to understand (1), along with fundamental properties of the operators involved. In Sect. 3 we make precise how equation (1) fits into the framework of [32], as a problem posed on the torus $\mathbb {T}^3$. In Sect. 4 we consider the vorticity form of equation (1) as a problem posed on a bounded domain of $\mathbb {R}^3$. We again justify the assumptions in [32] to prove existence and uniqueness of maximal solutions to this equation. Additional details for the proofs are given in Sect. 5, along with the results of the partnering paper [32].

2 Preliminaries

2.1 Elementary Notation

In the following $\mathcal {O}$ can represent both the 3-dimensional torus $\mathbb {T}^3$ and a smooth bounded domain $\mathscr {O} \subset \mathbb {R}^3$. We consider Banach Spaces as measure spaces equipped with the Borel $\sigma$-algebra, and use $\lambda$ to represent the Lebesgue Measure. Let $(\mathcal {X},\mu )$ denote a general measure space, $(\mathcal {Y},\lVert \cdot \rVert _{\mathcal {Y}})$ and $(\mathcal {Z},\lVert \cdot \rVert _{\mathcal {Z}})$ be Banach Spaces, and $(\mathcal {U},\langle \cdot , \cdot \rangle _{\mathcal {U}})$, $(\mathcal {H},\langle \cdot , \cdot \rangle _{\mathcal {H}})$ be general Hilbert spaces. $\mathcal {O}$ is equipped with Euclidean norm.

• $L^p(\mathcal {X};\mathcal {Y})$ is the class of measurable p-integrable functions from $\mathcal {X}$ into $\mathcal {Y}$, $1 \leq p < \infty$, which is a Banach space with norm

  $$\displaystyle \begin{aligned} \lVert \phi \rVert _{L^p(\mathcal{X};\mathcal{Y})}^p := \int_{\mathcal{X}}\lVert \phi(x) \rVert ^p_{\mathcal{Y}}\mu(dx).\end{aligned}$$

  In particular $L^2(\mathcal {X}; \mathcal {Y})$ is a Hilbert Space when $\mathcal {Y}$ itself is Hilbert, with the standard inner product

  $$\displaystyle \begin{aligned} \langle \phi, \psi \rangle _{L^2(\mathcal{X}; \mathcal{Y})} = \int_{\mathcal{X}}\langle \phi(x), \psi(x) \rangle _{\mathcal{Y}} \mu(dx).\end{aligned}$$

  In the case $\mathcal {X} = \mathcal {O}$ and $\mathcal {Y} = \mathbb {R}^3$ note that

  

  We denote $\lVert \cdot \rVert _{L^p(\mathcal {O};\mathbb {R}^3)}$ by $\lVert \cdot \rVert _{L^p}$ and $\lVert \cdot \rVert _{L^2(\mathcal {O};\mathbb {R}^3)}$ by $\lVert \cdot \rVert$.

• $L^{\infty }(\mathcal {X};\mathcal {Y})$ is the class of measurable functions from $\mathcal {X}$ into $\mathcal {Y}$ which are essentially bounded, which is a Banach Space when equipped with the norm

  $$\displaystyle \begin{aligned} \lVert \phi \rVert _{L^{\infty}(\mathcal{X};\mathcal{Y})} := \inf\{C \geq 0: \lVert \phi(x) \rVert _Y \leq C \mathrm{ for }\mu\mathrm{-}a.e.\ x \in \mathcal{X}\}.\end{aligned}$$

• $L^{\infty }(\mathcal {O};\mathbb {R}^3)$ is the class of measurable functions from $\mathcal {O}$ into $\mathbb {R}^3$ such that $\phi ^l \in L^{\infty }(\mathcal {O};\mathbb {R})$ for $l=1,\dots ,N$, which is a Banach Space when equipped with the norm

  $$\displaystyle \begin{aligned} \lVert \phi \rVert _{L^{\infty}}:= \sup_{l \leq N}\lVert \phi^l \rVert _{L^{\infty}(\mathcal{O};\mathbb{R})}.\end{aligned}$$

• $C(\mathcal {X};\mathcal {Y})$ is the space of continuous functions from $\mathcal {X}$ into $\mathcal {Y}$.

• $C^m(\mathcal {O};\mathbb {R})$ is the space of $m \in \mathbb {N}$ times continuously differentiable functions from $\mathcal {O}$ to $\mathbb {R}$, that is $\phi \in C^m(\mathcal {O};\mathbb {R})$ if and only if for every N dimensional multi index $\alpha = \alpha _1, \dots , \alpha _N$ with $\lvert \alpha \rvert \leq m$, $D^\alpha \phi \in C(\mathcal {O};\mathbb {R})$ where $D^\alpha$ is the corresponding classical derivative operator $\partial _{x_1}^{\alpha _1} \dots \partial _{x_N}^{\alpha _N}$.

• $C^\infty (\mathcal {O};\mathbb {R})$ is the intersection over all $m \in \mathbb {N}$ of the spaces $C^m(\mathcal {O};\mathbb {R})$.

• $C^m_0(\mathscr {O};\mathbb {R})$ for $m \in \mathbb {N}$ or $m = \infty$ is the subspace of $C^m(\mathscr {O};\mathbb {R})$ of functions which have compact support.

• $C^m(\mathcal {O};\mathbb {R}^3), C^m_0(\mathscr {O};\mathbb {R}^3)$ for $m \in \mathbb {N}$ or $m = \infty$ is the space of functions from $\mathcal {O}, \mathscr {O}$ to $\mathbb {R}^3$ whose N component mappings each belong to $C^m(\mathcal {O};\mathbb {R}), C^m_0(\mathscr {O};\mathbb {R})$.

• $W^{m,p}(\mathcal {O}; \mathbb {R})$ for $1 \leq p < \infty$ is the sub-class of $L^p(\mathcal {O}, \mathbb {R})$ which has all weak derivatives up to order $m \in \mathbb {N}$ also of class $L^p(\mathcal {O}, \mathbb {R})$. This is a Banach space with norm

  $$\displaystyle \begin{aligned} \lVert \phi \rVert ^p_{W^{m,p}(\mathcal{O}, \mathbb{R})} := \sum_{\lvert \alpha\rvert \leq m}\lVert D^\alpha \phi \rVert _{L^p(\mathcal{O}; \mathbb{R})}^p\end{aligned}$$

  where $D^\alpha$ is the corresponding weak derivative operator. In the case $p=2$ the space $W^{m,2}(\mathcal {O}, \mathbb {R})$ is Hilbert with inner product

  $$\displaystyle \begin{aligned} \langle \phi, \psi \rangle _{W^{m,2}(\mathcal{O}; \mathbb{R})} := \sum_{\lvert \alpha\rvert \leq m} \langle D^\alpha \phi, D^\alpha \psi \rangle _{L^2(\mathcal{O}; \mathbb{R})}.\end{aligned}$$

• $W^{m,\infty }(\mathcal {O};\mathbb {R})$ for $m \in \mathbb {N}$ is the sub-class of $L^\infty (\mathcal {O}, \mathbb {R})$ which has all weak derivatives up to order $m \in \mathbb {N}$ also of class $L^\infty (\mathcal {O}, \mathbb {R})$. This is a Banach space with norm

  $$\displaystyle \begin{aligned} \lVert \phi \rVert _{W^{m,\infty}(\mathcal{O}, \mathbb{R})} := \sup_{\lvert \alpha\rvert \leq m}\lVert D^{\alpha}\phi \rVert _{L^{\infty}(\mathcal{O};\mathbb{R}^3)}.\end{aligned}$$

• $W^{m,p}(\mathcal {O}; \mathbb {R}^3)$ for $1 \leq p < \infty$ is the sub-class of $L^p(\mathcal {O}, \mathbb {R}^3)$ which has all weak derivatives up to order $m \in \mathbb {N}$ also of class $L^p(\mathcal {O}, \mathbb {R}^3)$. This is a Banach space with norm

  $$\displaystyle \begin{aligned} \lVert \phi \rVert ^p_{W^{m,p}} := \sum_{l=1}^3\lVert \phi^l \rVert _{W^{m,p}(\mathcal{O}; \mathbb{R})}^p.\end{aligned}$$

  In the case $p=2$ the space $W^{m,2}(\mathcal {O}, \mathbb {R}^3)$ is Hilbertian with inner product

  $$\displaystyle \begin{aligned} \langle \phi, \psi \rangle _{W^{m,2}} := \sum_{l=1}^3 \langle \phi^l, \psi^l \rangle _{W^{m,2}(\mathcal{O}; \mathbb{R})}.\end{aligned}$$

• $W^{m,\infty }(\mathcal {O}; \mathbb {R}^3)$ is the sub-class of $L^\infty (\mathcal {O}, \mathbb {R}^3)$ which has all weak derivatives up to order $m \in \mathbb {N}$ also of class $L^\infty (\mathcal {O}, \mathbb {R}^3)$. This is a Banach space with norm

  $$\displaystyle \begin{aligned} \lVert \phi \rVert _{W^{m,\infty}(\mathcal{O}, \mathbb{R}^3)} := \sup_{l \leq N}\lVert \phi^l \rVert _{W^{m,\infty}(\mathcal{O}; \mathbb{R})}.\end{aligned}$$

• $\dot {L}^2(\mathbb {T}^3;\mathbb {R}^3)$ is the subset of $L^2(\mathbb {T}^3;\mathbb {R}^3)$ of functions $\phi$ such that

  $$\displaystyle \begin{aligned} \int_{\mathbb{T}^3}\phi \ d\lambda = 0.\end{aligned}$$

• $\dot {W}^{m,2}(\mathbb {T}^3;\mathbb {R}^3)$ is simply the intersection $W^{m,2}(\mathbb {T}^3;\mathbb {R}^3) \cap \dot {L}^2(\mathbb {T}^3;\mathbb {R}^3)$.

• $W^{m,p}_0(\mathscr {O};\mathbb {R}), W^{m,p}_0(\mathscr {O};\mathbb {R}^3)$ for $m \in N$ and $1 \leq p \leq \infty$ is the closure of $C^\infty _0(\mathscr {O};\mathbb {R}),C^\infty _0(\mathscr {O};\mathbb {R}^3)$ in $W^{m,p}(\mathscr {O};\mathbb {R}), W^{m,p}(\mathscr {O};\mathbb {R}^3)$.

• $\mathscr {L}(\mathcal {Y};\mathcal {Z})$ is the space of bounded linear operators from $\mathcal {Y}$ to $\mathcal {Z}$. This is a Banach Space when equipped with the norm

  $$\displaystyle \begin{aligned} \lVert F \rVert _{\mathscr{L}(\mathcal{Y};\mathcal{Z})} = \sup_{\lVert y \rVert _{\mathcal{Y}}=1}\lVert Fy \rVert _{\mathcal{Z}}\end{aligned}$$

  and is simply the dual space $\mathcal {Y}^*$ when $\mathcal {Z}=\mathbb {R}$, with operator norm $\lVert \cdot \rVert _{\mathcal {Y}^*}.$

• $\mathscr {L}^2(\mathcal {U};\mathcal {H})$ is the space of Hilbert-Schmidt operators from $\mathcal {U}$ to $\mathcal {H}$, defined as the elements $F \in \mathscr {L}(\mathcal {U};\mathcal {H})$ such that for some basis $(e_i)$ of $\mathcal {U}$,

  $$\displaystyle \begin{aligned} \sum_{i=1}^\infty \lVert Fe_i \rVert _{\mathcal{H}}^2 < \infty.\end{aligned}$$

  This is a Hilbert space with inner product

  $$\displaystyle \begin{aligned} \langle F, G \rangle _{\mathscr{L}^2(\mathcal{U};\mathcal{H})} = \sum_{i=1}^\infty \langle Fe_i, Ge_i \rangle _{\mathcal{H}}\end{aligned}$$

  which is independent of the choice of basis.

We will consider a partial ordering on the $3-$dimensional multi-indices by $\alpha \leq \beta$ if and only if for all $l =1, 2, 3$ we have that $\alpha _l \leq \beta _l$. We extend this to notation $<$ by $\alpha < \beta$ if and only if $\alpha \leq \beta$ and for some $l = 1, 2, 3$, $\alpha _l < \beta _l$.

We also now introduce some less familiar spaces in slightly greater detail. We recall notation that $\mathscr {O}$ represents a smooth bounded domain in $\mathbb {R}^3$ which we now fix, $\mathbb {T}^3$ is the $3-$dimensional torus, and $\mathcal {O}$ freely denotes both $\mathbb {T}^3$ and $\mathscr {O}$.

Definition 2.1

We define $C^{\infty }_{0,\sigma }(\mathscr {O};\mathbb {R}^3)$ as the subset of $C^{\infty }_0(\mathscr {O};\mathbb {R}^3)$ of functions which are divergence-free. $L^2_\sigma (\mathcal {O};\mathbb {R}^3)$ is defined as the completion of $C^{\infty }_{0,\sigma }(\mathscr {O};\mathbb {R}^3)$ in $L^2(\mathscr {O};\mathbb {R}^3)$, whilst we introduce $W^{1,2}_\sigma (\mathscr {O};\mathbb {R}^3)$ as the intersection of $W^{1,2}_0(\mathscr {O};\mathbb {R}^3)$ with $L^2_\sigma (\mathscr {O};\mathbb {R}^3)$ and $W^{2,2}_{\sigma } (\mathscr {O};\mathbb {R}^3)$ as the intersection of $W^{2,2}(\mathscr {O};\mathbb {R}^3)$ with $W^{1,2}_\sigma (\mathscr {O};\mathbb {R}^3)$.

Recall that any function $f \in L^2(\mathbb {T}^3;\mathbb {R}^3)$ admits the representation

$$\displaystyle \begin{aligned} {}f(x) = \sum_{k \in \mathbb{Z}^3}f_ke^{ik\cdot x}\end{aligned} $$

(3)

whereby each $f_k \in \mathbb {C}^3$ is such that $f_k = \mkern 1.5mu\overline {\mkern -1.5muf_{-k}\mkern -1.5mu}\mkern 1.5mu$ and the infinite sum is defined as a limit in $L^2(\mathbb {T}^3;\mathbb {R}^3)$, see e.g. [44] Subsection 1.5 for details.

Definition 2.2

We define $L^2_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)$ as the subset of $\dot {L}^2(\mathbb {T}^3;\mathbb {R}^3)$ of functions f whereby for all ), $k \cdot f_k = 0$ with $f_k$ as in (3). For general $m \in \mathbb {N}$ we introduce $W^{m,2}_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)$ as the intersection of $W^{m,2}(\mathbb {T}^3;\mathbb {R}^3)$ respectively with $L^2_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)$.

Note that $W^{1,2}_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)$ is precisely the subspace of $W^{1,2}(\mathbb {T}^3;\mathbb {R}^3)$ consisting of zero-average divergence free functions. Similarly $W^{1,2}_{\sigma }(\mathscr {O};\mathbb {R}^3)$ is precisely the subspace of $W^{1,2}_0(\mathscr {O};\mathbb {R}^3)$ consisting of divergence free functions. Moreover, $W^{1,2}_{\sigma }(\mathscr {O};\mathbb {R}^3)$ is the completion of $C^{\infty }_{0,\sigma }(\mathscr {O};\mathbb {R}^3)$ in $W^{1,2}(\mathscr {O};\mathbb {R}^3)$. The general space $W^{1,2}_{\sigma }(\mathcal {O};\mathbb {R}^3)$ thus incorporates the divergence-free and zero-average/zero-trace condition.

As for the stochastic set up, let $(\Omega ,\mathcal {F},(\mathcal {F}_t), \mathbb {P})$ be a fixed filtered probability space satisfying the usual conditions of completeness and right continuity. We take $\mathcal {W}$ to be a cylindrical Brownian Motion over some Hilbert Space $\mathfrak {U}$ with orthonormal basis $(e_i)$. Recall ([30], Subsection 1.4) that $\mathcal {W}$ admits the representation $\mathcal {W}_t = \sum _{i=1}^\infty e_iW^i_t$ as a limit in $L^2(\Omega ;\mathfrak {U}')$ whereby the $(W^i)$ are a collection of i.i.d. standard real valued Brownian Motions and $\mathfrak {U}'$ is an enlargement of the Hilbert Space $\mathfrak {U}$ such that the embedding ) is Hilbert-Schmidt and $\mathcal {W}$ is a $JJ^*-$cylindrical Brownian Motion over $\mathfrak {U}'$. Given a process ) progressively measurable and such that $F \in L^2\left (\Omega \times [0,T];\mathscr {L}^2(\mathfrak {U};\mathscr {H})\right )$, for any $0 \leq t \leq T$ we define the stochastic integral

$$\displaystyle \begin{aligned} \int_0^tF_sd\mathcal{W}_s:=\sum_{i=1}^\infty \int_0^tF_s(e_i)dW^i_s\end{aligned}$$

where the infinite sum is taken in $L^2(\Omega ;\mathscr {H})$. We can extend this notion to processes F which are such that $F(\omega ) \in L^2\left ( [0,T];\mathscr {L}^2(\mathfrak {U};\mathscr {H})\right )$ for $\mathbb {P}-a.e.$ $\omega$ via the traditional localisation procedure. In this case the stochastic integral is a local martingale in $\mathscr {H}$.²

2.2 Functional Framework

We now recap the classical functional framework for the study of the deterministic Navier-Stokes Equation. Firstly we briefly comment on the pressure term $\nabla \rho$, which will not play any role in our analysis. $\rho$ does not come with an evolution equation and is simply chosen to ensure the incompressibility (divergence-free) condition; moreover we will ignore this term via a suitable projection (in Sect. 3 we even consider a different form of the equation) and treat the projected equation, with the understanding to append a pressure to it later. This procedure is well discussed in [44] Sect. 5 and [40], and an explicit form for the pressure for the SALT Euler Equation is given in [45] Subsection 3.3.

The mapping $\mathcal {L}$ is defined for sufficiently regular functions ) by $\mathcal {L}_fg:= \sum _{j=1}^3f^j\partial _jg.$ Here and throughout the text we make no notational distinction between differential operators acting on a vector valued function or a scalar valued one; that is, we understand $\partial _jg$ by its component mappings $(\partial _lg)^l := \partial _jg^l$. We now give some clarification as to ‘sufficiently regular’, by stating basic properties of this mapping. For any $m \geq 1$, the mapping ) defined by $f \mapsto \mathcal {L}_ff$ is continuous. Additionally there exists a constant c such that for any $f,g \in W^{k,2}(\mathcal {O};\mathbb {R}^3)$ for $k \in \mathbb {N}$ as appropriate, we have the bounds:

$$\displaystyle \begin{aligned} {} \lVert \mathcal{L}_{f}{g} \rVert + \lVert \mathcal{L}_{g}{f} \rVert &\leq c\lVert g \rVert _{W^{1,2}}\lVert f \rVert _{W^{2,2}}; \end{aligned} $$

(4)

$$\displaystyle \begin{aligned} {} \lVert \mathcal{L}_{g}{f} \rVert _{W^{1,2}} &\leq c\lVert g \rVert _{W^{1,2}}\lVert f \rVert _{W^{3,2}}; \end{aligned} $$

(5)

$$\displaystyle \begin{aligned} {} \lVert \mathcal{L}_{g}{f} \rVert _{W^{1,2}} &\leq c\lVert g \rVert _{W^{2,2}}\lVert f \rVert _{W^{2,2}}. \end{aligned} $$

(6)

We introduce the Leray Projector $\mathcal {P}$ as the orthogonal projection in $L^2(\mathcal {O};\mathbb {R}^3)$ onto $L^2_{\sigma }(\mathcal {O};\mathbb {R}^3)$. It is well known (see e.g. [47] Remark 1.6.) that for any $m \in \mathbb {N}$, $\mathcal {P}$ is continuous as a mapping ). In fact, the complement space of $L^2_{\sigma }(\mathcal {O};\mathbb {R}^3)$ can be characterised (this is the so called Helmholtz-Weyl decomposition), a result that we state explicitly as we will need to exploit the precise structure in future arguments.

Lemma 2.3

Define the space

$$\displaystyle \begin{aligned} L^{2, \perp}_{\sigma}(\mathcal{O};\mathbb{R}^3):= \{\psi \in L^{2}(\mathcal{O};\mathbb{R}^3): \psi = \nabla g \mathrm{ for some } g \in W^{1,2}(\mathcal{O};\mathbb{R}) \}.\end{aligned}$$

Then indeed $L^{2, \perp }_{\sigma }(\mathcal {O};\mathbb {R}^3)$ is orthogonal to $L^{2}_{\sigma }(\mathcal {O};\mathbb {R}^3)$ in $L^{2}(\mathcal {O};\mathbb {R}^3)$ , i.e. for any $\phi \in L^{2}_{\sigma }(\mathcal {O};\mathbb {R}^3)$ and $\psi \in L^{2, \perp }_{\sigma }(\mathcal {O};\mathbb {R}^3)$ we have that

$$\displaystyle \begin{aligned} \langle \phi, \psi \rangle =0.\end{aligned}$$

Moreover, every $f \in L^{2}(\mathscr {O};\mathbb {R}^3)$ has the unique decomposition

$$\displaystyle \begin{aligned} {}f = \phi + \psi\end{aligned} $$

(7)

for some $\phi \in L^{2}_{\sigma }(\mathcal {O};\mathbb {R}^3)$, $\psi \in L^{2, \perp }_{\sigma }(\mathcal {O};\mathbb {R}^3)$ and every $f \in L^{2}(\mathbb {T}^3;\mathbb {R}^3)$ has the unique decomposition

$$\displaystyle \begin{aligned} {} f = \phi + \psi + c \end{aligned} $$

(8)

where $\phi \in L^{2}_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)$, $\psi \in L^{2, \perp }_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)$ and c is a constant function: that is, there exists $k \in \mathbb {R}^3$ such that each component mapping $c^j$ is identically equal to $k^j$, $j=1, 2, 3$.

Proof

See [47] Theorems 1.4, 1.5 and [44] Theorem 2.6 . □

Corollary 2.3.1

Every $f \in L^{2}(\mathscr {O};\mathbb {R}^3)$ admits the representation

$$\displaystyle \begin{aligned} {}f = \mathcal{P}f + \nabla g\end{aligned} $$

(9)

for some $g\in W^{1,2} (\mathscr {O};\mathbb {R})$ . Similarly every $f \in L^{2}(\mathbb {T}^3;\mathbb {R}^3)$ admits the representation

$$\displaystyle \begin{aligned} {}f = \mathcal{P}f + \nabla g + c\end{aligned} $$

(10)

for some $g\in W^{1,2} (\mathbb {T}^3;\mathbb {R})$ and constant function c.

Proof

It is an immediate property of the orthogonal projection that $\mathcal {P}f$ is the unique element $\phi \in L^{2}_{\sigma }(\mathcal {O};\mathbb {R}^3)$ of (7) and (8). □

Through $\mathcal {P}$ we define the Stokes Operator ) by $A:= -\mathcal {P}\Delta$. Once more we understand the Laplacian as an operator on vector valued functions through the component mappings, $(\Delta f)^l := \Delta f^l$. From the continuity of $\mathcal {P}$ we have immediately that for $m \in \{0\} \cup \mathbb {N}$, ) is continuous. We remark that $\Delta$ leaves the complement space $L^{2,\perp }_{\sigma }(\mathcal {O};\mathbb {R}^3)$ invariant, so $A\mathcal {P}$ is equal to A on $W^{2,2}(\mathcal {O};\mathbb {R}^3)$. Moreover (see [44] Theorem 2.24) there exists a collection of functions $(a_k)$, $a_k \in W^{1,2}_{\sigma }(\mathcal {O};\mathbb {R}^3) \cap C^{\infty }(\mkern 1.5mu\overline {\mkern -1.5mu\mathcal {O}\mkern -1.5mu}\mkern 1.5mu;\mathbb {R}^3)$ such that the $(a_k)$ are eigenfunctions of A, are an orthonormal basis in $L^2_{\sigma }(\mathcal {O};\mathbb {R}^3)$ and an orthogonal basis in $W^{1,2}_{\sigma }(\mathcal {O};\mathbb {R}^3)$. The eigenvalues $(\lambda _k)$ are strictly positive and approach infinity as ). Therefore every $f \in L^2_{\sigma }(\mathcal {O};\mathbb {R}^3)$ admits the representation

$$\displaystyle \begin{aligned} {} f = \sum_{k=1}^\infty f_ka_k \end{aligned} $$

(11)

where $f_k = \langle f, a_k \rangle$, as a limit in $L^2(\mathcal {O};\mathbb {R}^3)$.

Definition 2.4

For $m \in \mathbb {N}$ we introduce the spaces $D(A^{m/2})$ as the subspaces of $L^2_{\sigma }(\mathcal {O};\mathbb {R}^3)$ consisting of functions f such that

$$\displaystyle \begin{aligned} \sum_{k=1}^\infty \lambda_k^m f_k^2 < \infty\end{aligned}$$

for $f_k$ as in (11). Then ) is defined by

$$\displaystyle \begin{aligned} A^{m/2}:f \mapsto \sum_{k=1}^\infty \lambda_k^{m/2}f_ka_k.\end{aligned}$$

We present some fundamental properties regarding these spaces, which are justified in [9] Proposition 4.12, [44] Exercises 2.12, 2.13 and the discussion in Subsection 2.3.

• 1. $D(A^{m/2}) \subset W^{m,2}(\mathcal {O};\mathbb {R}^3) \cap W^{1,2}_{\sigma }(\mathcal {O};\mathbb {R}^3)$ and the bilinear form

  $$\displaystyle \begin{aligned} \langle f, g \rangle _m:= \langle A^{m/2}f, A^{m/2}g \rangle\end{aligned}$$

  is an inner product on $D(A^{m/2})$;

• 2. For m even the induced norm $\lVert \cdot \rVert _m^2 = \langle \cdot , \cdot \rangle _m$ is equivalent to the $W^{m,2}(\mathcal {O};\mathbb {R}^3)$ norm, and for m odd there is a constant c such that

  $$\displaystyle \begin{aligned} \lVert \cdot \rVert _{W^{m,2}} \leq c\lVert \cdot \rVert _m;\end{aligned}$$

• 3. $D(A) = W^{2,2}_{\sigma }(\mathcal {O};\mathbb {R}^3)$ and $D(A^{1/2}) = W^{1,2}_{\sigma }(\mathcal {O};\mathbb {R}^3)$ with the additional property that $\lVert \cdot \rVert _1$ is equivalent to $\lVert \cdot \rVert _{W^{1,2}}$ on this space.

It can be directly shown that for any $p,q \in \mathbb {N}$ with $p \leq q$, $p + q = 2m$ and $f \in D(A^{m/2})$, $g \in D(A^{q/2})$ we have that

$$\displaystyle \begin{aligned} {}\langle f, g \rangle _m = \langle A^{p/2}f, A^{q/2}g \rangle .\end{aligned} $$

(12)

From here we can also see that the collection of functions $(a_k)$ form an orthogonal basis of $W^{1,2}_{\sigma }(\mathcal {O};\mathbb {R}^3)$ equipped with the $\langle \cdot , \cdot \rangle _1$ inner product. In addition to using these spaces defined by powers of the Stokes Operator, we also use the basis to consider finite dimensional approximations of these spaces.

Definition 2.5

We define $\mathcal {P}_n$ as the orthogonal projection onto $\mathrm {span}\{a_1, \dots , a_n\}$ in $L^2(\mathcal {O};\mathbb {R}^3)$. That is $\mathcal {P}_n$ is given by

$$\displaystyle \begin{aligned} \mathcal{P}_n:f \mapsto \sum_{k=1}^n\langle f, a_k \rangle a_k\end{aligned}$$

for $f \in L^2(\mathcal {O};\mathbb {R}^3)$.

The restriction of $\mathcal {P}_n$ to $D(A^{m/2})$ is self-adjoint for the $\langle \cdot , \cdot \rangle _m$ inner product, and there exists a constant c independent of n such that for all $f\in D(A^{m/2})$,

$$\displaystyle \begin{aligned} {} \lVert \mathcal{P}_nf \rVert _{W^{m,2}} \leq c\lVert f \rVert _{W^{m,2}}, \end{aligned} $$

(13)

see [44] Lemma 4.1 for details. Similar ideas justify that for all $f \in W^{1,2}_{\sigma }(\mathcal {O};\mathbb {R}^3)$, $g \in W^{2,2}_{\sigma }(\mathcal {O};\mathbb {R}^3)$,

$$\displaystyle \begin{aligned} \lVert (I-\mathcal{P}_n)f \rVert ^2 \leq \frac{1}{\lambda_n}\lVert f \rVert _1^2, \qquad \lVert (I-\mathcal{P}_n)g \rVert _1^2 \leq \frac{1}{\lambda_n}\lVert g \rVert _2^2 \end{aligned} $$

where I represents the identity operator in the relevant spaces. To conclude this subsection we discuss briefly bounds related to the nonlinear term, which will be used in our analysis. For every $\phi \in W^{1,2}_{\sigma }(\mathcal {O};\mathbb {R}^3)$ and $f, g \in W^{1,2}(\mathcal {O};\mathbb {R}^3)$, we have that

$$\displaystyle \begin{aligned} {}\langle \mathcal{L}_{\phi}f, g \rangle &= -\langle f, \mathcal{L}_{\phi}g \rangle \end{aligned} $$

(14)

$$\displaystyle \begin{aligned} {} \langle \mathcal{L}_{\phi}f, f \rangle &= 0.\end{aligned} $$

(15)

2.3 The SALT Operator

Having established the relevant function spaces and some fundamental properties of the operators involved in the deterministic Navier-Stokes Equation, we now address the operator B appearing in the Stratonovich integral of (1). As in [30] Subsection 2.2, the operator B is defined by its action on the basis vectors $(e_i)$ of $\mathfrak {U}$. We shall show in Sect. 3.3 that B does indeed satisfy Assumption 2.2.2 of [30] for the spaces to $V,H,U,X$ to be defined. With the notation of [30], each $B_i$ is defined relative to the correlations $\xi _i$ for sufficiently regular f by the mapping

$$\displaystyle \begin{aligned} B_i:f \mapsto \mathcal{L}_{\xi_i}f + \mathcal{T}_{\xi_i}f\end{aligned}$$

where $\mathcal {L}$ is as before, and $\mathcal {T}$ is a new operator that we introduce defined by

$$\displaystyle \begin{aligned} \mathcal{T}_{g}f := \sum_{j=1}^3 f^j\nabla g^j.\end{aligned}$$

We shall assume throughout that each $\xi _i$ belongs to the space $W^{1,2}_{\sigma }(\mathcal {O};\mathbb {R}^3)$. If for some fixed $m \in \mathbb {N}$ we have $\xi _i \in W^{m+2,\infty }(\mathcal {O};\mathbb {R}^3)$ then for all $k = 0, \dots , m+1$,

$$\displaystyle \begin{aligned} {}\lVert \mathcal{T}_{\xi_i}f \rVert _{W^{k,2}}^2 &\leq c \lVert \xi_i \rVert ^2_{W^{k+1,\infty}}\lVert f \rVert ^2_{W^{k,2}} \end{aligned} $$

(16)

$$\displaystyle \begin{aligned} {} \lVert \mathcal{L}_{\xi_i}f \rVert _{W^{k,2}}^2 &\leq c\lVert \xi_i \rVert ^2_{W^{k,\infty}}\lVert f \rVert ^2_{W^{k+1,2}} \end{aligned} $$

(17)

$$\displaystyle \begin{aligned} {} \lVert B_if \rVert _{W^{k,2}}^2 &\leq c\lVert \xi_i \rVert ^2_{W^{k+1,\infty}}\lVert f \rVert ^2_{W^{k+1,2}}. \end{aligned} $$

(18)

Moreover $\mathcal {T}_{\xi _i}$ is a bounded linear operator on $L^2(\mathcal {O};\mathbb {R}^3)$ so has adjoint $\mathcal {T}_{\xi _i}^*$ satisfying the same boundedness. In conjunction with property (14), $\mathcal {L}_{\xi _i}$ is a densely defined operator in $L^2(\mathcal {O};\mathbb {R}^3)$ with domain of definition $W^{1,2}(\mathcal {O};\mathbb {R}^3)$, and has adjoint $\mathcal {L}_{\xi _i}^*$ in this space given by $-\mathcal {L}_{\xi _i}$ with same dense domain of definition. Likewise then $B_i^*$ is the densely defined adjoint $-\mathcal {L}_{\xi _i} + \mathcal {T}_{\xi _i}^*$. Our techniques centre around energy estimates, where the key idea as to how we preserve these estimates in the case of a transport type noise owes to the following proposition.

Proposition 2.6

There exists a constant c such that for each i and for all $f \in W^{k+2,2}(\mathcal {O};\mathbb {R}^3)$ with $k=0, \dots , m$ , we have the bounds

$$\displaystyle \begin{aligned} \langle B_i^2f, f \rangle _{W^{k,2}} + \lVert B_if \rVert _{W^{k,2}}^2 &\leq c\lVert \xi_i \rVert _{W^{k+2,\infty}}^2\lVert f \rVert _{W^{k,2}}^2 {}, \end{aligned} $$

(19)

$$\displaystyle \begin{aligned} \langle B_if, f \rangle _{W^{k,2}}^2 &\leq c\lVert \xi_i \rVert ^2_{W^{k+1,\infty}}\lVert f \rVert ^4_{W^{k,2}}. {} \end{aligned} $$

(20)

Proof

See Sect. 5.1. □

Another valuable result is given now, which will be necessary in showing comparable estimates to Proposition 2.6 in the $\langle \cdot , \cdot \rangle _k$ inner product for appropriate k.

Lemma 2.7

We have that

and moreover that $\mathcal {P}B_i = \mathcal {P}B_i\mathcal {P}$ on $W^{1,2}(\mathcal {O};\mathbb {R}^3)$.

Proof

See Sect. 5.1. □

We note that this result holds true only in the presence of the additional $\mathcal {T}_{\xi _i}$ term in the operator, highlighting the significance of considering a noise which is not purely transport. The Leray Projector does pose difficulties in the presence of a boundary though, which we state here.

Remark 1

The Leray Projector does not preserve the space $W^{1,2}_0(\mathscr {O};\mathbb {R}^3)$, and so we cannot say that ). The issues arising from this operator not satisfying the zero-trace property are fundamentally why we only treat the Torus for the velocity form in Sect. 3.

3 The Velocity Equation on the Torus

In this section we restrict ourselves to the Torus $\mathbb {T}^3$, leaving a treatment of the bounded domain to Sect. 4. We also now fix our assumptions on the $(\xi _i)$, assuming that each $\xi _i \in W^{1,2}_{\sigma }(\mathbb {T}^3;\mathbb {R}^3) \cap W^{3,\infty }(\mathbb {T}^3;\mathbb {R}^3)$ and they collectively satisfy

$$\displaystyle \begin{aligned} {} \sum_{i=1}^\infty\lVert \xi_i \rVert _{W^{3,\infty}}^2 < \infty. \end{aligned} $$

(21)

3.1 Definitions and Results

Here we state the key definitions and results of this section. To facilitate our analysis we work with an equation projected by the Leray Projector as discussed at the start of Sect. 2.2. Thus we consider the new equation

$$\displaystyle \begin{aligned} {} u_t - u_0 + \int_0^t\mathcal{P}\mathcal{L}_{u_s}u_s\,ds + \nu\int_0^t A u_s\, ds + \int_0^t \mathcal{P}Bu_s \circ d\mathcal{W}_s = 0 \end{aligned} $$

(22)

obtained at a heuristic level by projecting all terms of (1). Having not defined solutions of (1) we cannot be too formal here, but the idea is that we require solutions in $L^2_{\sigma }(\mathcal {O};\mathbb {R}^3)$ with initial condition also in this space so they are invariant under $\mathcal {P}$, and $\mathcal {P}$ is a bounded linear operator so can be taken through the integrals (see [30] Corollary 1.6.12.1 for this result in Itô integration, understanding the Stratonovich integral as the sum of an Itô and a Bochner integral).

Theorem 3.1

For any given $\mathcal {F}_0-$measurable ) there exists a pair $(u,\tau )$ such that: $\tau$ is a ) positive stopping time and u is a process whereby for ) $\omega$, $u_{\cdot }(\omega ) \in C\left ([0,T];W^{2,2}_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)\right )$ and ) for all $T>0$ with ) progressively measurable in $W^{3,2}_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)$, and moreover satisfying the identity

$$\displaystyle \begin{aligned} u_t = u_0 - \int_0^{t\wedge \tau}\mathcal{P}\mathcal{L}_{u_s}u_s\,ds -\nu\int_0^{t\wedge \tau} A u_s\, ds - \int_0^{t \wedge \tau} \mathcal{P}Bu_s \circ d\mathcal{W}_s \end{aligned}$$

) in $L^2_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)$ for all $t \geq 0$.

Theorem 3.1 will be proved as a consequence of the following proposition.

Proposition 3.2

Suppose that $(u,\tau )$ are such that: $\tau$ is a ) positive stopping time and u is a process whereby for ) $\omega$, $u_{\cdot }(\omega ) \in C\left ([0,T];W^{2,2}_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)\right )$ and ) for all $T>0$ with ) progressively measurable in $W^{3,2}_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)$, and moreover satisfying the identity

$$\displaystyle \begin{aligned} u_t &= u_0 - \int_0^{t\wedge \tau}\mathcal{P}\mathcal{L}_{u_s}u_s\ ds - \nu\int_0^{t\wedge \tau} A u_s\, ds \\ & \quad + \frac{1}{2}\int_0^{t\wedge \tau}\sum_{i=1}^\infty \mathcal{P}B_i^2u_s ds - \int_0^{t\wedge \tau} \mathcal{P}Bu_s d\mathcal{W}_s \end{aligned} $$

) in $W^{1,2}_{\sigma }(\mathscr {O};\mathbb {R}^3)$ for all $t \geq 0$. Then the pair $(u,\tau )$ satisfies the identity

$$\displaystyle \begin{aligned} u_t = u_0 - \int_0^{t\wedge \tau}\mathcal{P}\mathcal{L}_{u_s}u_s\,ds -\nu\int_0^{t\wedge \tau} A u_s\, ds - \int_0^{t \wedge \tau} \mathcal{P}Bu_s \circ d\mathcal{W}_s \end{aligned}$$

) in $L^2_{\sigma }(\mathscr {O};\mathbb {R}^3)$ for all $t \geq 0$.

Proposition 3.2 motivates studying the converted equation

$$\displaystyle \begin{aligned} {} u_t = u_0 - \int_0^t\mathcal{P}\mathcal{L}_{u_s}u_s\ ds - \nu\int_0^t A u_s\, ds + \frac{1}{2}\int_0^t\sum_{i=1}^\infty \mathcal{P}B_i^2u_s ds - \int_0^t \mathcal{P}Bu_s d\mathcal{W}_s. \end{aligned} $$

(23)

Once we convert to the Itô Form though, starting with an initial condition in $W^{2,2}_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)$ is not optimal in the sense that, at least according to the deterministic theory, we should be able to construct a solution (satisfying the identity in $L^2_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)$ as is natural) for only a $W^{1,2}_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)$ initial condition. To this end we give the following definitions and the main result of this section. Definition 3.3 is stated for an arbitrary $\mathcal {F}_0-$measurable ).

Definition 3.3

A pair $(u,\tau )$ where $\tau$ is a $\mathbb {P}-a.s.$ positive stopping time and u is a process such that for $\mathbb {P}-a.e.$ $\omega$, $u_{\cdot }(\omega ) \in C\left ([0,T];W^{1,2}_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)\right )$ and $u_{\cdot }(\omega ){\mathbf {1}}_{\cdot \leq \tau (\omega )} \in L^2\left ([0,T];W^{2,2}_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)\right )$ for all $T>0$ with $u_{\cdot }{\mathbf {1}}_{\cdot \leq \tau }$ progressively measurable in $W^{2,2}_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)$, is said to be a local strong solution of the equation (23) if the identity

$$\displaystyle \begin{aligned} {} u_t &= u_0 - \int_0^{t\wedge \tau}\mathcal{P}\mathcal{L}_{u_s}u_s\ ds - \nu\int_0^{t\wedge \tau} A u_s\, ds \\ & \quad + \frac{1}{2}\int_0^{t\wedge \tau}\sum_{i=1}^\infty \mathcal{P}B_i^2u_s ds - \int_0^{t\wedge \tau} \mathcal{P}Bu_s d\mathcal{W}_s \end{aligned} $$

(24)

holds $\mathbb {P}-a.s.$ in $L^2_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)$ for all $t \geq 0$.

Remark 2

If $(u,\tau )$ is a local strong solution of the equation (23), then $u_\cdot = u_{ \cdot \wedge \tau }$. The time integrals in (24) are well defined as Bochner integrals in $L^2_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)$, and the Itô integral in $W^{1,2}_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)$. These integrals are shown to be well defined in the abstract case in [30] Definition 2.2.3.

Definition 3.4

A pair $(u,\Theta )$ such that there exists a sequence of stopping times $(\theta _j)$ which are $\mathbb {P}-a.s.$ monotone increasing and convergent to $\Theta$, whereby $(u_{\cdot \wedge \theta _j},\theta _j)$ is a local strong solution of the equation (23) for each j, is said to be a maximal strong solution of the equation (23) if for any other pair $(v,\Gamma )$ with this property then $\Theta \leq \Gamma$ $\mathbb {P}-a.s.$ implies $\Theta = \Gamma$ $\mathbb {P}-a.s.$.

Definition 3.5

A maximal strong solution $(u,\Theta )$ of the equation (23) is said to be pathwise unique if for any other such solution $(v,\Gamma )$, then $\Theta = \Gamma$ $\mathbb {P}-a.s.$ and for all $t \in [0,\Theta )$,

$$\displaystyle \begin{aligned} \mathbb{P}\left(\left\{\omega \in \Omega: u_{t}(\omega) = v_{t}(\omega) \right\} \right) = 1. \end{aligned}$$

Theorem 3.6

For any given $\mathcal {F}_0-$ measurable ), there exists a pathwise unique maximal strong solution $(u,\Theta )$ of the equation (23). Moreover at $\mathbb {P}-a.e.$ $\omega$ for which $\Theta (\omega )<\infty$, we have that

$$\displaystyle \begin{aligned} {}\sup_{r \in [0,\Theta(\omega))}\lVert u_r(\omega) \rVert _{1}^2 + \int_0^{\Theta(\omega)}\lVert u_r(\omega) \rVert _2^2dr = \infty.\end{aligned} $$

(25)

3.2 Operator Bounds

In this subsection we state some intermediary results regarding control on the operators involved. In the following and throughout this section c will represent a generic constant changing from line to line, $c(\varepsilon )$ will be a generic constant dependent on a fixed $\varepsilon$, f and g will be arbitrary elements of $W^{3,2}_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)$ and $f_n \in \mathrm {span}\{a_1, \cdots , a_n\}$. The proofs of these lemmas can be found in Sect. 5.1.

Lemma 3.7

For any $\varepsilon > 0$ , we have that

$$\displaystyle \begin{aligned} {} \left\vert\langle \mathcal{P}_n\mathcal{P}\mathcal{L}_{f_n}f_n, f_n \rangle _2\right\vert &\leq c(\varepsilon)\lVert f_n \rVert ^4_2 + \varepsilon \lVert f_n \rVert _{3}^2; \end{aligned} $$

(26)

$$\displaystyle \begin{aligned} {} \langle \mathcal{P}_n\mathcal{P}B_i^2f_n, f_n \rangle _1 + \langle \mathcal{P}_n\mathcal{P}B_if_n, \mathcal{P}_n\mathcal{P}B_if_n \rangle _1 &\leq c(\varepsilon)\lVert \xi_i \rVert _{W^{3,\infty}}^2\lVert f_n \rVert _1^2 \\ + \quad \varepsilon \lVert \xi_i \rVert _{W^{3,\infty}}^2\lVert f_n \rVert _2^2; \end{aligned} $$

(27)

$$\displaystyle \begin{aligned} {} \langle \mathcal{P}_n\mathcal{P}B_i^2f_n, f_n \rangle _2 + \langle \mathcal{P}_n\mathcal{P}B_if_n, \mathcal{P}_n\mathcal{P}B_if_n \rangle _2 &\leq c(\varepsilon)\lVert \xi_i \rVert ^2_{W^{3,\infty}}\lVert f_n \rVert _2^2 \\ & \quad + \varepsilon \lVert \xi_i \rVert _{W^{3,\infty}}^2\lVert f_n \rVert _{3}^2. \end{aligned} $$

(28)

Remark 3

The algebra property of $W^{k,2}(\mathbb {T}^3;\mathbb {R}^3)$ when $k=2$ is fundamental in proving (26); a result of the form

$$\displaystyle \begin{aligned} \left\vert\langle \mathcal{P}_n\mathcal{P}\mathcal{L}_{f_n}f_n, f_n \rangle _1\right\vert \leq c(\varepsilon)\lVert f_n \rVert ^4_1 + \varepsilon \lVert f_n \rVert _{2}^2\end{aligned}$$

is unavailable to us given that the algebra property does not hold for $k=1$. This is revisited in Remark 4.

Lemma 3.8

For any $\varepsilon > 0$ , we have that

$$\displaystyle \begin{aligned} \left\vert\langle \mathcal{P}\mathcal{L}_{f}f - \mathcal{P}\mathcal{L}_{g}g, f-g \rangle _1\right\vert \leq c(\varepsilon)\left(\lVert g \rVert ^4_1 + \lVert f \rVert _2^2\right) \lVert f-g \rVert _1^2 + \varepsilon \lVert f-g \rVert _2^2. \end{aligned} $$

Lemma 3.9

For any $\varepsilon > 0$ , we have that

$$\displaystyle \begin{aligned} \left\vert\langle \mathcal{P}\mathcal{L}_{f}f - \mathcal{P}\mathcal{L}_{g}g, f-g \rangle \right\vert \leq c(\varepsilon)\lVert f \rVert _2^2 \lVert f-g \rVert ^2 + \varepsilon \lVert f-g \rVert _1^2 . \end{aligned} $$

3.3 Proof of Proposition 3.2

We prove this result through the abstract procedure used in [30] Subsections 2.2 and 2.3, the result of which is stated in Sect. 5.2. Towards this goal we define the quartet of spaces

$$\displaystyle \begin{aligned} V&:= W^{3,2}_{\sigma}(\mathbb{T}^3;\mathbb{R}^3), \qquad H:= W^{2,2}_{\sigma}(\mathbb{T}^3;\mathbb{R}^3),\\ U&:= W^{1,2}_{\sigma}(\mathbb{T}^3;\mathbb{R}^3), \qquad X:= L^2_{\sigma}(\mathbb{T}^3;\mathbb{R}^3). \end{aligned} $$

We equip $L^2_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)$ with the usual $\langle \cdot , \cdot \rangle$ inner product, but then equip $W^{1,2}_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)$ and $W^{2,2}_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)$ with the $\langle \cdot , \cdot \rangle _1$ and $\langle \cdot , \cdot \rangle _2$ inner products respectively. In fact we also have that $D(A^{3/2}) = W^{3,2}_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)$ and that the $\langle \cdot , \cdot \rangle _3$ inner product is equivalent to the usual $\langle \cdot , \cdot \rangle _{W^{3,2}}$ one on this space (see [44] Theorem 2.27), so we endow $W^{3,2}_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)$ with $\langle \cdot , \cdot \rangle _3$. Our SPDE (22) takes the form of (68) for the operators

$$\displaystyle \begin{aligned} \mathcal{Q}:= -\left(\mathcal{P}\mathcal{L} + \nu A \right), \qquad \mathcal{G}:=-\mathcal{P}B\end{aligned} $$

where we rewrite $(-\mathcal {P}B_i)^2$ as $\mathcal {P}B_i^2$ firstly from the linearity of $\mathcal {P}B_i$ to deal with the minus and secondly using the property that $\mathcal {P}B_i = \mathcal {P}B_i\mathcal {P}$. It is worth appreciating here that we chose to project the equation and then convert it into Itô Form, but we may equally have chosen to convert the unprojected Stratonovich Form and then project the resulting Itô Equation. Without addressing the conversion of the unprojected equation in complete detail, we would directly arrive at (23) taking this approach. Indeed before projection our correction term would be of the form $\sum _{i=1}^\infty B_i^2$ plus a function in $L^{2,\perp }_{\sigma }(\mathbb {T}^3,\mathbb {R}^3)$ as defined in Lemma 2.7. Under projection this is $\mathcal {P}\sum _{i=1}^\infty B_i^2$ which is just $\sum _{i=1}^\infty \mathcal {P}B_i^2$ from the linearity and continuity. Therefore the property $\mathcal {P}B_i = \mathcal {P}B_i\mathcal {P}$ from Lemma 2.7 establishes the consistency between these approaches.

To prove the result we check the Assumptions 5.3 and 5.4. Starting with 5.3, we first of all have that $\nu A$ is continuous from $W^{3,2}_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)$ into $W^{1,2}_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)$, therefore it is measurable and as a linear operator too satisfies the boundedness. As for $\mathcal {P}\mathcal {L}$, measurability is satisfied in the same way and for the boundedness we have that

$$\displaystyle \begin{aligned} \lVert \mathcal{P}\mathcal{L}_{f}f \rVert _{1} \leq c\lVert \mathcal{P}\mathcal{L}_ff \rVert _{W^{1,2}} \leq c\lVert \mathcal{L}_ff \rVert _{W^{1,2}} \leq c\lVert f \rVert _{W^{1,2}}\lVert f \rVert _{W^{3,2}} \leq c\lVert f \rVert _{1}\lVert f \rVert _{3}\end{aligned}$$

for any $f\in W^{3,2}_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)$ where c is a generic constant, critically applying (5). This verifies Assumption 5.3 so we move on to Assumption 5.4, which is immediate from (18) and the linearity of $\mathcal {P}B$ to show continuity in all relevant spaces.

3.4 Proofs of Theorems 3.1 and 3.6.

We use the abstract results of [32] stated in Sects. 5.3 and 5.4. Definitions 3.3, 3.4, 3.5 and Theorem 3.6 are precisely Definitions 5.19, 5.20, 5.21 and Theorem 5.22 for the equation (23) with respect to the spaces $V,H,U,X$ as defined in Sect. 3.3. Indeed we would also prove Theorem 3.1 through Proposition 3.2 by showing the existence of a local solution with the regularity of Definition 5.12. Therefore we prove both Theorem 3.1 and 3.6 by showing that the assumptions of Sects. 5.3 and 5.4, are satisfied. We work with the operators

$$\displaystyle \begin{aligned} \mathcal{A}:= -\left(\mathcal{P}\mathcal{L} + \nu A \right) + \frac{1}{2}\sum_{i=1}^\infty \mathcal{P}B_i^2, \qquad \mathcal{G}:= -\mathcal{P}B \end{aligned} $$

which were addressed to be measurable mappings into the required spaces in Sect. 3.3. We now prove Theorem 3.1 by justifying the assumptions of Sect. 5.3.

Proof of Theorem 3.1

First note that the density of the spaces is immediately inherited from the density of the usual Sobolev Spaces and the equivalence of the norms. The bilinear form satisfying (70) is chosen to be

$$\displaystyle \begin{aligned} \langle f, g \rangle _{U \times V}:= \langle A^{1/2}f, A^{3/2}g \rangle\end{aligned}$$

which reduces to the $\langle \cdot , \cdot \rangle _2$ inner product from (12). The remainder of the proof is in treating the numbered assumptions.

Assumption 5.6::

We use the system $(a_k)$ of eigenfunctions of the Stokes Operator.

Assumption 5.7::

Once more (74) follows from the discussion in Sect. 3.3. For (75) we treat the different operators in $\mathcal {A}$ individually, starting from the nonlinear term:

$$\displaystyle \begin{aligned} \lVert \mathcal{P}\mathcal{L}_ff - \mathcal{P}\mathcal{L}_gg \rVert _1 &= \lVert \mathcal{P}\mathcal{L}_f(f-g) + \mathcal{P}\mathcal{L}_{f-g}g \rVert _1\\ &\leq \lVert \mathcal{P}\mathcal{L}_f(f-g) \rVert _1 + \lVert \mathcal{P}\mathcal{L}_{f-g}g \rVert _1\\ &\leq c\lVert \mathcal{L}_f(f-g) \rVert _{W^{1,2}} + c\lVert \mathcal{L}_{f-g}g \rVert _{W^{1,2}}\\ &\leq c\lVert f \rVert _{W^{1,2}}\lVert f-g \rVert _{W^{3,2}} + c\lVert f-g \rVert _{W^{1,2}}\lVert g \rVert _{W^{3,2}}\\ &\leq c\left(\lVert f \rVert _{W^{1,2}} + \lVert g \rVert _{W^{3,2}}\right)\lVert f-g \rVert _{W^{3,2}}\\ &\leq c\left(\lVert f \rVert _{1} + \lVert g \rVert _{3}\right)\lVert f-g \rVert _{3} \end{aligned} $$

having applied (5). From the linearity of $\nu A$ and $\frac {1}{2}\sum _{i=1}^\infty \mathcal {P}B_i^2$ then the corresponding result follows immediately from (74) and this subsequently justifies (75). Additionally (76) follows immediately from the already justified Assumption 5.4.

Assumption 5.8::

(26) and (28) will be our basis of showing (77). The task is to control

$$\displaystyle \begin{aligned} 2\left \langle \mathcal{P}_n\left(-\mathcal{P}\mathcal{L} -\nu A +\frac{1}{2}\sum_{i=1}^\infty \mathcal{P}B_i^2\right) f_n , f_n \right \rangle_2 + \sum_{i=1}^\infty \lVert \mathcal{P}_n\mathcal{P}B_if_n \rVert _2^2\end{aligned}$$

which we rewrite as

$$\displaystyle \begin{aligned} {}&-2\langle \mathcal{P}_n\mathcal{P}\mathcal{L}_{f_n}f_n, f_n \rangle _2 - 2\nu\langle \mathcal{P}_nAf_n, f_n \rangle _2 \\ & \quad + \sum_{i=1}^\infty\left(\langle \mathcal{P}_n\mathcal{P}B_i^2f_n, f_n \rangle _2 + \lVert \mathcal{P}_n\mathcal{P}B_if_n \rVert _2^2 \right).\end{aligned} $$

(29)

Recalling the assumption (21) and (26), (28), we have that for any $\varepsilon > 0$,

$$\displaystyle \begin{aligned} \mbox{(29)} &\leq - 2\nu\langle \mathcal{P}_nAf_n, f_n \rangle _2 + c(\varepsilon)\lVert f_n \rVert ^4_2 + \varepsilon \lVert f_n \rVert _{3}^2 \\ & \quad + \sum_{i=1}^\infty\left(c(\varepsilon)\lVert \xi_i \rVert ^2_{W^{3,\infty}}\lVert f_n \rVert _2^2 + \varepsilon \lVert \xi_i \rVert _{W^{3,\infty}}^2\lVert f_n \rVert _{3}^2 \right)\\ &= - 2\nu\langle Af_n, f_n \rangle _2 + \left[ c(\varepsilon)\lVert f_n \rVert _2^4 + c(\varepsilon)\sum_{i=1}^\infty \lVert \xi_i \rVert _{W^{3,\infty}}^2\lVert f_n \rVert _2^2\right] \\ & \quad + \varepsilon\left[1 + \sum_{i=1}^\infty \lVert \xi_i \rVert _{W^{3,\infty}}^2\right]\lVert f_n \rVert _3^2\\ &\leq - 2\nu\langle A^{1/2}Af_n, A^{3/2}f_n \rangle _2 + c(\varepsilon)\left[1 + \lVert f_n \rVert _2^2\right]\lVert f_n \rVert _2^2\\ & \quad + \varepsilon\left[1 + \sum_{i=1}^\infty \lVert \xi_i \rVert _{W^{3,\infty}}^2\right]\lVert f_n \rVert _3^2\\ &= - 2\nu\lVert f_n \rVert _3^2 + c(\varepsilon)\left[1 + \lVert f_n \rVert _2^2\right]\lVert f_n \rVert _2^2 + \varepsilon\left[1 + \sum_{i=1}^\infty \lVert \xi_i \rVert _{W^{3,\infty}}^2\right]\lVert f_n \rVert _3^2 \end{aligned} $$

where we have embedded the $\sum _{i=1}^\infty \lVert \xi _i \rVert _{W^{3,\infty }}^2$ into the constant $c(\varepsilon )$. Therefore by choosing

$$\displaystyle \begin{aligned} \varepsilon:= \frac{\nu}{1 + \sum_{i=1}^\infty \lVert \xi_i \rVert _{W^{3,\infty}}^2}\end{aligned}$$

then (77) is satisfied for $\kappa := \nu$. Moving on to (78), we are interested in the term

$$\displaystyle \begin{aligned} \sum_{i=1}^\infty \langle \mathcal{P}_n\mathcal{P}B_if_n, f_n \rangle _2^2.\end{aligned}$$

Observe that

$$\displaystyle \begin{aligned} \langle \mathcal{P}_n\mathcal{P}B_if_n, f_n \rangle _2^2 &= \langle \mathcal{P}B_if_n, f_n \rangle _2^2\\ &= \langle \mathcal{P}[\Delta, B_i]f_n + \mathcal{P}B_i \Delta f_n, Af_n \rangle ^2\\ &\leq 2\langle \mathcal{P}[\Delta, B_i]f_n, Af_n \rangle ^2 + 2\langle \mathcal{P}B_i \Delta f_n, Af_n \rangle ^2\\ &= 2\langle [\Delta, B_i]f_n, Af_n \rangle ^2 + 2\langle B_i A f_n, Af_n \rangle ^2. \end{aligned} $$

The first of these terms is dealt with through a simple Cauchy-Schwarz, as

$$\displaystyle \begin{aligned} \langle [\Delta, B_i]f_n, Af_n \rangle ^2 \leq \lVert [\Delta, B_i]f_n \rVert ^2\lVert Af_n \rVert ^2 \leq c\lVert \xi_i \rVert ^2_{W^{3,\infty}}\lVert f_n \rVert ^4_{2} \end{aligned} $$

using Proposition 5.2, and the second comes directly from (20) as

$$\displaystyle \begin{aligned} \langle B_i A f_n, Af_n \rangle ^2 \leq c\lVert \xi_i \rVert ^2_{W^{1,\infty}}\lVert Af_n \rVert ^4 \leq c\lVert \xi_i \rVert ^2_{W^{3,\infty}}\lVert f_n \rVert _2^4. \end{aligned} $$

Summing up the two terms and over all i gives that

$$\displaystyle \begin{aligned} \sum_{i=1}^\infty \langle \mathcal{P}_n\mathcal{P}B_if_n, f_n \rangle _2^2 \leq \left(c\sum_{i=1}^\infty\lVert \xi_i \rVert ^2_{W^{3,\infty}} \right)\lVert f_n \rVert _2^4\end{aligned}$$

which justifies (78) and Assumption 5.8.

Assumption 5.9::

For (79) we must bound the term

$$\displaystyle \begin{aligned} & 2\left \langle\left(-\mathcal{P}\mathcal{L} -\nu A +\frac{1}{2}\sum_{i=1}^\infty \mathcal{P}B_i^2\right) f - \left(-\mathcal{P}\mathcal{L} -\nu A +\frac{1}{2}\sum_{i=1}^\infty \mathcal{P}B_i^2\right) g, f-g \right \rangle_1\\ & \quad + \sum_{i=1}^\infty \lVert \mathcal{P}B_if - \mathcal{P}B_ig \rVert _1^2 \end{aligned} $$

which we simply rewrite as

$$\displaystyle \begin{aligned} &-2\langle \mathcal{P}\mathcal{L}_ff - \mathcal{P}\mathcal{L}_gg, f-g \rangle _1 -2\nu \langle A(f-g), f-g \rangle _1 \\ & \quad + \sum_{i=1}^\infty \left(\langle \mathcal{P}B_i^2(f-g), f-g \rangle _1 + \lVert \mathcal{P}B_i(f-g) \rVert _1^2 \right) \end{aligned} $$

and inspect the distinct items individually. Firstly from Lemma 3.8 we have that for any $\varepsilon >0$,

$$\displaystyle \begin{aligned} {}-2\langle \mathcal{P}\mathcal{L}_ff - \mathcal{P}\mathcal{L}_gg, f-g \rangle _1 \leq c(\varepsilon)\left(\lVert g \rVert ^4_1 + \lVert f \rVert _2^2\right) \lVert f-g \rVert _1^2 + \varepsilon \lVert f-g \rVert _2^2.\end{aligned} $$

(30)

Similarly to the justification of Assumption 5.8 we also see that

$$\displaystyle \begin{aligned} {}-2\nu \langle A(f-g), f-g \rangle _1 = -2\nu \lVert f-g \rVert _2^2. \end{aligned} $$

(31)

Shifting focus to the final term, note that in (27) we in fact showed that

$$\displaystyle \begin{aligned} \langle \mathcal{P}B_i^2f_n, f_n \rangle _1 + \langle \mathcal{P}B_if_n, \mathcal{P}B_if_n \rangle _1 \leq c(\varepsilon)\lVert \xi_i \rVert _{W^{3,\infty}}^2\lVert f_n \rVert _1^2 + \varepsilon \lVert \xi_i \rVert _{W^{3,\infty}}^2\lVert f_n \rVert _2^2\end{aligned}$$

and scanning the proof we see that all arguments hold for arbitrary $f_n \in W^{3,2}_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)$ so we can deduce directly the bound

$$\displaystyle \begin{aligned} {}\langle \mathcal{P}B_i^2(f-g), f-g \rangle _1 &+ \lVert \mathcal{P}B_i(f-g) \rVert _1^2 \leq c(\varepsilon)\lVert \xi_i \rVert _{W^{3,\infty}}^2\lVert f-g \rVert _1^2 \\ & + \varepsilon \lVert \xi_i \rVert _{W^{3,\infty}}^2\lVert f-g \rVert _2^2.\end{aligned} $$

(32)

Summing over (30), (31) and all i in (32), we deduce a bound by

$$\displaystyle \begin{aligned} & -2\nu \lVert f-g \rVert _2^2 + c(\varepsilon)\left[\lVert g \rVert ^4_1 + \lVert f \rVert _2^2 + \sum_{i=1}^\infty\lVert \xi_i \rVert _{W^{3,\infty}}^2 \right]\lVert f-g \rVert _1^2 \\ & \quad + \varepsilon\left[1 + \sum_{i=1}^\infty\lVert \xi_i \rVert _{W^{3,\infty}}^2 \right]\lVert f-g \rVert _2^2\end{aligned} $$

so again a choice of

$$\displaystyle \begin{aligned} {}\varepsilon:= \frac{\nu}{1 + \sum_{i=1}^\infty \lVert \xi_i \rVert _{W^{3,\infty}}^2}\end{aligned} $$

(33)

ensures (79) is satisfied for $\kappa := \nu$. Moving on to (80), we are interested in the term

$$\displaystyle \begin{aligned} {} \sum_{i=1}^\infty \langle \mathcal{P}B_i(f-g), f-g \rangle _1^2, \end{aligned} $$

(34)

noting that

$$\displaystyle \begin{aligned} \langle \mathcal{P}B_i(f-g), f-g \rangle _1^2 &= \langle A\mathcal{P}B_i(f-g), f-g \rangle ^2\\ &\leq N\sum_{k=1}^3\langle \partial_k B_i(f-g), \partial_k(f - g) \rangle ^2. \end{aligned} $$

We have

$$\displaystyle \begin{aligned} \partial_kB_i(f-g) = B_{\partial_k\xi_i}(f-g) + B_i\partial_k(f-g)\end{aligned}$$

so

$$\displaystyle \begin{aligned} \langle \partial_k B_i(f-g), \partial_k(f - g) \rangle ^2 & \leq 2\langle B_{\partial_k\xi_i}(f-g), \partial_k(f - g) \rangle ^2\\ & \quad + 2\langle B_i\partial_k(f-g), \partial_k(f-g) \rangle ^2.\end{aligned} $$

Now from (18),

$$\displaystyle \begin{aligned} \langle B_{\partial_k\xi_i}(f-g), \partial_k(f - g) \rangle ^2 & \leq \lVert B_{\partial_k\xi_i}(f-g) \rVert ^2\lVert \partial_k(f - g) \rVert ^2\\ &\leq c\lVert \xi_i \rVert ^2_{W^{3,\infty}}\lVert f-g \rVert ^4_1 \end{aligned} $$

and from (15),

$$\displaystyle \begin{aligned} \langle B_i\partial_k(f-g), \partial_k(f-g) \rangle ^2 &= \langle \mathcal{T}_{\xi_i}\partial_k(f-g), \partial_k(f-g) \rangle ^2\\ & \leq c\lVert \xi_i \rVert ^2_{W^{3,\infty}}\lVert f-g \rVert ^4_1. \end{aligned} $$

By summing both terms, over all $k =1, \dots , N$ and $i \in \mathbb {N}$, we have shown that

$$\displaystyle \begin{aligned} {}\sum_{i=1}^\infty\langle \mathcal{P}B_i(f-g), f-g \rangle _1^2 \leq \left(c\sum_{i=1}^\infty \lVert \xi_i \rVert ^2_{W^{3,\infty}}\right)\lVert f-g \rVert ^4_1\end{aligned} $$

(35)

demonstrating (80) and hence Assumption 5.9.

Assumption 5.10::

For (81) we must bound the term

$$\displaystyle \begin{aligned} 2\left \langle\left(-\mathcal{P}\mathcal{L} -\nu A +\frac{1}{2}\sum_{i=1}^\infty \mathcal{P}B_i^2\right) f, f \right \rangle_1 + \sum_{i=1}^\infty \lVert \mathcal{P}B_if \rVert _1^2 \end{aligned} $$

which we simply rewrite as

$$\displaystyle \begin{aligned} -2\langle \mathcal{P}\mathcal{L}_ff, f \rangle _1 -2\nu \langle Af, f \rangle _1 + \sum_{i=1}^\infty \left(\langle \mathcal{P}B_i^2f, f \rangle _1 + \lVert \mathcal{P}B_if \rVert _1^2 \right). {} \end{aligned} $$

(36)

The nonlinear term can be controlled precisely as seen in Lemma 3.8 to deduce that for any $\varepsilon > 0$,

$$\displaystyle \begin{aligned} \left\vert\langle \mathcal{P}\mathcal{L}_ff, f \rangle _1\right\vert \leq c(\varepsilon)\lVert f \rVert _1^6 + \varepsilon \lVert f \rVert _2^2.\end{aligned}$$

Meanwhile across (31) and (32) we have that

$$\displaystyle \begin{aligned} -2\nu \langle Af, f \rangle _1 &+ \sum_{i=1}^\infty \left(\langle \mathcal{P}B_i^2f, f \rangle _1 + \lVert \mathcal{P}B_if \rVert _1^2 \right)\\ & \quad \leq -2\nu\lVert f \rVert _2^2 + c(\varepsilon)\left[\sum_{i=1}^3\lVert \xi_i \rVert _{W^{3,\infty}}^2\right]\lVert f \rVert _1^2 \\ & \quad + \varepsilon\left[\sum_{i=1}^\infty \lVert \xi_i \rVert _{W^{3,\infty}}^2\right]\lVert f \rVert _2^2 \end{aligned} $$

so with the familiar choice of $\varepsilon$ (33) we see that

$$\displaystyle \begin{aligned} \mbox{(36)} \leq c\left(1 + \lVert f \rVert _1^6 \right) - \nu\lVert f \rVert _2^2\end{aligned}$$

which is more than sufficient to show (81). (82) follows immediately from (35), concluding the justification.

Assumption 5.11::

For any $\eta \in W^{2,2}_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)$ we must bound the term

$$\displaystyle \begin{aligned} \left \langle\left(-\mathcal{P}\mathcal{L} -\nu A +\frac{1}{2}\sum_{i=1}^\infty \mathcal{P}B_i^2\right) f - \left(-\mathcal{P}\mathcal{L} -\nu A +\frac{1}{2}\sum_{i=1}^\infty \mathcal{P}B_i^2\right) g, \eta \right \rangle_1 \end{aligned} $$

which we simply rewrite as

$$\displaystyle \begin{aligned} -2\langle \mathcal{P}\mathcal{L}_ff - \mathcal{P}\mathcal{L}_gg, \eta \rangle _1 -2\nu \langle A(f-g), \eta \rangle _1 + \sum_{i=1}^\infty \langle \mathcal{P}B_i^2(f-g), \eta \rangle _1 \end{aligned} $$

and further by

$$\displaystyle \begin{aligned} -2\langle \mathcal{P}\mathcal{L}_ff - \mathcal{P}\mathcal{L}_gg, A\eta \rangle -2\nu \langle A(f-g), A\eta \rangle + \sum_{i=1}^\infty \langle \mathcal{P}B_i^2(f-g), A\eta \rangle . \end{aligned} $$

Through Cauchy-Schwarz this is controlled by

$$\displaystyle \begin{aligned} \lVert \eta \rVert _2\left( 2\lVert \mathcal{P}\mathcal{L}_ff - \mathcal{P}\mathcal{L}_gg \rVert + 2\nu \lVert A(f-g) \rVert + \sum_{i=1}^\infty \lVert \mathcal{P}B_i^2(f-g) \rVert \right). \end{aligned} $$

so our problem is reduced to bounding the bracketed terms. The linear terms are trivial when recalling (18), and for the nonlinear term we revert back to (4) to see that

$$\displaystyle \begin{aligned} \lVert \mathcal{P}\mathcal{L}_ff - \mathcal{P}\mathcal{L}_gg \rVert \leq \lVert \mathcal{L}_{f-g}f \rVert +\lVert \mathcal{L}_g(f-g) \rVert \leq c\left(\lVert f \rVert _2+\lVert g \rVert _1\right)\lVert f-g \rVert _2 \end{aligned} $$

comfortably justifying the assumption.

□

Proof of Theorem 3.1

The new space X will again be $L^2_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)$ as laid out in Sect. 3.3. We choose the bilinear form $\langle \cdot , \cdot \rangle _{X \times H}$ to be given by

$$\displaystyle \begin{aligned} {}\langle f, g \rangle _{X \times H}:= \langle f, Ag \rangle \end{aligned} $$

(37)

noting that the property (86) follows from (12). Noting also that the system $(a_k)$ is an orthogonal basis of $W^{1,2}_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)$, and that the operators were shown to be measurable into the relevant spaces in Sect. 3.3, we are in the setting of Sect. 5.4. We now proceed to justify the assumptions required to apply Theorem 5.22.

Assumption 5.16::

This follows identically to Assumption 5.7, referring again to Sect. 3.3 and (4).

Assumption 5.17::

Continuing to consider the distinct terms, we have that

$$\displaystyle \begin{aligned} -2\nu \langle A(f-g), f-g \rangle = -2\nu\lVert f-g \rVert _1^2\end{aligned}$$

and

$$\displaystyle \begin{aligned} & \langle \mathcal{P}B_i^2(f-g), f-g \rangle + \lVert \mathcal{P}B_i(f-g) \rVert ^2\\ &\quad \leq \langle B_i^2(f-g), f-g \rangle + \lVert B_i(f-g) \rVert ^2 \leq c\lVert \xi_i \rVert _{W^{2,\infty}}^2\lVert f-g \rVert ^2.\end{aligned} $$

from (19). With these components in place, the proof of (89) then follows identically to that of (79). (90) is a direct consequence of (20), concluding the justification.

Assumption 5.18::

This stronger Assumption was in fact already verified in the address of Assumption 5.10.

□

4 The Vorticity Equation on a Bounded Main

In order to address the well-posedness problem of the SALT Navier-Stokes Equations on bounded domains, we now pose it in vorticity form. The analysis conducted in Sect. 3.4 was done with reference to the properties derived across Sects. 2.2 and 2.3, applicable to the bounded domain as well as the torus. The issue in studying the velocity form is that our operators do not map into the correct spaces in order to use these properties: in particular, the Leray Projector does not preserve the zero trace property and so the operators do not map into the necessary $W^{k,2}_{\sigma }(\mathscr {O};\mathbb {R}^3)$ spaces (see Remark 1). The motivation behind the vorticity form is to circumvent the necessity of Leray Projection.

Our attentions shall be decidedly on the bounded domain $\mathscr {O}$, though the results for the vorticity form carry over seamlessly to the torus $\mathbb {T}^3$. For this section we impose new constraints on the $\xi _i$, which are such that for each $i \in \mathbb {N}$, $\xi _i \in W^{1,2}_{\sigma }(\mathscr {O};\mathbb {R}^3) \cap W^{2,2}_{0}(\mathscr {O};\mathbb {R}^3) \cap W^{3,\infty }(\mathscr {O};\mathbb {R}^3)$ and they collectively satisfy

$$\displaystyle \begin{aligned} {} \sum_{i=1}^\infty \lVert \xi_i \rVert _{W^{3,\infty}}^2 < \infty. \end{aligned} $$

(38)

4.1 Deriving the Equation

The vorticity form of the equation is derived through taking the curl of the velocity form, where the curl operator is defined for $f \in W^{1,2}(\mathscr {O};\mathbb {R}^3)$ by

$$\displaystyle \begin{aligned} \mathrm{curl}f := \begin{pmatrix} \partial_2 f^3 - \partial_3 f^2\\ \partial_3f^1 - \partial_1f^3\\ \partial_1f^2 - \partial_2f^1 \end{pmatrix}. \end{aligned}$$

We introduce the Lie Bracket operator $\mathscr {L}$ defined on sufficiently regular functions ) by

$$\displaystyle \begin{aligned} {} \mathscr{L}_fg := \mathcal{L}_fg - \mathcal{L}_gf. \end{aligned} $$

(39)

In [44] Subsection 12.1 it is shown that, with notation $\phi := \mathrm {curl}f$,

$$\displaystyle \begin{aligned} \mathrm{curl}\left(\mathcal{L}_ff\right) = \mathscr{L}_f \phi\end{aligned}$$

where it is also observed that the curl of elements of $W^{1,2}(\mathscr {O};\mathbb {R}^3) \cap L^{2,\perp }_{\sigma }(\mathscr {O};\mathbb {R}^3)$ is null. It is clear that the Laplacian commutes with the curl operation, and we now consider how the curl operation interacts with the stochastic term.

Lemma 4.1

We have that

$$\displaystyle \begin{aligned} {} \mathrm{curl}(B_if) = \mathscr{L}_{\xi_i}\phi \end{aligned} $$

(40)

where once more $\phi := \mathrm {curl}f.$

Proof

We shall show only that the identity (40) holds in its first component, with the others following similarly. To this end we calculate the first component of the left hand side of (40):

$$\displaystyle \begin{aligned} [\mathrm{curl}(B_if)]^1 &= \partial_2[B_if]^3 - \partial_3[B_if]^2 \\ &= \partial_2\left(\sum_{j=1}^3\xi_i^j\partial_jf^3 + f^j\partial_3\xi_i^j\right) - \partial_3\left(\sum_{j=1}^3\xi_i^j\partial_jf^2 + f^j\partial_2\xi_i^j\right)\\ &= \sum_{j=1}^3\left(\partial_2\xi_i^j\partial_jf^3 + \xi_i^j\partial_2\partial_jf^3 + \partial_2f^j\partial_3\xi_i^j + f^j\partial_2\partial_3\xi_i^j\right)\\ & \quad - \sum_{j=1}^3\left( \partial_3\xi_i^j\partial_jf^2 + \xi_i^j\partial_3\partial_jf^2 + \partial_3f^j\partial_2\xi_i^j + f^j\partial_3\partial_2\xi_i^j\right)\\ &= \sum_{j=1}^3\left(\partial_2\xi_i^j\partial_jf^3 + \xi_i^j\partial_2\partial_jf^3 + \partial_2f^j\partial_3\xi_i^j - \partial_3\xi_i^j\partial_jf^2 \right.\\ & \qquad \left. - \xi_i^j\partial_3\partial_jf^2 - \partial_3f^j\partial_2\xi_i^j \right)\\ &= \sum_{j=1}^3\xi_i^j\partial_j(\partial_2f^3 - \partial_3f^2) + \sum_{j=1}^3\left(\partial_2\xi_i^j\partial_jf^3 + \partial_2f^j\partial_3\xi_i^j \right.\\ & \quad \left.- \partial_3\xi_i^j\partial_jf^2 - \partial_3f^j\partial_2\xi_i^j \right)\\ &= [\mathcal{L}_{\xi_i}\phi]^1 + \sum_{j=1}^3\left(\partial_2\xi_i^j\partial_jf^3 + \partial_2f^j\partial_3\xi_i^j - \partial_3\xi_i^j\partial_jf^2 - \partial_3f^j\partial_2\xi_i^j \right). \end{aligned} $$

Therefore it remains to show that

$$\displaystyle \begin{aligned} {} \sum_{j=1}^3\left(\partial_2\xi_i^j\partial_jf^3 + \partial_2f^j\partial_3\xi_i^j - \partial_3\xi_i^j\partial_jf^2 - \partial_3f^j\partial_2\xi_i^j \right) = -[\mathcal{L}_{\phi}\xi_i]^1. \end{aligned} $$

(41)

We expand the sum in (41) to

$$\displaystyle \begin{aligned} &\left(\partial_2\xi_i^1\partial_1f^3 + \partial_2f^1\partial_3\xi_i^1 - \partial_3\xi_i^1\partial_1f^2 - \partial_3f^1\partial_2\xi_i^1\right) \\ & \quad + \left(\partial_2\xi_i^2\partial_2f^3 + \partial_2f^2\partial_3\xi_i^2 - \partial_3\xi_i^2\partial_2f^2 - \partial_3f^2\partial_2\xi_i^2 \right)\\ & \quad + \left(\partial_2\xi_i^3\partial_3f^3 + \partial_2f^3\partial_3\xi_i^3 - \partial_3\xi_i^3\partial_3f^2 - \partial_3f^3\partial_2\xi_i^3 \right) \end{aligned} $$

achieving some immediate cancellation in the second two brackets to

$$\displaystyle \begin{aligned} & \left(\partial_2\xi_i^1\partial_1f^3 + \partial_2f^1\partial_3\xi_i^1 - \partial_3\xi_i^1\partial_1f^2 - \partial_3f^1\partial_2\xi_i^1\right) \\ & \quad + \left(\partial_2\xi_i^2\partial_2f^3 - \partial_3f^2\partial_2\xi_i^2 \right)+ \left( \partial_2f^3\partial_3\xi_i^3 - \partial_3\xi_i^3\partial_3f^2 \right). \end{aligned} $$

We now simply rewrite the above by combining like terms, into

$$\displaystyle \begin{aligned} \partial_2\xi_i^1(\partial_1f^3 - \partial_3f^1) + \partial_3\xi_i^1(\partial_2f^1 - \partial_1f^2) + (\partial_2\xi_i^2 + \partial_3\xi_i^3)(\partial_2f^3 - \partial_3f^2) \end{aligned}$$

or more succinctly as

$$\displaystyle \begin{aligned} -\partial_2\xi_i^1\phi^2 - \partial_3\xi_i^1\phi^3 + (\partial_2\xi_i^2 + \partial_3\xi_i^3)\phi^1\end{aligned}$$

to which we add and subtract $\partial _1\xi _i^1\phi ^1$ to arrive at

$$\displaystyle \begin{aligned} -\sum_{j=1}^3 \phi^j\partial_j\xi_i^1 +\sum_{j=1}^3\left(\partial_j\xi_i^j \right)\phi^1.\end{aligned}$$

The first term is precisely $-[\mathcal {L}_{\phi }\xi _i]^1$ as we wished to show, appreciating that the second term is zero given the divergence free condition on $\xi _i$. □

From this point forwards we adopt the notation of $\mathscr {L}_i := \mathscr {L}_{\xi _i}$. Writing the Stratonovich integral of (1) in its component form over the basis vectors of $\mathfrak {U}$, and introducing the notation $w:= \mathrm {curl}u$, at a heuristic level we can take the curl of (1) to obtain

$$\displaystyle \begin{aligned} {} w_t - w_0 + \int_0^t\mathscr{L}_{u_s}w_s\,ds - \nu\int_0^t \Delta w_s\, ds + \sum_{i=1}^\infty \int_0^t \mathscr{L}_iw \circ dW^i_s = 0.\end{aligned} $$

(42)

Having already rigorously demonstrated the Itô conversion of the velocity form (22) we shall make the conversion without explicit reference to the conditions of Sect. 5.2, though this can be precisely shown in the same way. At least again then at the heuristic level, the Itô form is

$$\displaystyle \begin{aligned} w_t = w_0 - \int_0^t\mathscr{L}_{u_s}w_s\,ds + \nu\int_0^t \Delta w_s\, ds + \frac{1}{2}\int_0^t\sum_{i=1}^\infty \mathscr{L}_i^2w_sds- \sum_{i=1}^\infty \int_0^t \mathscr{L}_iw \, dW^i_s\end{aligned}$$

which can again be projected³ to the equation

$$\displaystyle \begin{aligned} {} w_t &= w_0 - \int_0^t\mathcal{P}\mathscr{L}_{u_s}w_s\,ds - \nu\int_0^t A w_s\, ds \\ & \quad + \frac{1}{2}\int_0^t\sum_{i=1}^\infty\mathcal{P} \mathscr{L}_i^2w_sds- \sum_{i=1}^\infty \int_0^t \mathcal{P}\mathscr{L}_iw \, dW^i_s.\end{aligned} $$

(43)

Having motivated this section by an avoidance of the Leray Projection this may seem counter intuitive, however we shall shortly show that the projection is not felt in the noise (where it becomes problematic in velocity form); that is to say that for sufficiently regular w, $\mathcal {P}\mathscr {L}_iw = \mathscr {L}_iw$. The goal is to deduce the existence of a unique maximal solution of (43), a task which requires some clarification. Having reached (43) from the velocity form, we now look to solve the equation for vorticity which demands a representation of the velocity u in terms of the vorticity w. For this we quote Theorem 1 of [20] (or in fact, a slightly relaxed version).

Theorem 4.2

There exists a mapping ) such that for every $\phi \in W^{1,2}_{\sigma }(\mathscr {O};\mathbb {R}^3) \cap W^{k,p}(\mathscr {O};\mathbb {R}^3)$ where $k \in \mathbb {N} \cup \{0\}$, $1 < p <\infty$, the function ) defined $\lambda -a.e.$ by

$$\displaystyle \begin{aligned} {}(BS_K\phi)(x) = \int_{\mathscr{O}}K(x,y)\phi(y)dy\end{aligned} $$

(44)

is such that:

• 1. $BS_K\phi \in L^{2}_{\sigma }(\mathscr {O};\mathbb {R}^3) \cap W^{k+1,p}(\mathscr {O};\mathbb {R}^3)$;

• 2. $\mathrm {curl}(BS_K\phi ) = \phi$;

• 3. There exists a constant C independent of $\phi$ (but dependent on $k,p$ ) such that

  $$\displaystyle \begin{aligned} \lVert BS_K\phi \rVert _{W^{k+1,p}} \leq C\lVert \phi \rVert _{W^{k,p}}.\end{aligned}$$

It should immediately be noted that such a K is not claimed to be unique, and in [20] is explicitly shown to be non-unique, however it does allow us to identify a velocity from a given vorticity satisfying the divergence-free and boundary conditions (2). From this point forwards we fix a specific K from the class of admissable integral kernels postulated in Theorem 4.2. We thus understand the nonlinear term as a mapping

$$\displaystyle \begin{aligned} \phi \mapsto \mathscr{L}_{BS_K\phi} \phi.\end{aligned}$$

This mapping shall at times be simply denoted by $\mathscr {L}_{BS_K}$. The Eq. (43) is thus closed in w.

4.2 Definitions and Results

We now state and prove the existence, uniqueness and maximality results for (43). We recall that to solve the velocity form (23) we used the extended criterion of Theorem 5.22, requiring the space $V:=W^{3,2}_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)$ to prove Theorem 3.6. This arose naturally in first showing Theorem 3.1, where we considered solutions explicitly in terms of the original Stratonovich form. For (43), however, solutions can be obtained for the natural choice of $w_0 \in W^{1,2}_{\sigma }(\mathscr {O};\mathbb {R}^3)$ with an application only of Theorem 5.15 in Sect. 5.3 (see Remark 4). We note that such a choice is natural as the identity is satisfied in $L^2(\mathscr {O};\mathbb {R}^3)$. Thus we do not take the detour of considering a fourth Hilbert Space to rigorously define solutions of the Stratonovich form (42), although this can be done similarly. Notions of local strong solution, maximal strong solution and pathwise uniqueness can then be defined identically to Definitions 3.3, 3.4 and 3.5 for the new identity (43). The result is then the following.

Theorem 4.3

For any given $\mathcal {F}_0-$ measurable ), there exists a unique maximal strong solution $(w,\Theta )$ of the equation (43). Moreover at $\mathbb {P}-a.e.$ $\omega$ for which $\Theta (\omega )<\infty$, we have that

$$\displaystyle \begin{aligned} \sup_{r \in [0,\Theta(\omega))}\lVert w_r(\omega) \rVert _{1}^2 + \int_0^{\Theta(\omega)}\lVert w_r(\omega) \rVert _2^2dr = \infty.\end{aligned}$$

4.3 Operator Bounds

Just as in Sect. 3.2, we state some intermediary results regarding the operators involved. In the following and throughout this section c will represent a generic constant changing from line to line, $c(\varepsilon )$ will be a generic constant dependent on a fixed $\varepsilon$, $\phi$ and $\psi$ will be arbitrary elements of $W^{2,2}_{\sigma }(\mathscr {O};\mathbb {R}^3)$ and $\phi _n \in \mathrm {span}\{a_1, \cdots , a_n\}$. The mapping $\mathscr {L}_i$ satisfies the same boundedness as (18), and also the following.

Lemma 4.4

There exists a constant c such that for each i and for all $f \in W^{k+2,2}(\mathcal {O};\mathbb {R}^3)$ with $k=0, 1$ , we have the bounds

$$\displaystyle \begin{aligned} \langle \mathscr{L}_i^2f, f \rangle _{W^{k,2}} + \lVert \mathscr{L}_if \rVert _{W^{k,2}}^2 &\leq c\lVert \xi_i \rVert _{W^{k+2,\infty}}^2\lVert f \rVert _{W^{k,2}}^2 {}, \end{aligned} $$

(45)

$$\displaystyle \begin{aligned} {} \langle \mathscr{L}_if, f \rangle _{W^{k,2}}^2 &\leq c\lVert \xi_i \rVert ^2_{W^{k+1,\infty}}\lVert f \rVert ^4_{W^{k,2}}, \end{aligned} $$

(46)

$$\displaystyle \begin{aligned} \lVert [\Delta,\mathscr{L}_i]f \rVert ^2 &\leq c\lVert \xi_i \rVert _{W^{3,\infty}}^2\lVert f \rVert _{W^{2,2}}^2 {} \end{aligned} $$

(47)

where $[\Delta ,\mathscr {L}_i]$ is the commutator

$$\displaystyle \begin{aligned} [\Delta,\mathscr{L}_i]:= \Delta \mathscr{L}_i - \mathscr{L}_i\Delta.\end{aligned}$$

Proof

See Sect. 5.1. □

Lemma 4.5

For any $\varepsilon > 0$ we have the bound

$$\displaystyle \begin{aligned} \langle \mathcal{P}_n\mathcal{P}\mathscr{L}_i^2\phi_n, \phi_n \rangle _1 & + \langle \mathcal{P}_n\mathcal{P}\mathscr{L}_i\phi_n, \mathcal{P}_n\mathcal{P}\mathscr{L}_i\phi_n \rangle _1\\ & \quad \leq c(\varepsilon)\lVert \xi_i \rVert _{W^{3,\infty}}^2\lVert \phi_n \rVert _1^2 + \varepsilon \lVert \xi_i \rVert _{W^{3,\infty}}^2\lVert \phi_n \rVert _2^2.\end{aligned} $$

Proof

This now follows precisely as for (27). □

Lemma 4.6

For any $\varepsilon > 0$ we have the bound

$$\displaystyle \begin{aligned} \left\vert \langle \mathcal{P}_n\mathcal{P}\mathscr{L}_{BS_K\phi_n}\phi_n, \phi_n \rangle _1\right\vert \leq c(\varepsilon)\lVert \phi_n \rVert _1^4 + \varepsilon\lVert \phi_n \rVert _2^2.\end{aligned}$$

Proof

See Sect. 5.1. □

Remark 4

Recalling Remark 3, the nonlinear term in velocity form fails this estimate. It is satisfied in the vorticity form due to the additional regularity that $f_n$ has compared to $\phi _n$. This difference is what enables us to apply Theorem 5.15 directly in the case $H:= W^{1,2}_{\sigma }$ for the vorticity form, whereas for the velocity form the appropriate estimate is only satisfied for $H:= W^{2,2}_{\sigma }$ (see Assumption 5.8) hence the need for Theorem 5.22 in velocity form.

In the following g is defined by

$$\displaystyle \begin{aligned} g(x) = \int_{\mathscr{O}}K(x,y)\psi(y)dy\end{aligned}$$

as in Theorem 4.2. The subsequent lemma is in analogy with Lemma 3.9.

Lemma 4.7

For any $\varepsilon > 0$ , we have that

$$\displaystyle \begin{aligned} \left\vert \langle \mathcal{P}\mathscr{L}_{BS_K\phi}\phi - \mathcal{P}\mathscr{L}_{BS_K\psi}\psi, \phi - \psi \rangle \right\vert \leq c(\varepsilon)\left[\lVert \phi \rVert _1^2 +\lVert \psi \rVert _1^2\right]\lVert \phi-\psi \rVert ^2 + \varepsilon \lVert \phi-\psi \rVert ^2_1\end{aligned}$$

Proof

See Sect. 5.1. □

4.4 Proof of Theorem 4.3

As discussed we apply Theorem 5.15, which we do for the spaces

$$\displaystyle \begin{aligned} V:= W^{2,2}_{\sigma}(\mathscr{O};\mathbb{R}^3), \qquad H:= W^{1,2}_{\sigma}(\mathscr{O};\mathbb{R}^3), \qquad U:= L^{2}_{\sigma}(\mathscr{O};\mathbb{R}^3).\end{aligned}$$

Proof of Theorem 4.3

The density relations are clear as $C^{\infty }_{0,\sigma }(\mathscr {O};\mathbb {R}^3) \subset W^{2,2}_{\sigma }(\mathscr {O};\mathbb {R}^3)$ is dense in both $W^{1,2}_{\sigma }(\mathscr {O};\mathbb {R}^3)$ and $L^{2}_{\sigma }(\mathscr {O};\mathbb {R}^3)$. The bilinear form (70) is simply again (37). Now we shift attentions to checking that the operators are measurable into the correct spaces. We note that $\mathscr {L}_{BS_K}$ has improved regularity properties over $\mathcal {L}$ given item 3 of Theorem 4.2, so retains the continuity with measurability following. There is no change to the Stokes Operator from Sect. 3. As for $\mathcal {P}\mathscr {L}_i$, we in fact first show that $\mathscr {L}_i \in C\left (W^{2,2}_{\sigma }(\mathscr {O};\mathbb {R}^3);W^{1,2}_{\sigma }(\mathscr {O};\mathbb {R}^3)\right )$ (and hence is invariant under $\mathcal {P}$⁴). This consists of three parts: showing that it is continuous as a mapping into $W^{1,2}(\mathscr {O};\mathbb {R}^3)$, showing the divergence free property and then the zero trace property. In fact with the appropriate regularity, it follows identically to (18) that we again have

$$\displaystyle \begin{aligned} {}\lVert \mathscr{L}_i\phi \rVert _{W^{k,2}}^2 \leq c\lVert \xi_i \rVert _{W^{k+1,\infty}}^2\lVert \phi \rVert _{W^{k+1,2}}^2\end{aligned} $$

(48)

which addresses the continuity. The fact that $\mathscr {L}_i \phi$ is divergence free comes immediately from the relation $\mathscr {L}_i \phi = \mathrm {curl}\left (B_i[BS_K\phi ]\right )$ and the well established fact the divergence of a curl is zero. For the zero trace property it is sufficient to show the existence of a sequence of compactly supported $\eta _n \in W^{1,2}(\mathscr {O};\mathbb {R}^3)$ which converge to $\mathscr {L}_i\phi$ in $W^{1,2}(\mathscr {O};\mathbb {R}^3)$. By definition of the property that $\xi _i \in W^{2,2}_0(\mathscr {O}\mathbb {R}^3)$ there is a sequence $(\gamma _n)$, $\gamma _n \in C^{\infty }_0(\mathscr {O};\mathbb {R}^3)$ such that ) in $W^{2,2}(\mathscr {O}\mathbb {R}^3)$. Evidently $\eta _n:=\mathscr {L}_{\gamma _n}\phi$ is again compactly supported, and observe that

$$\displaystyle \begin{aligned} \lVert \mathscr{L}_{\gamma_n}\phi - \mathscr{L}_i\phi \rVert _{W^{1,2}} = \lVert \mathscr{L}_{\gamma_n-\xi_i}\phi \rVert _{W^{1,2}} \leq c\lVert \gamma_n-\xi_i \rVert _{W^{2,2}}\lVert \phi \rVert _{W^{2,2}} \end{aligned} $$

from (6), which converges to zero as required to justify the zero trace property. The fact that $\mathcal {P}\mathscr {L}_i^2 \in C\left (W^{2,2}_{\sigma }(\mathscr {O};\mathbb {R}^3); L^2_{\sigma }(\mathscr {O};\mathbb {R}^3)\right )$ again follows from the linearity, (48) and continuity of $\mathcal {P}$. We now proceed to justify the numbered assumptions of Sect. 5.3.

Assumption 5.6::

We use the system $(a_k)$ of eigenfunctions of the Stokes Operator.

Assumption 5.7::

Items (74), (75) follow identically to the justification of (87), (88) for (23) given the increased regularity of $\mathscr {L}$ over $\mathcal {L}$ and the corresponding boundedness of the noise term (48). With the linearity of $\mathscr {L}_i$ then (76) follows trivially from (48).

Assumption 5.8::

(77) now follows from Lemmas 4.5 and 4.6 in the same manner as for the velocity equation. (78) is shown exactly as (82) was for the velocity equation, using that the $\mathcal {P}_n$ are orthogonal projections in $W^{1,2}_{\sigma }(\mathscr {O};\mathbb {R}^3)$.

Assumption 5.9::

The justification now comes together exactly as in the proof for Assumption 5.17 in the velocity case, noting again that (19) holds for $\mathscr {L}_i$ as well, and using Lemma 4.7.

Assumption 5.10::

There is very little to demonstrate here, as the linear terms follow from Assumption 5.9 so we just briefly address the nonlinear term. Through the same process as in Lemma 4.7, we have that

$$\displaystyle \begin{aligned} \left\vert \langle \mathcal{P}\mathscr{L}_f\phi, \phi \rangle \right\vert &\leq \left\vert \langle \mathcal{L}_f\phi, \phi \rangle + \langle \mathcal{L}_{\phi}f, \phi \rangle \right\vert\\ &\leq c\left[\lVert f \rVert _2\lVert \phi \rVert _1 + \lVert \phi \rVert _1\lVert f \rVert _2 \right]\lVert \phi \rVert \\ &\leq c\lVert \phi \rVert \lVert \phi \rVert _1^2 \end{aligned} $$

where the rest simply follows as in Assumption 5.9.

Assumption 5.11::

We consider the different operators in turn, starting with the nonlinear term and using that

$$\displaystyle \begin{aligned} \left\vert \langle \mathcal{P}\mathscr{L}_f\phi - \mathcal{P}\mathscr{L}_{g}\psi, \eta \rangle \right\vert &= \left\vert \langle \mathcal{L}_f\phi - \mathcal{L}_{\phi}f - \mathcal{L}_{g}\psi + \mathcal{L}_{\psi}g, \eta \rangle \right\vert\\ &\!=\! \left\vert \langle \mathcal{L}_{f-g}\phi \!+\! \mathcal{L}_{g}(\phi - \psi) \!-\! \mathcal{L}_{\phi - \psi}f - \mathcal{L}_{\psi}(f-g), \eta \rangle \right\vert \end{aligned} $$

where exactly as in Lemma 4.7 we have that

$$\displaystyle \begin{aligned} \lVert \mathcal{L}_{f-g}\phi + \mathcal{L}_{g}(\phi - \psi) - \mathcal{L}_{\phi - \psi}f - \mathcal{L}_{\psi}(f-g) \rVert \leq c\left[\lVert \phi \rVert _1 +\lVert \psi \rVert _1\right]\lVert \phi-\psi \rVert _1\end{aligned}$$

so in particular

$$\displaystyle \begin{aligned} {} \left\vert \langle \mathcal{P}\mathscr{L}_f\phi - \mathcal{P}\mathscr{L}_{g}\psi, \eta \rangle \right\vert \leq \lVert \eta \rVert \left(c\left[\lVert \phi \rVert _1 +\lVert \psi \rVert _1\right]\lVert \phi-\psi \rVert _1 \right). \end{aligned} $$

(49)

For the Stokes Operator we simply apply Proposition (12) to see that

$$\displaystyle \begin{aligned} {} \lvert \langle A\phi -A\psi, \eta \rangle \rvert = \lvert \langle A(\phi -\psi), \eta \rangle \rvert = \langle \phi - \psi, \eta \rangle _1 \leq \lVert \phi-\psi \rVert _1\lVert \eta \rVert _1 \end{aligned} $$

(50)

and for the $\mathscr {L}_i^2$ term we use the adjoint characterisation to observe that

$$\displaystyle \begin{aligned} {} \lvert \langle \mathscr{L}_i^2\phi -\mathscr{L}_i^2\psi, \eta \rangle \rvert &= \lvert \langle \mathscr{L}_i^2(\phi -\psi), \eta \rangle \rvert = \lvert \langle \mathscr{L}_i(\phi -\psi), B_i^*\eta \rangle \rvert \\ & \leq c\lVert \xi_i \rVert ^2_{W^{1,\infty}}\lVert \phi-\psi \rVert _1\lVert \eta \rVert _1. \end{aligned} $$

(51)

Combining (49), (50) and (51) gives the result.

□

5 Appendices

5.1 Proofs from Sects. 2.3, 3.2, and 4.3

We begin with the proofs from Sect. 2.3.

Proof of Proposition 2.6

We begin with showing (19), tasked to control

$$\displaystyle \begin{aligned} {}\langle B_i^2f, f \rangle _{W^{k,2}} + \lVert B_if \rVert _{W^{k,2}}^2\end{aligned} $$

(52)

and do so with each derivative in the sum for the inner product: that is, we are looking at

$$\displaystyle \begin{aligned} {}\langle D^\alpha B_i^2f, D^\alpha f \rangle + \langle D^\alpha B_if, D^\alpha B_if \rangle \end{aligned} $$

(53)

where we simplify the derivatives by writing

$$\displaystyle \begin{aligned} D^\alpha B_{\xi_i}f &= {}\sum_{\alpha' \leq \alpha}B_{D^{\alpha-\alpha'} \xi_i}D^{\alpha'}f\\ &= \sum_{\alpha' < \alpha}B_{D^{\alpha-\alpha'} \xi_i}D^{\alpha'}f + B_{\xi_i}D^{\alpha}f.\end{aligned} $$

(54)

Plugging this result in, we also see that

$$\displaystyle \begin{aligned} D^\alpha B^2_{\xi_i}f &= D^\alpha B_{\xi_i}\big(B_{\xi_i}f\big)\\ &= \sum_{\alpha' < \alpha}B_{D^{\alpha-\alpha'} \xi_i}D^{\alpha'}B_{\xi_i}f + B_{ \xi_i}D^{\alpha}B_{\xi_i}f \end{aligned} $$

which will use in our analysis of (53), reducing the expression to

$$\displaystyle \begin{aligned} \Big\langle \sum_{\alpha' < \alpha}B_{D^{\alpha-\alpha'} \xi_i}D^{\alpha'}B_{\xi_i}f + B_{\xi_i}D^\alpha B_{\xi_i}f,D^\alpha f\Big\rangle + \langle D^\alpha B_{\xi_i}f, D^\alpha B_{\xi_i}f \rangle\end{aligned}$$

which we further break up in terms of the adjoint $B_{\xi _i}^*$:

$$\displaystyle \begin{aligned} {}\Big\langle \sum_{\alpha' < \alpha} B_{D^{\alpha-\alpha'} \xi_i}D^{\alpha'}B_{\xi_i}f,D^\alpha f\Big\rangle + \langle D^\alpha B_{\xi_i}f, B^*_{\xi_i} D^\alpha f \rangle + \langle D^\alpha B_{\xi_i}f, D^\alpha B_{\xi_i}f \rangle \end{aligned} $$

(55)

requiring that $D^\alpha f \in W^{1,2}(\mathcal {O};\mathbb {R}^3)$, which is satisfied by the assumption $f \in W^{k+2,2}(\mathcal {O};\mathbb {R}^3)$. By summing the second and third inner products and using (54), this becomes

$$\displaystyle \begin{aligned} \Big\langle \sum_{\alpha' < \alpha} B_{D^{\alpha-\alpha'} \xi_i}D^{\alpha'}B_{\xi_i}f,D^\alpha f\Big\rangle + \Big\langle D^\alpha B_{\xi_i}f,B^*_{\xi_i} D^\alpha f + \sum_{\alpha' < \alpha}B_{D^{\alpha-\alpha'} \xi_i}D^{\alpha'}f + B_{\xi_i}D^\alpha f\Big\rangle\end{aligned}$$

which we look to simplify by combining $B_{\xi _i}^*$ and $B_{\xi _i}$, noting that

$$\displaystyle \begin{aligned} B_i^*+B_i= \mathcal{L}_{\xi_i}^* + \mathcal{T}_i^* + \mathcal{L}_{\xi_i} + \mathcal{T}_i = \mathcal{T}_i^* + \mathcal{T}_i.\end{aligned}$$

Indeed we arrive at the expression

$$\displaystyle \begin{aligned} \Big\langle \sum_{\alpha' < \alpha}B_{D^{\alpha-\alpha'} \xi_i}D^{\alpha'}B_{\xi_i}f,D^\alpha f\Big\rangle + \Big\langle D^\alpha B_{\xi_i}f,\big(\mathcal{T}_{\xi_i} + \mathcal{T}_{\xi_i}^*\big) D^\alpha f + \sum_{\alpha' < \alpha}B_{D^{\alpha-\alpha'} \xi_i}D^{\alpha'}f\Big\rangle.\end{aligned}$$

As we are looking to achieve control with respect to the $W^{k,2}(\mathcal {O};\mathbb {R}^3)$ norm of f, then it is the terms with differential operators of order greater than k that concern us. Of course this was the motivating factor behind combining $B_{\xi _i}$ and its adjoint, nullifying the additional derivative coming from $\mathcal {L}_{\xi _i}$. There are more higher order terms to go though, and the strategy will be to write these in terms of commutators with a differential operator of controllable order. This will involve considering different aspects of our sum in tandem, which will be helped with (54) reducing our expression again to

$$\displaystyle \begin{aligned} \Big\langle \sum_{\alpha' < \alpha}B_{D^{\alpha-\alpha'} \xi_i}D^{\alpha'}B_{\xi_i}f,D^\alpha f\Big\rangle &+ \Big\langle \sum_{\beta < \alpha} B_{D^{\alpha-\beta}\xi_i}D^\beta f + B_{\xi_i}D^{\alpha}f,\big(\mathcal{T}_{\xi_i} + \mathcal{T}_{\xi_i}^*\big) D^\alpha f\\ & \quad + \sum_{\alpha' < \alpha}B_{D^{\alpha-\alpha'} \xi_i}D^{\alpha'}f\Big\rangle.\end{aligned} $$

Ultimately the terms in the summand are split up into

$$\displaystyle \begin{aligned} {}& \langle B_{\xi_i}D^\alpha f, \big(\mathcal{T}_{\xi_i}+\mathcal{T}^*_{\xi_i}\big) D^\alpha f \rangle & \end{aligned} $$

(56)

$$\displaystyle \begin{aligned} {}+ &\Big\langle \sum_{\beta < \alpha} B_{D^{\alpha-\beta} \xi_i}D^{\beta}f ,\big(\mathcal{T}_{\xi_i} + \mathcal{T}_{\xi_i}^*\big) D^\alpha f + \sum_{\alpha' < \alpha} B_{D^{\alpha-\alpha'} \xi_i}D^{\alpha'}f\Big\rangle \end{aligned} $$

(57)

$$\displaystyle \begin{aligned} {}+ & \sum_{\alpha' < \alpha}\bigg(\langle B_{D^{\alpha-\alpha'} \xi_i}D^{\alpha'}B_{\xi_i}f, D^\alpha f \rangle + \langle B_{\xi_i}D^\alpha f, B_{D^{\alpha-\alpha'} \xi_i}D^{\alpha'}f \rangle \bigg) \end{aligned} $$

(58)

with the intention of controlling each one individually. Firstly for a treatment of (56),

$$\displaystyle \begin{aligned} \mbox{(56)} &=\langle (\mathcal{L}_{\xi_i} + \mathcal{T}_{\xi_i})D^{\alpha}f, (\mathcal{T}_{\xi_i}^* + \mathcal{T}_{\xi_i})D^{\alpha}f \rangle \\ &= \langle \mathcal{L}_{\xi_i}D^{\alpha}f, \mathcal{T}_{\xi_i}^*D^{\alpha}f \rangle + \langle \mathcal{L}_{\xi_i}D^{\alpha}f, \mathcal{T}_{\xi_i}D^{\alpha}f \rangle \\ & \quad + \langle \mathcal{T}_{\xi_i}D^{\alpha}f, \mathcal{T}_{\xi_i}^*D^{\alpha}f \rangle + \langle \mathcal{T}_{\xi_i}D^{\alpha}f, \mathcal{T}_{\xi_i}D^{\alpha}f \rangle \\ &= \Big(\langle \mathcal{T}_{\xi_i}\mathcal{L}_{\xi_i}D^{\alpha}f, D^{\alpha}f \rangle + \langle \mathcal{L}_{\xi_i}D^{\alpha}f, \mathcal{T}_{\xi_i}D^{\alpha}f \rangle \Big)\\ & \quad + \Big(\langle \mathcal{T}_{\xi_i}^2D^{\alpha}f, D^{\alpha}f \rangle + \langle \mathcal{T}_{\xi_i}D^{\alpha}f, \mathcal{T}_{\xi_i}D^{\alpha}f \rangle \Big). \end{aligned} $$

We now bound the brackets in terms of $\lVert D^{\alpha }f \rVert ^2$ separately, starting with the latter one as

$$\displaystyle \begin{aligned} \langle \mathcal{T}_{\xi_i}^2D^{\alpha}f, D^{\alpha}f \rangle & \leq \lVert \mathcal{T}_{\xi_i}^2D^{\alpha}f \rVert \lVert D^{\alpha}f \rVert \leq c\lVert \xi_i \rVert _{W^{1,\infty}}\lVert \mathcal{T}_{\xi_i}D^{\alpha}f \rVert \lVert D^{\alpha}f \rVert \\ & \leq c\lVert \xi_i \rVert ^2_{W^{1,\infty}}\lVert D^{\alpha}f \rVert ^2 \end{aligned} $$

and similarly

$$\displaystyle \begin{aligned} \langle \mathcal{T}_{\xi_i}D^{\alpha}f, \mathcal{T}_{\xi_i}D^{\alpha}f \rangle \leq \lVert \mathcal{T}_{\xi_i}D^{\alpha}f \rVert \lVert \mathcal{T}_{\xi_i}D^{\alpha}f \rVert \leq c\lVert \xi_i \rVert ^2_{W^{1,\infty}}\lVert D^{\alpha}f \rVert ^2.\end{aligned}$$

Now for the first bracket, we add and subtract a term to have an expression through the commutator of the operators:

$$\displaystyle \begin{aligned} &\langle \mathcal{T}_{\xi_i}\mathcal{L}_{\xi_i}D^{\alpha}f, D^{\alpha}f \rangle + \langle \mathcal{L}_{\xi_i}D^{\alpha}f, \mathcal{T}_{\xi_i}D^{\alpha}f \rangle \\ = & \langle (\mathcal{T}_{\xi_i}\mathcal{L}_{\xi_i} - \mathcal{L}_{\xi_i}\mathcal{T}_{\xi_i}) D^{\alpha}f, D^{\alpha}f \rangle + \langle \mathcal{L}_{\xi_i}\mathcal{T}_{\xi_i}D^{\alpha}f, D^{\alpha}f \rangle + \langle \mathcal{L}_{\xi_i}D^{\alpha}f, \mathcal{T}_{\xi_i}D^{\alpha}f \rangle \\ = & \langle (\mathcal{T}_{\xi_i}\mathcal{L}_{\xi_i} - \mathcal{L}_{\xi_i}\mathcal{T}_{\xi_i}) D^{\alpha}f, D^{\alpha}f \rangle + \langle \mathcal{T}_{\xi_i}D^{\alpha}f, \mathcal{L}_{\xi_i}^*D^{\alpha}f \rangle + \langle \mathcal{L}_{\xi_i}D^{\alpha}f, \mathcal{T}_{\xi_i}D^{\alpha}f \rangle \\ = & \langle (\mathcal{T}_{\xi_i}\mathcal{L}_{\xi_i} - \mathcal{L}_{\xi_i}\mathcal{T}_{\xi_i}) D^{\alpha}f, D^{\alpha}f \rangle . \end{aligned} $$

The commutator term is given explicitly through

$$\displaystyle \begin{aligned} \mathcal{T}_{\xi_i}\mathcal{L}_{\xi_i}D^{\alpha}f &= \mathcal{T}_{\xi_i}\Big(\sum_{j=1}^3\xi_i^j\partial_jD^{\alpha}f\Big)\\ &= \sum_{k=1}^3\Big(\sum_{j=1}^3\xi_i^j\partial_jD^{\alpha}f\Big)^k\nabla \xi_i^k\\ &= \sum_{k=1}^3\sum_{j=1}^3\xi_i^j\partial_jD^{\alpha}f^k\nabla \xi_i^k \end{aligned} $$

and

$$\displaystyle \begin{aligned} \mathcal{L}_{\xi_i}\mathcal{T}_{\xi_i}D^{\alpha}f &= \mathcal{L}_{\xi_i}\Big(\sum_{k=1}^3D^{\alpha}f^k\nabla \xi_i^k\Big)\\ &= \sum_{j=1}^3\xi_i^j\partial_j\Big(\sum_{k=1}^3D^{\alpha}f^k\nabla \xi_i^k\Big)\\ &= \sum_{j=1}^3\sum_{k=1}^3\xi_i^j\partial_j \big(D^{\alpha}f^k\nabla \xi_i^k\big)\\ &= \sum_{j=1}^3\sum_{k=1}^3\Big(\xi_i^j\partial_jD^{\alpha}f^k \nabla \xi_i^k + \xi_i^jD^{\alpha}f^k\partial_j\nabla\xi_i^k\Big) \end{aligned} $$

such that

$$\displaystyle \begin{aligned} (\mathcal{T}_{\xi_i}\mathcal{L}_{\xi_i} - \mathcal{L}_{\xi_i}\mathcal{T}_{\xi_i}) D^{\alpha}f = -\sum_{j=1}^3\sum_{k=1}^3\xi_i^jD^{\alpha}f^k\partial_j\nabla\xi_i^k.\end{aligned}$$

Therefore

$$\displaystyle \begin{aligned} \lVert (\mathcal{T}_{\xi_i}\mathcal{L}_{\xi_i} - \mathcal{L}_{\xi_i}\mathcal{T}_{\xi_i}) D^{\alpha}f \rVert &\leq c\sum_{j=1}^3\sum_{k=1}^3\lVert \xi_i^jD^{\alpha}f^k\partial_j\nabla\xi_i^k \rVert \\ &\leq c\sum_{j=1}^3\sum_{k=1}^3\sum_{l=1}^3\lVert \xi_i^jD^{\alpha}f^k\partial_j\partial_l\xi_i^k \rVert _{L^2(\mathcal{O};\mathbb{R})}\\ &\leq c\sum_{j=1}^3\sum_{k=1}^3\sum_{l=1}^3\lVert \xi_i^j\partial_j\partial_l\xi_i^k \rVert _{L^\infty(\mathcal{O};\mathbb{R})}\lVert D^{\alpha}f^k \rVert _{L^2(\mathcal{O};\mathbb{R})}\\ &\leq c\lVert \xi_i \rVert ^2_{W^{2,\infty}}\sum_{j=1}^3\sum_{k=1}^3\sum_{l=1}^3\lVert D^{\alpha}f^k \rVert _{L^2(\mathcal{O};\mathbb{R})}\\ &\leq c\lVert \xi_i \rVert ^2_{W^{2,\infty}}\lVert D^{\alpha}f \rVert \end{aligned} $$

and

$$\displaystyle \begin{aligned} \langle (\mathcal{T}_{\xi_i}\mathcal{L}_{\xi_i} - \mathcal{L}_{\xi_i}\mathcal{T}_{\xi_i}) D^{\alpha}f, D^{\alpha}f \rangle & \leq \lVert (\mathcal{T}_{\xi_i}\mathcal{L}_{\xi_i} - \mathcal{L}_{\xi_i}\mathcal{T}_{\xi_i}) D^{\alpha}f \rVert \lVert D^{\alpha}f \rVert \\ & \leq c\lVert \xi_i \rVert ^2_{W^{2,\infty}}\lVert D^{\alpha}f \rVert ^2.\end{aligned} $$

Combining these inequalities we determine the bound

$$\displaystyle \begin{aligned} \mbox{(56)} \leq c\lVert \xi_i \rVert ^2_{W^{2,\infty}}\lVert D^\alpha f \rVert ^2. \end{aligned} $$

As for (57) we look to use Cauchy-Schwarz and bound each item in the inner product. Indeed straight from (18) in the $L^2(\mathcal {O};\mathbb {R}^3)$ setting, by simply replacing $\xi _i$ with $D^{\alpha -\beta }\xi _i$, we have that

$$\displaystyle \begin{aligned} \lVert B_{D^{\alpha-\beta}\xi_i}D^{\beta}f \rVert ^2 \leq c\lVert D^{\alpha-\beta}\xi_i \rVert ^2_{W^{1,\infty}}\lVert D^\beta f \rVert ^2_{W^{1,2}} \leq c\lVert \xi_i \rVert ^2_{W^{k+1,\infty}}\lVert f \rVert ^2_{W^{k,2}}.\end{aligned} $$

Moreover,

$$\displaystyle \begin{aligned} \bigg\vert\bigg\vert \sum_{\beta < \alpha}B_{D^{\alpha-\beta}\xi_i}D^{\beta}f\bigg\vert\bigg\vert^2 \leq c\sum_{\beta < \alpha}\lVert B_{D^{\alpha-\beta}\xi_i}D^{\beta}f \rVert ^2 \leq c\lVert \xi_i \rVert ^2_{W^{k+1,\infty}}\lVert f \rVert ^2_{W^{k,2}}. \end{aligned} $$

In addition to this,

$$\displaystyle \begin{aligned} \lVert \big(\mathcal{T}_{\xi_i}+\mathcal{T}^*_{\xi_i}\big) D^\alpha f \rVert \leq \lVert \mathcal{T}_{\xi_i} D^\alpha f \rVert + \lVert \mathcal{T}^*_{\xi_i} D^\alpha f \rVert \leq c\lVert \xi_i \rVert _{W^{k+1,\infty}}\lVert D^\alpha f \rVert\end{aligned}$$

using the equivalence in operator norm of the adjoint. Together this amounts to

$$\displaystyle \begin{aligned} \mbox{(57)} &\leq \bigg\vert\bigg\vert \sum_{\beta < \alpha}B_{D^{\alpha-\beta}\xi_i}D^{\beta}f\bigg\vert \bigg\vert \cdot \bigg\vert\bigg\vert \big(\mathcal{T}_{\xi_i}+\mathcal{T}^*_{\xi_i}\big) D^\alpha f + \sum_{\alpha' < \alpha} B_{D^{\alpha-\alpha'}\xi_i}D^{\alpha'}f\bigg\vert\bigg\vert\\ &\leq \bigg\vert\bigg\vert \sum_{\beta < \alpha}B_{D^{\alpha-\beta}\xi_i}D^{\beta}f\bigg\vert\bigg\vert \Bigg(\lVert \big(\mathcal{T}_{\xi_i}+\mathcal{T}^*_{\xi_i}\big) D^\alpha f \rVert + \bigg\vert\bigg\vert \sum_{\alpha' < \alpha} B_{D^{\alpha-\alpha'}\xi_i}D^{\alpha'}f\bigg\vert\bigg\vert\Bigg)\\ &\leq c\lVert \xi_i \rVert _{W^{k+1,\infty}}\lVert f \rVert _{W^{k,2}}\Big(c\lVert \xi_i \rVert _{W^{k+1,\infty}}\lVert D^\alpha f \rVert + c\lVert \xi_i \rVert _{W^{k+1,\infty}}\lVert f \rVert _{W^{k,2}}\Big)\\ &\leq c\lVert \xi_i \rVert _{W^{k+1,\infty}}^2\lVert f \rVert _{W^{k,2}}^2. \end{aligned} $$

Let’s now turn our attentions to (58), which for each $\alpha '$ in the sum we rewrite as

$$\displaystyle \begin{aligned} {}\langle D^\alpha f, B_{D^{\alpha-\alpha'}\xi_i}D^{\alpha'}B_{\xi_i}f + B^*_{\xi_i}B_{D^{\alpha-\alpha'} \xi_i}D^{\alpha'}f \rangle \end{aligned} $$

(59)

and employing (54) again we see this becomes

$$\displaystyle \begin{aligned} & \Big\langle D^\alpha f,B_{D^{\alpha-\alpha'}\xi_i}\bigg(\sum_{\beta < \alpha'}B_{D^{\alpha'-\beta}\xi_i}D^\beta f + B_{\xi_i}D^{\alpha'}f\bigg)+ B^*_{\xi_i}B_{D^{\alpha-\alpha'} \xi_i}D^{\alpha'}f\Big\rangle\\ & \quad = \Big\langle D^\alpha f,\sum_{\beta < \alpha'}B_{D^{\alpha-\alpha'}\xi_i}B_{D^{\alpha'-\beta}\xi_i}D^\beta f\Big\rangle \\ & \quad + \langle D^\alpha f, B_{D^{\alpha-\alpha'}\xi_i}B_{\xi_i}D^{\alpha'}f + B^*_{\xi_i}B_{D^{\alpha-\alpha'} \xi_i}D^{\alpha'}f \rangle . \end{aligned} $$

We have split up these terms to make our approach clearer, as the two right hand sides of the inner products will be considered separately. For the first inner product, two applications of (18) give that

$$\displaystyle \begin{aligned} \lVert B_{D^{\alpha-\alpha'}\xi_i}B_{D^{\alpha'-\beta}\xi_i}D^\beta f \rVert ^2 &\leq c\lVert D^{\alpha-\alpha'}\xi_i \rVert ^2_{W^{1,\infty}}\lVert B_{D^{\alpha'-\beta}\xi_i}D^\beta f \rVert ^2_{W^{1,2}}\\ &\leq c\lVert D^{\alpha-\alpha'}\xi_i \rVert ^2_{W^{1,\infty}}\Big(c\lVert D^{\alpha'-\beta}\xi_i \rVert ^2_{W^{2,\infty}}\lVert D^\beta f \rVert ^2_{W^{2,2}}\Big)\\ &\leq c\lVert \xi_i \rVert _{W^{k+1,\infty}}^4\lVert f \rVert ^2_{W^{k,2}} \end{aligned} $$

Moreover,

$$\displaystyle \begin{aligned} \bigg\vert\bigg\vert \sum_{\beta < \alpha'}B_{D^{\alpha-\alpha'}\xi_i}B_{D^{\alpha'-\beta}\xi_i}D^\beta f\bigg\vert\bigg\vert^2 & \leq c\lVert B_{D^{\alpha-\alpha'}\xi_i}B_{D^{\alpha'-\beta}\xi_i}D^\beta f \rVert ^2\\ & \leq c\lVert \xi_i \rVert _{W^{k+1,\infty}}^4\lVert f \rVert ^2_{W^{k,2}}. \end{aligned} $$

As for the second inner product, we rewrite the right side as

$$\displaystyle \begin{aligned} B_{D^{\alpha-\alpha'} \xi_i}\big((\mathcal{L}_{\xi_i} + \mathcal{T}_{\xi_i})D^{\alpha'}f\big) + (\mathcal{L}^*_{\xi_i} + \mathcal{T}^*_{\xi_i})B_{D^{\alpha-\alpha'} \xi_i}D^{\alpha'}f\end{aligned}$$

and further

$$\displaystyle \begin{aligned} {}\big(B_{D^{\alpha -\alpha'} \xi_i}\mathcal{L}_{\xi_i} - \mathcal{L}_{\xi_i}B_{D^{\alpha-\alpha'} \xi_i}\big)D^{\alpha'}f + B_{D^{\alpha-\alpha'} \xi_i}\mathcal{T}_{\xi_i}D^{\alpha'}f + \mathcal{T}_{\xi_i}^*B_{D^{\alpha-\alpha'} \xi_i}D^{\alpha'}f.\end{aligned} $$

(60)

Starting with the latter two terms,

$$\displaystyle \begin{aligned} \lVert B_{D^{\alpha-\alpha'} \xi_i}\mathcal{T}_{\xi_i}D^{\alpha'}f \rVert ^2 &\leq c\lVert D^{\alpha-\alpha'} \xi_i \rVert ^2_{W^{1,\infty}}\lVert \mathcal{T}_{\xi_i}D^{\alpha'}f \rVert ^2_{W^{1,2}}\\ &\leq c\lVert D^{\alpha-\alpha'} \xi_i \rVert ^2_{W^{1,\infty}} \Big(c\lVert \xi_i \rVert ^2_{W^2,\infty}\lVert D^{\alpha'}f \rVert ^2_{W^{1,2}}\Big)\\ &\leq c\lVert \xi_i \rVert ^4_{W^{k+1,\infty}}\lVert f \rVert ^2_{W^{k,2}} \end{aligned} $$

and likewise

$$\displaystyle \begin{aligned} \lVert \mathcal{T}_{\xi_i}^*B_{D^{\alpha-\alpha'} \xi_i}D^{\alpha'}f \rVert ^2 &\leq c\lVert \xi_i \rVert ^2_{W^{1,\infty}}\lVert B_{D^{\alpha-\alpha'} \xi_i}D^{\alpha'}f \rVert ^2\\ &\leq c\lVert \xi_i \rVert ^2_{W^{1,\infty}}\Big(c\lVert D^{\alpha-\alpha'} \xi_i \rVert ^2_{W^{1,\infty}}\lVert D^{\alpha'}f \rVert ^2_{W^{1,2}}\Big)\\ &\leq c\lVert \xi_i \rVert ^4_{W^{k+1,\infty}}\lVert f \rVert ^2_{W^{k,2}}. \end{aligned} $$

Now we show explicitly that the commutator in

$$\displaystyle \begin{aligned} {} \big(B_{D^{\alpha -\alpha'} \xi_i}\mathcal{L}_{\xi_i} - \mathcal{L}_{\xi_i}B_{D^{\alpha-\alpha'} \xi_i}\big)D^{\alpha'}f \end{aligned} $$

(61)

from (60) is of first order (so of $k^{\mathrm {th}}$ order when composed with $D^{\alpha '}$), through the expressions

$$\displaystyle \begin{aligned} B_{D^{\alpha-\alpha'} \xi_i}\mathcal{L}_{\xi_i}D^{\alpha'}f &= \sum_{j=1}^3\bigg(D^{\alpha-\alpha'}\xi_i^j \partial_j\Big(\sum_{k=1}^3\xi_i^k\partial_k D^{\alpha'}f\Big) \\ & \quad + \Big(\sum_{k=1}^3\xi_i^k\partial_k D^{\alpha'}f\Big)^j\nabla D^{\alpha-\alpha'} \xi_i^j\bigg)\\ &= \sum_{j=1}^3\sum_{k=1}^3\bigg(D^{\alpha-\alpha'}\xi_i^j\partial_j\xi_i^k\partial_kD^{\alpha'}f \\ & \quad + D^{\alpha-\alpha'}\xi_i^j\xi_i^k\partial_j\partial_kD^{\alpha'}f + \xi_i^k\partial_k D^{\alpha'}f^j \nabla D^{\alpha-\alpha'} \xi_i^j\bigg) \end{aligned} $$

and

$$\displaystyle \begin{aligned} \mathcal{L}_{\xi_i}B_{D^{\alpha-\alpha'} \xi_i}D^{\alpha'}f &= \sum_{k=1}^3\xi_i^k\partial_k\bigg(\sum_{j=1}^3D^{\alpha-\alpha'} \xi_i^j\partial_jD^{\alpha'}f + D^{\alpha'}f^j \nabla D^{\alpha-\alpha'} \xi_i^j\bigg)\\ &= \sum_{j=1}^3\sum_{k=1}^3\bigg(\xi_i^k\partial_kD^{\alpha-\alpha'} \xi_i^j \partial_jD^{\alpha'}f + \xi_i^kD^{\alpha-\alpha'}\xi_i^j\partial_k\partial_jD^{\alpha'}f \\ & \quad + \xi_i^k\partial_kD^{\alpha'}f^j \nabla D^{\alpha-\alpha'} \xi_i^j + \xi_i^kD^{\alpha'}f^j\partial_k \nabla D^{\alpha-\alpha'} \xi_i^j\bigg) \end{aligned} $$

such that

$$\displaystyle \begin{aligned} (\mbox{61}) & = \sum_{j=1}^3\sum_{k=1}^3\bigg(D^{\alpha-\alpha'}\xi_i^j\partial_j\xi_i^k\partial_kD^{\alpha'}f - \xi_i^k\partial_kD^{\alpha-\alpha'} \xi_i^j \partial_jD^{\alpha'}f\\ & \quad - \xi_i^kD^{\alpha'}f^j\partial_k \nabla D^{\alpha-\alpha'} \xi_i^j\bigg)\end{aligned} $$

and in particular

$$\displaystyle \begin{aligned} \lVert \mbox{(61)} \rVert ^2 &\leq c\sum_{j=1}^3\sum_{k=1}^3\bigg(\lVert D^{\alpha-\alpha'}\xi_i^j\partial_j\xi_i^k\partial_kD^{\alpha'}f \rVert ^2 + \lVert \xi_i^k\partial_kD^{\alpha-\alpha'} \xi_i^j \partial_jD^{\alpha'}f \rVert ^2 \\ & \quad + \lVert \xi_i^kD^{\alpha'}f^j\partial_k \nabla D^{\alpha-\alpha'} \xi_i^j \rVert ^2\bigg)\\ &= c\sum_{j=1}^3\sum_{k=1}^3\sum_{l=1}^3\bigg(\lVert D^{\alpha-\alpha'}\xi_i^j\partial_j\xi_i^k\partial_kD^{\alpha'}f^l \rVert ^2_{L^2(\mathcal{O};\mathbb{R})} \\ & \quad + \lVert \xi_i^k\partial_kD^{\alpha-\alpha'} \xi_i^j \partial_jD^{\alpha'}f^l \rVert ^2_{L^2(\mathcal{O};\mathbb{R})} + \lVert \xi_i^kD^{\alpha'}f^j\partial_k \partial_l D^{\alpha-\alpha'} \xi_i^j \rVert ^2_{L^2(\mathcal{O};\mathbb{R})}\bigg)\\ & \leq c\lVert \xi_i \rVert ^4_{W^{k+2,\infty}}\sum_{j=1}^3\sum_{k=1}^3\sum_{l=1}^3\bigg(\lVert \partial_k D^{\alpha'}f^l \rVert ^2_{L^2(\mathcal{O};\mathbb{R})} \\ & \quad + \lVert \partial_j D^{\alpha'}f^l \rVert ^2_{L^2(\mathcal{O};\mathbb{R})} + \lVert D^{\alpha'}f^l \rVert ^2_{L^2(\mathcal{O};\mathbb{R})}\bigg)\\ &\leq c\lVert \xi_i \rVert ^4_{W^{k+2,\infty}}\bigg(\sum_{j=1}^3\lVert f \rVert ^2_{W^{k,2}} + \sum_{k=1}^3\lVert f \rVert ^2_{W^{k,2}} + \sum_{j=1}^3\sum_{k=1}^3\lVert D^{\alpha'}f \rVert ^2\bigg)\\ &\leq c\lVert \xi_i \rVert ^4_{W^{k+2,\infty}}\lVert f \rVert ^2_{W^{k,2}}. \end{aligned} $$

Finally now we can piece these four inequalities together to produce a bound on (59):

$$\displaystyle \begin{aligned} \lvert \mbox{(59)}\rvert &\leq \lVert D^\alpha f \rVert \bigg\vert\bigg\vert\sum_{\beta < \alpha'}B_{D^{\alpha-\alpha'}\xi_i}B_{D^{\alpha'-\beta}\xi_i}D^\beta f + \mbox{(60)}\bigg\vert\bigg\vert\\ &\leq \lVert D^\alpha f \rVert \bigg(\bigg\vert\bigg\vert\sum_{\beta < \alpha'}B_{D^{\alpha-\alpha'}\xi_i}B_{D^{\alpha'-\beta}\xi_i}D^\beta f\bigg\vert\bigg\vert + \lVert B_{D^{\alpha-\alpha'} \xi_i}\mathcal{T}_{\xi_i}D^{\alpha'}f \rVert \\ & \quad + \lVert \mathcal{T}_{\xi_i}^*B_{D^{\alpha-\alpha'} \xi_i}D^{\alpha'}f \rVert + \lVert \mbox{(61)} \rVert \bigg)\\ &\leq c\lVert D^\alpha f \rVert \bigg(\lVert \xi_i \rVert _{W^{k+1,\infty}}^2\lVert f \rVert _{W^{k,2}} + \lVert \xi_i \rVert ^2_{W^{k+1,\infty}}\lVert f \rVert _{W^{k,2}}\\ & \quad + \lVert \xi_i \rVert ^2_{W^{k+1,\infty}}\lVert f \rVert _{W^{k,2}} + \lVert \xi_i \rVert ^2_{W^{k+2,\infty}}\lVert f \rVert _{W^{k,2}}\bigg)\\ &\leq c\lVert D^\alpha f \rVert \Big(\lVert \xi_i \rVert _{W^{k+2,\infty}}^2\lVert f \rVert _{W^{k,2}}\Big)\\ &\leq c\lVert \xi_i \rVert _{W^{k+2,\infty}}^2\lVert f \rVert _{W^{k,2}}^2 \end{aligned} $$

and subsequently of (58):

$$\displaystyle \begin{aligned} \mbox{(58)} = \sum_{\alpha' < \alpha}\mbox{(59)} \leq c\sum_{\alpha' < \alpha}\lVert \xi_i \rVert _{W^{k+2,\infty}}^2\lVert f \rVert _{W^{k,2}}^2 \leq c\lVert \xi_i \rVert _{W^{k+2,\infty}}^2\lVert f \rVert _{W^{k,2}}^2. \end{aligned} $$

We can now conclude the proof of (19) by observing that

$$\displaystyle \begin{aligned} \mbox{(52)} &= \sum_{\lvert \alpha\rvert \leq k} \mbox{(53)}\\ &= \sum_{\lvert \alpha\rvert \leq k} \mbox{(56)} + \mbox{(57)} + \mbox{(58)}\\ &\leq c\sum_{\lvert \alpha\rvert \leq k} \bigg(\lVert \xi_i \rVert ^2_{W^{2,\infty}}\lVert D^\alpha f \rVert ^2 + \lVert \xi_i \rVert _{W^{k+1,\infty}}^2\lVert f \rVert _{W^{k,2}}^2 + \lVert \xi_i \rVert _{W^{k+2,\infty}}^2\lVert f \rVert _{W^{k,2}}^2\bigg)\\ &\leq c\sum_{\lvert \alpha\rvert \leq k} \lVert \xi_i \rVert _{W^{k+2,\infty}}^2\lVert f \rVert _{W^{k,2}}^2\\ &= c\lVert \xi_i \rVert _{W^{k+2,\infty}}^2\lVert f \rVert _{W^{k,2}}^2. \end{aligned} $$

As for (20), using (54) once more, we see that for each $\alpha$ in the sum for the inner product,

$$\displaystyle \begin{aligned} \lvert \langle D^\alpha B_i f, D^\alpha f \rangle \rvert &= \Big\vert\Big \langle \sum_{\alpha' < \alpha}B_{D^{\alpha-\alpha'}\xi_i}D^{\alpha'}f + B_{\xi_i}D^\alpha f, D^\alpha f \Big\rangle\Big\vert\\ &= \Big\vert\Big \langle \sum_{\alpha' < \alpha}B_{D^{\alpha-\alpha'}\xi_i}D^{\alpha'}f , D^\alpha f \Big\rangle + \langle B_{\xi_i}D^\alpha f, D^\alpha f \rangle \Big\vert\\ &\leq \Big\vert\Big \langle \sum_{\alpha' < \alpha}B_{D^{\alpha-\alpha'}\xi_i}D^{\alpha'}f , D^\alpha f \Big\rangle\Big\vert + \lvert \langle \mathcal{T}_{\xi_i}D^\alpha f, D^\alpha f \rangle \rvert \end{aligned} $$

using the cancellation from (15) to dispose of the order $k+1$ term. In our treatment of (57) in (19), we showed the bound

$$\displaystyle \begin{aligned} \bigg\vert\bigg\vert \sum_{\beta < \alpha}B_{D^{\alpha-\beta}\xi_i}D^{\beta}f\bigg\vert\bigg\vert &\leq c\lVert \xi_i \rVert _{W^{k+1,\infty}}\lVert f \rVert _{W^{k,2}} \end{aligned} $$

and therefore

$$\displaystyle \begin{aligned} &\Big\vert\Big \langle \sum_{\alpha' < \alpha}B_{D^{\alpha-\alpha'}\xi_i}D^{\alpha'}f , D^\alpha f \Big\rangle\Big\vert \leq c\lVert \xi_i \rVert _{W^{k+1,\infty}}\lVert f \rVert _{W^{k,2}}\lVert D^\alpha f \rVert\\ & \quad \leq c\lVert \xi_i \rVert _{W^{k+1,\infty}}\lVert f \rVert _{W^{k,2}}^2\end{aligned} $$

whilst a simple bound on the second term yields the result. □

Proof of Lemma 2.7

For $\nabla g \in L^{2, \perp }_{\sigma }(\mathcal {O};\mathbb {R}^3) \cap W^{1,2}(\mathcal {O};\mathbb {R}^3)$,

$$\displaystyle \begin{aligned} B_i(\nabla g) &= \mathcal{L}_{\xi_i}(\nabla g) + \mathcal{T}_{\xi_i}(\nabla g)\\ &= \sum_{j=1}^3\xi_i^j \partial_j(\nabla g) + \sum_{j=1}^3 \partial_jg \nabla \xi_i^j\\ &= \sum_{j=1}^3\xi_i^j (\nabla \partial_jg) + \sum_{j=1}^3 (\nabla \xi_i^j) \partial_jg\\ &= \nabla \sum_{j=1}^3\xi_i^j\partial_jg\\ &\in L^{2, \perp}_{\sigma}(\mathcal{O};\mathbb{R}^3) \end{aligned} $$

using in the last line the assumption that $\nabla g \in W^{1,2}(\mathcal {O};\mathbb {R}^3)$ and that $\xi _i \in W^{1,\infty }(\mathcal {O};\mathbb {R}^3)$. We now make a distinction between the settings of $\mathbb {T}^3$ and $\mathscr {O}$. For the bounded domain $\mathscr {O}$, we take any $f \in W^{1,2}(\mathscr {O};\mathbb {R}^3)$ and use the representation (9) to see that

$$\displaystyle \begin{aligned} \mathcal{P}B_if = \mathcal{P}B_i\left(\mathcal{P}f + \nabla g\right) = \mathcal{P}B_i\mathcal{P}f + \mathcal{P}(B_i\nabla g) = \mathcal{P}B_i\mathcal{P}f\end{aligned}$$

as required, using again the $W^{1,2}(\mathscr {O};\mathbb {R}^3)$ regularity of both components of the decomposition (9). In the case of the Torus we must address the constant term in the decomposition (10), appreciating that

$$\displaystyle \begin{aligned} B_ic = \mathcal{T}_{\xi_i}c = \sum_{j=1}^3c^j\nabla \xi_i^j = \nabla \sum_{j=1}^3c^j\xi_i^j \in L^{2, \perp}_{\sigma}(\mathbb{T}^3;\mathbb{R}^3)\end{aligned}$$

so the result follows in the same manner. □

We now state two results used in proving the Lemmas of Sects. 3.2 and 4.3. The first is derived from the Gagliardo-Nirenberg Inequality, whilst the second is proved below.

Proposition 5.1

There exists a constant c such that for any $f \in W^{1,2}_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)$ and $g \in W^{2,2}_{\sigma }(\mathbb {T}^3;\mathbb {R}^3)$,

$$\displaystyle \begin{aligned} {} \lVert \mathcal{L}_fg \rVert \leq c\lVert f \rVert _{1}\lVert g \rVert _{1}^{1/2}\lVert g \rVert _{2}^{1/2}.\end{aligned} $$

(62)

Meanwhile for $f \in W^{1,2}_{\sigma }(\mathscr {O};\mathbb {R}^3)$ and $g \in W^{2,2}_{\sigma }(\mathscr {O};\mathbb {R}^3)$ we have that

$$\displaystyle \begin{aligned} {} \lVert \mathcal{L}_fg \rVert \leq c\lVert f \rVert _{1}\left(\lVert g \rVert _{1}^{1/2}\lVert g \rVert _{2}^{1/2} + \lVert g \rVert _1\right).\end{aligned} $$

(63)

Proposition 5.2

There exists a constant c such that for every $f\in W^{3,2}(\mathcal {O};\mathbb {R}^3)$,

$$\displaystyle \begin{aligned} \lVert [\Delta,B_i]f \rVert ^2 \leq c\lVert \xi_i \rVert _{W^{3,\infty}}^2\lVert f \rVert _{W^{2,2}}^2\end{aligned}$$

where $[\Delta ,B_i]$ is the commutator

$$\displaystyle \begin{aligned} [\Delta,B_i]:= \Delta B_i - B_i\Delta.\end{aligned}$$

Proof

We fix any such f and first show that

$$\displaystyle \begin{aligned} {}[\Delta, B_i]f = \sum_{k=1}^3\sum_{j=1}^3\left(\partial_k^2\xi_i^j\partial_jf + 2 \partial_k\xi_i^j\partial_k\partial_jf + 2\partial_kf^j \partial_k\nabla \xi_i^j + f^j \partial_k^2\nabla \xi_i^j \right).\end{aligned} $$

(64)

Indeed

$$\displaystyle \begin{aligned} \Delta B_i f &= \sum_{k=1}^3\partial_k^2\left( \sum_{j=1}^3\left( \xi_i^j\partial_jf + f^j \nabla \xi_i^j\right)\right)\\ &= \sum_{k=1}^3\partial_k\left(\sum_{j=1}^3\left(\partial_k\xi_i^j\partial_jf + \xi_i^j\partial_k\partial_jf + \partial_kf^j \nabla \xi_i^j + f^j \partial_k \nabla \xi_i^j\right)\right)\\ &= \sum_{k=1}^3\sum_{j=1}^3\left(\partial_k^2\xi_i^j\partial_jf + 2 \partial_k\xi_i^j\partial_k\partial_jf + \xi_i^j\partial_k^2\partial_jf + \partial_k^2f^j \nabla \xi_i^j \right.\\ & \quad \left. + 2\partial_kf^j \partial_k\nabla \xi_i^j + f^j \partial_k^2\nabla \xi_i^j\right) \end{aligned} $$

and

$$\displaystyle \begin{aligned} B_i\Delta f &= \sum_{j=1}^3\left(\xi_i^j\partial_j\left(\sum_{k=1}^3\partial_k^2f\right) + \left(\sum_{k=1}^3\partial_k^2f\right)^j\nabla \xi_i^j\right)\\ &= \sum_{k=1}^3\sum_{j=1}^3\left(\xi_i^j\partial_k^2\partial_jf + \partial_k^2f^j\nabla \xi_i^j \right) \end{aligned} $$

therefore

$$\displaystyle \begin{aligned} [\Delta, B_i]f &= \Delta B_i f - B_i \Delta f\\ &= \sum_{k=1}^3\sum_{j=1}^3\left(\partial_k^2\xi_i^j\partial_jf + 2 \partial_k\xi_i^j\partial_k\partial_jf + 2\partial_kf^j \partial_k\nabla \xi_i^j + f^j \partial_k^2\nabla \xi_i^j\right) \end{aligned} $$

justifying (64). The result then follows with direct calculation:

$$\displaystyle \begin{aligned} \left \Vert [\Delta, B_i] f\right \Vert ^2 &\!=\! \left \Vert \sum_{k=1}^3\sum_{j=1}^3\left(\partial_k^2\xi_i^j\partial_jf \!+\! 2 \partial_k\xi_i^j\partial_k\partial_jf \!+\! 2\partial_kf^j \partial_k\nabla \xi_i^j \!+\! f^j \partial_k^2\nabla \xi_i^j\right) \right \Vert^2\\ &\leq c \sum_{k=1}^3\sum_{j=1}^3 \left \Vert\partial_k^2\xi_i^j\partial_jf + 2 \partial_k\xi_i^j\partial_k\partial_jf + 2\partial_kf^j \partial_k\nabla \xi_i^j + f^j \partial_k^2\nabla \xi_i^j \right \Vert ^2\\ &= c \sum_{k=1}^3\sum_{j=1}^3\sum_{l=1}^3 \left \Vert\partial_k^2\xi_i^j\partial_jf^l + 2 \partial_k\xi_i^j\partial_k\partial_jf^l + 2\partial_kf^j \partial_k \partial_l \xi_i^j \right. \\ & \qquad \left. + f^j \partial_k^2\partial_l\xi_i^j \right \Vert_{L^2(\mathcal{O};\mathbb{R})} ^2\\ &\leq c\sum_{k=1}^3\sum_{j=1}^3\sum_{l=1}^3 \Big(\lVert \partial_k^2\xi_i^j\partial_jf^l \rVert _{L^2(\mathcal{O};\mathbb{R})}^2 + \lVert 2 \partial_k\xi_i^j\partial_k\partial_jf^l \rVert _{L^2(\mathcal{O};\mathbb{R})}^2\\ & \qquad + \lVert 2\partial_kf^j \partial_k \partial_l \xi_i^j \rVert _{L^2(\mathcal{O};\mathbb{R})}^2 + \lVert f^j \partial_k^2\partial_l\xi_i^j \rVert _{L^2(\mathcal{O};\mathbb{R})}^2\Big)\\ &\leq c\lVert \xi_i \rVert _{W^{3,\infty}}^2\sum_{k=1}^3\sum_{j=1}^3\sum_{l=1}^3\Big(\lVert \partial_jf^l \rVert _{L^2(\mathcal{O};\mathbb{R})}^2 + \lVert \partial_k\partial_jf^l \rVert _{L^2(\mathcal{O};\mathbb{R})}^2\\ & \qquad + \lVert \partial_kf^j \rVert _{L^2(\mathcal{O};\mathbb{R})}^2 + \lVert f^j \rVert _{L^2(\mathcal{O};\mathbb{R})}^2\Big)\\ &\leq c\lVert \xi_i \rVert _{W^{3,\infty}}^2\lVert f \rVert ^2_{W^{2,2}}. \end{aligned} $$

□

We now prove Lemma 3.7 through the properties (26), (27) and (28) independently.

Proof of ( 26 )

We use the established result that for $k=2$ then the Sobolev Space $W^{k,2}(\mathcal {O};\mathbb {R})$ is an algebra (a result first presented in [46]), to deduce that

$$\displaystyle \begin{aligned} \lVert \mathcal{L}_{f_n}f_n \rVert _{W^{2,2}} = \left\Vert\sum_{j=1}^3f_n^j\partial_jf_n \right\Vert {}_{W^{2,2}} \leq c\lVert f_n \rVert _{2}\lVert f_n \rVert _{3}. \end{aligned} $$

From here we simply use that $\mathcal {P}_n$ is self-adjoint and Young’s Inequality to see that

$$\displaystyle \begin{aligned} \left\vert\langle \mathcal{P}_n\mathcal{P}\mathcal{L}_{f_n}f_n, f_n \rangle _2\right\vert = \left\vert\langle \mathcal{P}\mathcal{L}_{f_n}f_n, f_n \rangle _2\right\vert \leq c(\varepsilon)\lVert f_n \rVert ^4_2 + \varepsilon \lVert f_n \rVert _{3}^2. \end{aligned} $$

□

Proof of (27)

As the $\mathcal {P}_n$ are self-adjoint we can readily justify the inequality

$$\displaystyle \begin{aligned} \langle \mathcal{P}_n\mathcal{P}B_i^2f_n, f_n \rangle _1 + \langle \mathcal{P}_n\mathcal{P}B_if_n, \mathcal{P}_n\mathcal{P}B_if_n \rangle _1 \leq \langle \mathcal{P}B_i^2f_n, f_n \rangle _1 + \langle \mathcal{P}B_if_n, \mathcal{P}B_if_n \rangle _1\end{aligned}$$

and moreover from (12) that this is just

$$\displaystyle \begin{aligned} \langle \mathcal{P}B_i^2f_n, Af_n \rangle + \langle \mathcal{P}B_if_n, A\mathcal{P}B_if_n \rangle .\end{aligned}$$

We rewrite this as

$$\displaystyle \begin{aligned} \langle (\mathcal{P}B_i)^2f_n, Af_n \rangle + \langle \mathcal{P}B_if_n, AB_if_n \rangle\end{aligned}$$

and further as

$$\displaystyle \begin{aligned} {}\langle \mathcal{P}B_if_n, B_i^*Af_n \rangle - \langle \mathcal{P}B_if_n, \Delta B_if_n \rangle \end{aligned} $$

(65)

for the adjoint $B_i^* = \mathcal {L}_{\xi _i} + \mathcal {T}_{\xi _i}^*$. We look to commute the Laplacian and $B_i$, using Proposition 5.2 and subsequently the cancellation of the derivative in $B_i$ when considered with the adjoint. Indeed,

$$\displaystyle \begin{aligned} - \langle \mathcal{P}B_if_n, \Delta B_if_n \rangle &= - \langle \mathcal{P}B_if_n, ([\Delta, B_i] + B_i\Delta )f_n \rangle \\ &= - \langle \mathcal{P}B_if_n, [\Delta, B_i]f_n \rangle - \langle \mathcal{P}B_if_n, \mathcal{P}B_i\Delta f_n \rangle \\ &= - \langle \mathcal{P}B_if_n, [\Delta, B_i]f_n \rangle + \langle \mathcal{P}B_if_n, \mathcal{P}B_iAf_n \rangle \\ &= - \langle \mathcal{P}B_if_n, [\Delta, B_i]f_n \rangle + \langle \mathcal{P}B_if_n, B_iAf_n \rangle . \end{aligned} $$

Thus (65) becomes

$$\displaystyle \begin{aligned} \langle \mathcal{P}B_if_n, B_i^*Af_n \rangle - \langle \mathcal{P}B_if_n, [\Delta, B_i]f_n \rangle + \langle \mathcal{P}B_if_n, B_iAf_n \rangle\end{aligned}$$

or simply

$$\displaystyle \begin{aligned} {}\langle \mathcal{P}B_if_n, (\mathcal{T}_{\xi_i} + \mathcal{T}_{\xi_i}^*)Af_n - [\Delta, B_i]f_n \rangle \end{aligned} $$

(66)

which we look to bound through Cauchy-Schwarz and the results of (18) and Proposition 5.2 to see that

$$\displaystyle \begin{aligned} \mbox{(66)} &\leq\lVert \mathcal{P}B_if_n \rVert \left( \lVert (\mathcal{T}_{\xi_i} + \mathcal{T}_{\xi_i}^*)Af_n \rVert + \lVert [\Delta, B_i]f_n \rVert \right)\\ &\leq c\lVert \xi_i \rVert _{W^{1,\infty}}\lVert f_n \rVert _{W^{1,2}}\left(\lVert \xi_i \rVert _{W^{1,\infty}}\lVert Af_n \rVert + \lVert \xi_i \rVert _{W^{3,\infty}}\lVert f_n \rVert _{W^{2,2}}\right)\\ &\leq c\lVert \xi_i \rVert _{W^{1,\infty}}\lVert f_n \rVert _{1}\left(\lVert \xi_i \rVert _{W^{1,\infty}}\lVert f_n \rVert _{2} + \lVert \xi_i \rVert _{W^{3,\infty}}\lVert f_n \rVert _{2}\right)\\ &\leq c(\varepsilon)\lVert \xi_i \rVert _{W^{3,\infty}}^2\lVert f_n \rVert _{1}^2 + \varepsilon \lVert \xi_i \rVert _{W^{3,\infty}}^2\lVert f_n \rVert _{2}^2 \end{aligned} $$

as required. □

Proof of ( 28 )

As with (27) we can immediately say that

$$\displaystyle \begin{aligned} \langle \mathcal{P}_n\mathcal{P}B_i^2f_n, f_n \rangle _2 &+ \langle \mathcal{P}_n\mathcal{P}B_if_n, \mathcal{P}_n\mathcal{P}B_if_n \rangle _2 \\ & \qquad \leq \langle \mathcal{P}B_i^2f_n, f_n \rangle _2 + \langle \mathcal{P}B_if_n, \mathcal{P}B_if_n \rangle _2 {} \end{aligned} $$

(67)

which we again manipulate to give

$$\displaystyle \begin{aligned} \mbox{(67)} &= \langle A\mathcal{P}B_i^2f_n, Af_n \rangle + \langle A\mathcal{P}B_if_n, A\mathcal{P}B_if_n \rangle \\ & = -\langle \mathcal{P} \Delta B_i^2f_n, Af_n \rangle - \langle AB_if_n, \mathcal{P}\Delta B_if_n \rangle \\ &= - \langle \mathcal{P} [\Delta, B_i]B_if_n + \mathcal{P}B_i\Delta B_i f_n, Af_n \rangle - \langle AB_if_n, \mathcal{P}[\Delta, B_i]f_n + \mathcal{P}B_i\Delta f_n \rangle \\ &= - \langle \mathcal{P} [\Delta, B_i]B_if_n, Af_n \rangle + \langle \mathcal{P}B_i A B_i f_n, Af_n \rangle - \langle AB_if_n, \mathcal{P}[\Delta, B_i]f_n \rangle \\ & \quad + \langle AB_if_n, \mathcal{P}B_iA f_n \rangle \\ &= \langle B_i A B_i f_n, Af_n \rangle + \langle AB_if_n, B_iA f_n \rangle - \langle \mathcal{P} [\Delta, B_i]B_if_n, Af_n \rangle \\ & \quad - \langle AB_if_n, \mathcal{P}[\Delta, B_i]f_n \rangle \\ &=\langle AB_if_n, (B_i + B_i^*)A f_n \rangle - \langle \mathcal{P} [\Delta, B_i]B_if_n, Af_n \rangle-\langle AB_if_n, \mathcal{P}[\Delta, B_i]f_n \rangle \\ &= \langle AB_if_n, (\mathcal{T}_{\xi_i} + \mathcal{T}_{\xi_i}^*)A f_n \rangle - \langle \mathcal{P} [\Delta, B_i]B_if_n, Af_n \rangle - \langle AB_if_n, \mathcal{P}[\Delta, B_i]f_n \rangle . \end{aligned} $$

We shall treat each term individually using Cauchy-Schwarz, Young’s Inequality and Proposition 5.2 in the same manner as the proof of (27):

$$\displaystyle \begin{aligned} \langle AB_if_n, (\mathcal{T}_{\xi_i}+ \mathcal{T}_{\xi_i}^*)A f_n \rangle & \leq \lVert AB_if_n \rVert \lVert (\mathcal{T}_{\xi_i}+ \mathcal{T}_{\xi_i}^*)A f_n \rVert\\ & \leq c(\varepsilon)\lVert \xi_i \rVert ^2_{W^{3,\infty}}\lVert f_n \rVert _2^2 + \frac{\varepsilon}{3} \lVert \xi_i \rVert _{W^{3,\infty}}^2\lVert f_n \rVert _{3}^2 \end{aligned} $$

as well as

$$\displaystyle \begin{aligned} - \langle \mathcal{P} [\Delta, B_i]B_if_n, Af_n \rangle &\leq \lVert \mathcal{P} [\Delta, B_i]B_if_n \rVert \lVert Af_n \rVert \\ &\leq c\lVert [\Delta, B_i]B_if_n \rVert \lVert f_n \rVert _2\\ &\leq c\lVert \xi_i \rVert _{W^{3,\infty}}\lVert B_if_n \rVert _{W^{2,2}}\lVert f_n \rVert _2\\ &\leq c\lVert \xi_i \rVert _{W^{3,\infty}}^2\lVert f_n \rVert _{W^{3,2}}\lVert f_n \rVert _2\\ &\leq c(\varepsilon)\lVert \xi_i \rVert ^2_{W^{3,\infty}}\lVert f_n \rVert _2^2 + \frac{\varepsilon}{3} \lVert \xi_i \rVert _{W^{3,\infty}}^2\lVert f_n \rVert _{3}^2 \end{aligned} $$

and finally

$$\displaystyle \begin{aligned} - \langle AB_if_n, \mathcal{P}[\Delta, B_i]f_n \rangle &\leq \lVert AB_if_n \rVert \lVert \mathcal{P}[\Delta, B_i]f_n \rVert \\ &\leq c\lVert B_if_n \rVert _{W^{2,2}}\lVert [\Delta, B_i]f_n \rVert \\ &\leq c\lVert \xi_i \rVert _{W^{3,\infty}}^2\lVert f_n \rVert _{W^{3,2}}\lVert f_n \rVert _{W^{2,2}}\\ &\leq c(\varepsilon)\lVert \xi_i \rVert ^2_{W^{3,\infty}}\lVert f_n \rVert _2^2 + \frac{\varepsilon}{3} \lVert \xi_i \rVert _{W^{3,\infty}}^2\lVert f_n \rVert _{3}^2 \end{aligned} $$

Summing these up completes the proof. □

Proof of Lemma 3.8

Observe that

$$\displaystyle \begin{aligned} \langle \mathcal{P}\mathcal{L}_{f}f - \mathcal{P}\mathcal{L}_{g}g, f-g \rangle _1 &= \langle \mathcal{P}\mathcal{L}_{f}f - \mathcal{L}_{g}g, A(f-g) \rangle\\ &= \langle \mathcal{L}_{f - g}f + \mathcal{L}_{g}(f - g), A(f-g) \rangle \end{aligned} $$

and so it is sufficient to control the terms

$$\displaystyle \begin{aligned} \left\vert\langle \mathcal{L}_{f - g}f, A(f-g) \rangle \right\vert &\leq \lVert \mathcal{L}_{f - g}f \rVert \lVert A(f-g) \rVert\\ & \leq c(\varepsilon)\lVert f \rVert _2^2\lVert f-g \rVert _1^2 + \frac{\varepsilon}{2} \lVert f-g \rVert _2^2 \end{aligned} $$

and

$$\displaystyle \begin{aligned} \left\vert\langle \mathcal{L}_{g}(f - g), A(f-g) \rangle \right\vert&\leq \lVert \mathcal{L}_{g}(f - g) \rVert \lVert A(f-g) \rVert \\ &\leq c\lVert g \rVert _1\lVert f - g \rVert _1^{1/2}\lVert f - g \rVert _2^{1/2}\lVert f - g \rVert _2\\ &\leq c(\varepsilon)\lVert g \rVert ^4_1\lVert f - g \rVert _1^{2} + \frac{\varepsilon}{2}\lVert f - g \rVert _2^{2} \end{aligned} $$

using (62) and Young’s Inequality with conjugate exponents 4 and $4/3$. □

Proof of Lemma 3.9

As in Lemma 3.8, we use the inequality

$$\displaystyle \begin{aligned} \left\vert\langle \mathcal{P}\mathcal{L}_{f}f - \mathcal{P}\mathcal{L}_{g}g, f-g \rangle \right\vert \leq \left\vert\langle \mathcal{L}_{f-g}f, f-g \rangle \right\vert + \left\vert\langle \mathcal{L}_{g}(f-g), f-g \rangle \right\vert.\end{aligned}$$

For the first term, appealing to (4), observe that

$$\displaystyle \begin{aligned} \left\vert\langle \mathcal{L}_{f-g}f, f-g \rangle \right\vert \leq \lVert \mathcal{L}_{f-g}f \rVert \lVert f-g \rVert \leq c(\varepsilon)\lVert f \rVert _2^2 \lVert f-g \rVert ^2 + \varepsilon \lVert f-g \rVert _1^2. \end{aligned} $$

The second term is null due to (15), which concludes the proof. □

Proof of Lemma 4.4

We consider the proofs individually.

Assumption (45)::

We look to use the same approach as the one for (19), and given the similar structure of $\mathscr {L}_i$ to $B_i$ we provide only a sketch of the proof addressing the key properties. The essential component of the proof is in removing the derivative dependency coming from the transport type operator. Inspecting the proof of (19), there are three key properties that should be shown.

• 1. $\mathscr {L}_i^* + \mathscr {L}_i$ is of zeroth order.

• 2. Defining $\mathcal {Q}_i$ by $\mathcal {Q}_i: f \mapsto \mathcal {L}_f\xi _i$, then the commutator $[\mathcal {Q}_i, \mathcal {L}_{\xi _i}]$ is of zeroth order.

• 3. For $\lvert \alpha \rvert \leq 1$, the commutator $[\mathscr {L}_{D^{\alpha }\xi _i}, \mathcal {L}_{\xi _i}]$ is of first order.

We note that property 3 can be shown for arbitrary $\alpha$ if we assume sufficient regularity for $\xi _i$, as we did in Proposition 2.6. Property 1 is clear from the same structure $\mathscr {L}_i^*= -\mathcal {L}_{\xi _i} + \mathcal {Q}_i^*$ where $\mathcal {Q}_i, \mathcal {Q}_i^*$ are of zeroth order. We calculate the commutator in 2 explicitly, acting on $f \in W^{1,2}(\mathscr {O};\mathbb {R}^3)$:

$$\displaystyle \begin{aligned} \mathcal{Q}_i\mathcal{L}_{\xi_i}f &= \sum_{k=1}^3\left( \mathcal{L}_{\xi_i}f\right)^k\partial_k\xi_i = \sum_{k=1}^3\left( \sum_{j=1}^3\xi_i^j\partial_jf\right)^k\partial_k\xi_i\\ &= \sum_{k=1}^3\sum_{j=1}^3\xi_i^j\partial_jf^k\partial_k\xi_i \end{aligned} $$

and

$$\displaystyle \begin{aligned} \mathcal{L}_{\xi_i}\mathcal{Q}_if &= \sum_{j=1}^3\xi_i^j\partial_j\left( \mathcal{Q}_if\right) = \sum_{j=1}^3\xi_i^j\partial_j\left( \sum_{k=1}^3f^k\partial_k\xi_i\right)\\ &= \sum_{j=1}^3\sum_{k=1}^3\xi_i^j\left(\partial_j f^k\partial_k\xi_i +f^k \partial_j \partial_k\xi_i\right) \end{aligned} $$

hence

$$\displaystyle \begin{aligned} [\mathcal{Q}_i, \mathcal{L}_{\xi_i}] = -\sum_{j=1}^3\sum_{k=1}^3\xi_i^jf^k \partial_j \partial_k\xi_i\end{aligned}$$

which is of zeroth order. As for 3, the term which needs to be addressed is the one $[\mathcal {L}_{D^{\alpha }\xi _i},\mathcal {L}_{\xi _i}]$ which was already attended to in the original proof, so the result is concluded here.

Assumption (46)::

Comparing to the proof of (20), this is a consequence of the property (15) once more and the same boundedness of (18).

Assumption (47)::

Once more the critical term is $[\Delta , \mathcal {L}_{\xi _i}]$ which was addressed in the proof of Proposition 5.2.

□

Proof of Lemma 4.6

Note that

$$\displaystyle \begin{aligned} \left\vert \langle \mathcal{P}_n\mathcal{P}\mathscr{L}_{BS_K\phi_n}\phi_n, \phi_n \rangle _1\right\vert \!=\! \left\vert \langle \mathcal{P}_n\mathcal{P}\mathscr{L}_{BS_K\phi_n}\phi_n, A \phi_n \rangle \right\vert \!\leq\! c(\varepsilon)\lVert \mathscr{L}_{BS_K\phi_n}\phi_n \rVert ^2 + \varepsilon\lVert \phi_n \rVert _2^2\end{aligned}$$

and

$$\displaystyle \begin{aligned} \lVert \mathscr{L}_{BS_K\phi_n}\phi_n \rVert ^2 \leq 2\left(\lVert \mathcal{L}_{BS_K\phi_n}\phi_n \rVert ^2 + \lVert \mathcal{L}_{\phi_n}BS_K\phi_n \rVert ^2\right)\end{aligned}$$

so we look to control these two terms. Indeed,

$$\displaystyle \begin{aligned} \lVert \mathcal{L}_{BS_K\phi_n}\phi_n \rVert ^2 &\leq c\sum_{j=1}^3\sum_{k=1}^3\lVert BS_K\phi_n^j \rVert _{L^\infty(\mathscr{O};\mathbb{R})}^2\lVert \partial_j\phi_n^k \rVert _{L^2(\mathscr{O};\mathbb{R})}^2\\ &\leq c\sum_{j=1}^3\sum_{k=1}^3\lVert BS_K\phi_n^j \rVert _{W^{2,2}(\mathscr{O};\mathbb{R})}^2\lVert \partial_j\phi_n^k \rVert _{L^2(\mathscr{O};\mathbb{R})}^2\\ &\leq c\lVert BS_K\phi_n \rVert _{W^{2,2}}^2\lVert \phi_n \rVert _{W^{1,2}}^2\\ &\leq c\lVert \phi_n \rVert _{W^{1,2}}^4\\ &\leq c\lVert \phi_n \rVert _{1}^4 \end{aligned} $$

using the Sobolev Embedding of and item (3) of Theorem 4.2. Likewise observe that

$$\displaystyle \begin{aligned} \lVert \mathcal{L}_{\phi_n}BS_K\phi_n \rVert ^2 &\leq c\sum_{j=1}^3\sum_{k=1}^3\lVert \phi_n^j \rVert _{L^4(\mathscr{O};\mathbb{R})}^2\lVert \partial_jBS_K\phi_n^k \rVert _{L^4(\mathscr{O};\mathbb{R})}^2\\ &\leq c\sum_{j=1}^3\sum_{k=1}^3\lVert \phi_n^j \rVert _{W^{1,2}(\mathscr{O};\mathbb{R})}^2\lVert \partial_jBS_K\phi_n^k \rVert _{W^{1,2}(\mathscr{O};\mathbb{R})}^2\\ &\leq c\lVert \phi_n \rVert _{W^{1,2}}^2\lVert BS_K\phi_n \rVert _{W^{2,2}}^2\\ &\leq c\lVert \phi_n \rVert _{W^{1,2}}^4\\ &\leq c\lVert \phi_n \rVert _{1}^4. \end{aligned} $$

Summing these terms completes the proof. □

Proof of Lemma 4.7

We write out the left hand side in full:

$$\displaystyle \begin{aligned} &\left\vert \langle \mathcal{P}\mathscr{L}_{BS_K\phi}\phi - \mathcal{P}\mathscr{L}_{{BS_K\psi}}\psi, \phi - \psi \rangle \right\vert\\ & \qquad = \left\vert \langle \mathcal{L}_{BS_K\phi}\phi - \mathcal{L}_{\phi}{BS_K\phi} - \mathcal{L}_{{BS_K\psi}}\psi + \mathcal{L}_{\psi}{BS_K\psi}, \phi - \psi \rangle \right\vert\\ & \qquad = \left\vert \langle \mathcal{L}_{{BS_K\phi}-{BS_K\psi}}\phi + \mathcal{L}_{{BS_K\psi}}(\phi - \psi) - \mathcal{L}_{\phi - \psi}{BS_K\phi} \right.\\ & \qquad \left.- \mathcal{L}_{\psi}({BS_K\phi}-{BS_K\psi}), \phi - \psi \rangle \right\vert \end{aligned} $$

from which we shall split up the terms and control them individually. Firstly,

$$\displaystyle \begin{aligned} \left\vert \langle \mathcal{L}_{{BS_K\phi}-{BS_K\psi}}\phi, \phi - \psi \rangle \right\vert &\leq \lVert \mathcal{L}_{{BS_K\phi}-{BS_K\psi}}\phi \rVert \lVert \phi - \psi \rVert \\ &\leq c\lVert {BS_K(\phi -\psi)} \rVert _2\lVert \phi \rVert _1\lVert \phi-\psi \rVert \\ &\leq c\lVert \phi-\psi \rVert _1\lVert \phi \rVert _1\lVert \phi-\psi \rVert \\ &\leq c(\varepsilon)\lVert \phi \rVert _1^2\lVert \phi-\psi \rVert ^2 + \frac{\varepsilon}{3}\lVert \phi-\psi \rVert _1^2 \end{aligned} $$

using (4) and that $[{BS_K\phi }-{BS_K\psi }](x) = \int _{\mathscr {O}}K(x,y)[\phi -\psi ](y)dy$ is the solution specified by $BS_K(\phi - \psi )$ in Theorem 4.2 for $\phi -\psi$. Even more directly we have that

$$\displaystyle \begin{aligned} \langle \mathcal{L}_{{BS_K\psi}}(\phi - \psi), \phi - \psi \rangle = 0\end{aligned}$$

owing to (15), and for the final two terms the bounds

$$\displaystyle \begin{aligned} \left\vert \langle \mathcal{L}_{\phi - \psi}{BS_K\phi}, \phi - \psi \rangle \right\vert & \leq c\lVert \phi-\psi \rVert _1\lVert {BS_K\phi} \rVert _2\lVert \phi-\psi \rVert\\ & \leq c(\varepsilon)\lVert \phi \rVert _1^2\lVert \phi-\psi \rVert ^2 + \frac{\varepsilon}{3}\lVert \phi-\psi \rVert _1^2\end{aligned} $$

and

$$\displaystyle \begin{aligned} \left\vert \langle \mathcal{L}_{\psi}({BS_K\phi}-{BS_K\psi}), \phi - \psi \rangle \right\vert &\leq c\lVert \psi \rVert _1\lVert {BS_K\phi}-{BS_K\psi} \rVert _2\lVert \phi-\psi \rVert \\ & \leq c(\varepsilon)\lVert \psi \rVert _1^2\lVert \phi-\psi \rVert ^2 + \frac{\varepsilon}{3}\lVert \phi-\psi \rVert _1^2.\end{aligned} $$

Summing these terms concludes the proof. □

5.2 A Conversion from Stratonovich to Itô

This theory is taken from [30] Subsections 2.2 and 2.3, and is provided here for simplicity to apply in Sect. 3.3. We work with a quartet of embedded Hilbert Spaces

$$\displaystyle \begin{aligned} V \hookrightarrow H \hookrightarrow U \hookrightarrow X\end{aligned}$$

where the embedding is assumed as a continuous linear injection. We start from an SPDE

$$\displaystyle \begin{aligned} {} \boldsymbol{\Psi}_t = \boldsymbol{\Psi}_0 + \int_0^t \mathcal{Q}\boldsymbol{\Psi}_sds + \int_0^t\mathcal{G}\boldsymbol{\Psi}_s \circ d\mathcal{W}_s. \end{aligned} $$

(68)

where the mappings $\mathcal {Q}$, $\mathcal {G}$ satisfy the following conditions, with the general operator ) defined by

$$\displaystyle \begin{aligned} \tilde{K}(\phi):= c\left(1 + \lVert \phi \rVert _U^p + \lVert \phi \rVert _H^q\right)\end{aligned}$$

for any constants $c,p,q$ independent of $\phi$.

Assumption 5.3

) is measurable and for any $\phi \in V$,

$$\displaystyle \begin{aligned} \lVert \mathcal{Q}\phi \rVert _U \leq \tilde{K}(\phi)[1 + \lVert \phi \rVert _V^2].\end{aligned}$$

Assumption 5.4

$\mathcal {G}$ is understood as a measurable mapping

defined over $\mathfrak {U}$ by its action on the basis vectors

$$\displaystyle \begin{aligned} \mathcal{G}(\cdot, e_i):= \mathcal{G}_i(\cdot).\end{aligned}$$

In addition each $\mathcal {G}_i$ is linear and there exists constants $c_i$ such that for all $\phi \in V$, $\psi \in H$, $\eta \in U$:

$$\displaystyle \begin{aligned} \lVert \mathcal{G}_i\phi \rVert _{H} \leq c_i \lVert \phi \rVert _V, \quad \lVert \mathcal{G}_i\psi \rVert _{U} \leq c_i \lVert \psi \rVert _H, \quad \lVert \mathcal{G}_i\eta \rVert _{X} \leq c_i \lVert \eta \rVert _U, \quad \sum_{i=1}^\infty c_i^2 < \infty. \end{aligned} $$

In this setting, we have the following result ([30] Theorem 2.3.1 and Corollary 2.3.1.1).

Theorem 5.5

Suppose that $(\boldsymbol {\Psi },\tau )$ are such that: $\tau$ is a ) positive stopping time and $\boldsymbol {\Psi }$ is a process whereby for ) $\omega$, $\boldsymbol {\Psi }_{\cdot }(\omega ) \in C\left ([0,T];H\right )$ and ) for all $T>0$ with ) progressively measurable in V , and moreover satisfy the identity

$$\displaystyle \begin{aligned} \boldsymbol{\Psi}_{t} = \boldsymbol{\Psi}_0 + \int_0^{t\wedge \tau} \left(\mathcal{Q} + \frac{1}{2}\sum_{i=1}^\infty \mathcal{G}_i^2\right)\boldsymbol{\Psi}_sds + \int_0^{t \wedge \tau}\mathcal{G}\boldsymbol{\Psi}_s d\mathcal{W}_s \end{aligned}$$

) in U for all $t \geq 0$. Then the pair $(\boldsymbol {\Psi },\tau )$ satisfies the identity

$$\displaystyle \begin{aligned} \boldsymbol{\Psi}_{t} = \boldsymbol{\Psi}_0 + \int_0^{t\wedge \tau} \mathcal{Q}\boldsymbol{\Psi}_sds + \int_0^{t \wedge \tau}\mathcal{G}\boldsymbol{\Psi}_s \circ d\mathcal{W}_s\end{aligned}$$

) in X for all $t \geq 0$.

The mapping $\frac {1}{2}\sum _{i=1}^\infty \mathcal {G}_i^2$ is understood as a pointwise limit, which is justified in [30] Subsection 2.3.

Remark 5

Practically, Theorem 5.5 provides an Itô equation from a Stratonovich one in the sense that solving this Itô equation is sufficient to satisfy the identity in Stratonovich form. To discuss an equivalence between the equations we would need to formally define a solution of the Stratonovich equation, which we do not do here. To make sense of the Stratonovich integral one would have to impose that the solution is a local semimartingale in U, in which case the two notions are genuinely equivalent.

5.3 Abstract Solution Criterion I

The result is given in the context of an Itô SPDE

$$\displaystyle \begin{aligned} {} \boldsymbol{\Psi}_t = \boldsymbol{\Psi}_0 + \int_0^t \mathcal{A}(s,\boldsymbol{\Psi}_s)ds + \int_0^t\mathcal{G} (s,\boldsymbol{\Psi}_s) d\mathcal{W}_s. \end{aligned} $$

(69)

We state the assumptions for a triplet of embedded Hilbert Spaces

$$\displaystyle \begin{aligned} V \hookrightarrow H \hookrightarrow U\end{aligned}$$

and ask that there is a continuous bilinear form ) such that for $\phi \in H$ and $\psi \in V$,

$$\displaystyle \begin{aligned} {} \langle \phi, \psi \rangle _{U \times V} = \langle \phi, \psi \rangle _{H}. \end{aligned} $$

(70)

The mappings $\mathcal {A},\mathcal {G}$ are such that for any $T>0$, ) are measurable. We assume that V  is dense in H which is dense in U.

Assumption 5.6

There exists a system $(a_n)$ of elements of V  such that, defining the spaces $V_n:= \mathrm {span}\left \{a_1, \dots , a_n \right \}$ and $\mathcal {P}_n$ as the orthogonal projection to $V_n$ in U, then:

• 1. There exists some constant c independent of n such that for all $\phi \in H$,

  $$\displaystyle \begin{aligned} {} \lVert \mathcal{P}_n \phi \rVert _H^2 \leq c\lVert \phi \rVert _H^2. \end{aligned} $$

  (71)

• 2. There exists a real valued sequence $(\mu _n)$ with ) such that for any $\phi \in H$,

  $$\displaystyle \begin{aligned} {} \lVert (I - \mathcal{P}_n)\phi \rVert _U \leq \frac{1}{\mu_n}\lVert \phi \rVert _H \end{aligned} $$

  (72)

  where I represents the identity operator in U.

These conditions are of course supplemented by a series of assumptions on the mappings. We shall use general notation $c_t$ to represent a function ) bounded on $[0,T]$ for any $T > 0$, evaluated at the time t. Moreover we define functions K, $\tilde {K}$ relative to some non-negative constants $p,\tilde {p},q,\tilde {q}$. We use a generic notation to define the functions ), ), ) and ) by

$$\displaystyle \begin{aligned} K(\phi)&:= 1 + \lVert \phi \rVert _U^{p}, \qquad K(\phi,\psi):= 1+\lVert \phi \rVert _U^{p} + \lVert \psi \rVert _U^{q},\\ \tilde{K}(\phi) &:= K(\phi) + \lVert \phi \rVert _H^{\tilde{p}}, \qquad \tilde{K}(\phi,\psi) := K(\phi,\psi) + \lVert \phi \rVert _H^{\tilde{p}} + \lVert \psi \rVert _H^{\tilde{q}} \end{aligned} $$

Distinct use of the function K will depend on different constants but in no meaningful way in our applications, hence no explicit reference to them shall be made. In the case of $\tilde {K}$, when $\tilde {p}, \tilde {q} = 2$ then we shall denote the general $\tilde {K}$ by $\tilde {K}_2$. In this case no further assumptions are made on the $p,q$. That is, $\tilde {K}_2$ has the general representation

$$\displaystyle \begin{aligned} {}\tilde{K}_2(\phi,\psi) = K(\phi,\psi) + \lVert \phi \rVert _H^2 + \lVert \psi \rVert _H^2\end{aligned} $$

(73)

and similarly as a function of one variable.

We state the subsequent assumptions for arbitrary elements $\phi ,\psi \in V$, $\phi ^n \in V_n$, $\eta \in H$ and $t \in [0,\infty )$, and a fixed $\kappa > 0$. Understanding $\mathcal {G}$ as a mapping), we introduce the notation $\mathcal {G}_i(\cdot ,\cdot ):= \mathcal {G}(\cdot ,\cdot ,e_i)$.

Assumption 5.7

$$\displaystyle \begin{aligned} {} \lVert \mathcal{A}(t,\boldsymbol{\phi}) \rVert ^2_U +\sum_{i=1}^\infty \lVert \mathcal{G}_i(t,\boldsymbol{\phi}) \rVert ^2_H &\leq c_t K(\boldsymbol{\phi})\left[1 + \lVert \boldsymbol{\phi} \rVert _V^2\right], \end{aligned} $$

(74)

$$\displaystyle \begin{aligned} {} \lVert \mathcal{A}(t,\boldsymbol{\phi}) - \mathcal{A}(t,\boldsymbol{\psi}) \rVert _U^2 &\leq c_t\left[K(\phi,\psi) + \lVert \phi \rVert _V^2 + \lVert \psi \rVert _V^2\right]\lVert \phi-\psi \rVert _V^2, \end{aligned} $$

(75)

$$\displaystyle \begin{aligned} {} \sum_{i=1}^\infty \lVert \mathcal{G}_i(t,\boldsymbol{\phi}) - \mathcal{G}_i(t,\boldsymbol{\psi}) \rVert _U^2 &\leq c_tK(\phi,\psi)\lVert \phi-\psi \rVert _H^2. \end{aligned} $$

(76)

Assumption 5.8

$$\displaystyle \begin{aligned} {} 2\langle \mathcal{P}_n\mathcal{A}(t,\boldsymbol{\phi}^n), \boldsymbol{\phi}^n \rangle _H + \sum_{i=1}^\infty\lVert \mathcal{P}_n\mathcal{G}_i(t,\boldsymbol{\phi}^n) \rVert _H^2 &\leq c_t\tilde{K}_2(\boldsymbol{\phi}^n)\left[1 + \lVert \boldsymbol{\phi}^n \rVert _H^2\right] - \kappa\lVert \boldsymbol{\phi}^n \rVert _V^2, \end{aligned} $$

(77)

$$\displaystyle \begin{aligned} {} \sum_{i=1}^\infty \langle \mathcal{P}_n\mathcal{G}_i(t,\boldsymbol{\phi}^n), \boldsymbol{\phi}^n \rangle ^2_H &\leq c_t\tilde{K}_2(\boldsymbol{\phi}^n)\left[1 + \lVert \boldsymbol{\phi}^n \rVert _H^4\right]. \end{aligned} $$

(78)

Assumption 5.9

$$\displaystyle \begin{aligned} 2\langle \mathcal{A}(t,\boldsymbol{\phi}) - \mathcal{A}(t,\boldsymbol{\psi}), \boldsymbol{\phi} - \boldsymbol{\psi} \rangle _U &+ \sum_{i=1}^\infty\lVert \mathcal{G}_i(t,\boldsymbol{\phi}) - \mathcal{G}_i(t,\boldsymbol{\psi}) \rVert _U^2\\ {} &\leq c_{t}\tilde{K}_2(\boldsymbol{\phi},\boldsymbol{\psi}) \lVert \boldsymbol{\phi}-\boldsymbol{\psi} \rVert _U^2 - \kappa\lVert \boldsymbol{\phi}-\boldsymbol{\psi} \rVert _H^2, \end{aligned} $$

(79)

$$\displaystyle \begin{aligned} {} \sum_{i=1}^\infty \langle \mathcal{G}_i(t,\boldsymbol{\phi}) - \mathcal{G}_i(t,\boldsymbol{\psi}), \boldsymbol{\phi}-\boldsymbol{\psi} \rangle ^2_U & \leq c_{t} \tilde{K}_2(\boldsymbol{\phi},\boldsymbol{\psi}) \lVert \boldsymbol{\phi}-\boldsymbol{\psi} \rVert _U^4. \end{aligned} $$

(80)

Assumption 5.10

$$\displaystyle \begin{aligned} {} 2\langle \mathcal{A}(t,\boldsymbol{\phi}), \boldsymbol{\phi} \rangle _U + \sum_{i=1}^\infty\lVert \mathcal{G}_i(t,\boldsymbol{\phi}) \rVert _U^2 &\leq c_tK(\boldsymbol{\phi})\left[1 + \lVert \boldsymbol{\phi} \rVert _H^2\right], \end{aligned} $$

(81)

$$\displaystyle \begin{aligned}{} \sum_{i=1}^\infty \langle \mathcal{G}_i(t,\boldsymbol{\phi}), \boldsymbol{\phi} \rangle ^2_U &\leq c_tK(\boldsymbol{\phi})\left[1 + \lVert \boldsymbol{\phi} \rVert _H^4\right]. \end{aligned} $$

(82)

Assumption 5.11

$$\displaystyle \begin{aligned} {} \langle \mathcal{A}(t,\phi)-\mathcal{A}(t,\psi), \eta \rangle _U \leq c_t(1+\lVert \eta \rVert _H)\left[K(\phi,\psi) + \lVert \phi \rVert _V + \lVert \psi \rVert _V\right]\lVert \phi-\psi \rVert _H. \end{aligned} $$

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With these assumptions in place we state the relevant definitions and results, first announced in [31] and proven in [32]. Definition 5.12 is stated for an $\mathcal {F}_0-$ measurable ).

Definition 5.12

A pair $(\boldsymbol {\Psi },\tau )$ where $\tau$ is a $\mathbb {P}-a.s.$ positive stopping time and $\boldsymbol {\Psi }$ is a process such that for $\mathbb {P}-a.e.$ $\omega$, $\boldsymbol {\Psi }_{\cdot }(\omega ) \in C\left ([0,T];H\right )$ and $\boldsymbol {\Psi }_{\cdot }(\omega ){\mathbf {1}}_{\cdot \leq \tau (\omega )} \in L^2\left ([0,T];V\right )$ for all $T>0$ with $\boldsymbol {\Psi }_{\cdot }{\mathbf {1}}_{\cdot \leq \tau }$ progressively measurable in V , is said to be an H-valued local strong solution of the Eq. (69) if the identity

$$\displaystyle \begin{aligned} {} \boldsymbol{\Psi}_{t} = \boldsymbol{\Psi}_0 + \int_0^{t\wedge \tau} \mathcal{A}(s,\boldsymbol{\Psi}_s)ds + \int_0^{t \wedge \tau}\mathcal{G} (s,\boldsymbol{\Psi}_s) d\mathcal{W}_s \end{aligned} $$

(84)

holds $\mathbb {P}-a.s.$ in U for all $t \geq 0$.

Definition 5.13

A pair $(\boldsymbol {\Psi },\Theta )$ such that there exists a sequence of stopping times $(\theta _j)$ which are $\mathbb {P}-a.s.$ monotone increasing and convergent to $\Theta$, whereby $(\boldsymbol {\Psi }_{\cdot \wedge \theta _j},\theta _j)$ is a $V-$valued local strong solution of the Eq. (69) for each j, is said to be an $H-$valued maximal strong solution of the Eq. (69) if for any other pair $(\boldsymbol {\Phi },\Gamma )$ with this property then $\Theta \leq \Gamma$ $\mathbb {P}-a.s.$ implies $\Theta = \Gamma$ $\mathbb {P}-a.s.$.

Definition 5.14

An $H-$valued maximal strong solution $(\boldsymbol {\Psi },\Theta )$ of the equation (69) is said to be unique if for any other such solution $(\boldsymbol {\Phi },\Gamma )$, then $\Theta = \Gamma$ $\mathbb {P}-a.s.$ and for all $t \in [0,\Theta )$,

$$\displaystyle \begin{aligned} \mathbb{P}\left(\left\{\omega \in \Omega: \boldsymbol{\Psi}_{t}(\omega) = \boldsymbol{\Phi}_{t}(\omega) \right\} \right) = 1. \end{aligned}$$

We now state the main theorem in this setting.

Theorem 5.15

Suppose that Assumptions 5.6–5.11 are satisfied in this framework. Then for any given $\mathcal {F}_0-$ measurable ), there exists a unique $H-$valued maximal strong solution $(\boldsymbol {\Psi },\Theta )$ of the equation (69). Moreover at $\mathbb {P}-a.e.$ $\omega$ for which $\Theta (\omega )<\infty$, we have that

$$\displaystyle \begin{aligned} {}\sup_{r \in [0,\Theta(\omega))}\lVert \boldsymbol{\Psi}_r(\omega) \rVert _H^2 + \int_0^{\Theta(\omega)}\lVert \boldsymbol{\Psi}_r(\omega) \rVert _V^2dr = \infty.\end{aligned} $$

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Proof

See [32] Theorem 3.15. □

5.4 Abstract Solution Criterion II

We extend the framework of Sect. 5.3, introducing now another Hilbert Space X which is such that . We ask that there is a continuous bilinear form ) such that for $\phi \in U$ and $\psi \in H$,

$$\displaystyle \begin{aligned} {} \langle \phi, \psi \rangle _{X \times H} = \langle \phi, \psi \rangle _{U}. \end{aligned} $$

(86)

Moreover it is now necessary that the system $(a_n)$ from Assumption 5.6 forms an orthogonal basis of U. We state the remaining assumptions now for arbitrary elements $\phi ,\psi \in H$ and $t \in [0,\infty )$, and continue to use the $c,K,\tilde {K}, \kappa$ notation of Assumption Set 1. We now further assume that for any $T>0$, ) and ) are measurable.

Assumption 5.16

$$\displaystyle \begin{aligned} {} \lVert \mathcal{A}(t,\boldsymbol{\phi}) \rVert ^2_X + \sum_{i=1}^\infty \lVert \mathcal{G}_i(t,\boldsymbol{\phi}) \rVert ^2_U &\leq c_tK(\boldsymbol{\phi})\left[1 + \lVert \boldsymbol{\phi} \rVert _H^2\right], \end{aligned} $$

(87)

$$\displaystyle \begin{aligned} {} \lVert \mathcal{A}(t,\boldsymbol{\phi}) - \mathcal{A}(t,\boldsymbol{\psi}) \rVert _X &\leq c_t\left[K(\phi,\psi) + \lVert \phi \rVert _H + \lVert \psi \rVert _H \right]\lVert \phi-\psi \rVert _H \end{aligned} $$

(88)

Assumption 5.17

$$\displaystyle \begin{aligned} & 2\langle \mathcal{A}(t,\boldsymbol{\phi}) - \mathcal{A}(t,\boldsymbol{\psi}), \boldsymbol{\phi} - \boldsymbol{\psi} \rangle _X \\ & \quad + \sum_{i=1}^\infty\lVert \mathcal{G}_i(t,\boldsymbol{\phi}) - \mathcal{G}_i(t,\boldsymbol{\psi}) \rVert _X^2 \leq {} c_{t}\tilde{K}_2(\boldsymbol{\phi},\boldsymbol{\psi}) \lVert \boldsymbol{\phi}-\boldsymbol{\psi} \rVert _X^2, \end{aligned} $$

(89)

$$\displaystyle \begin{aligned} & \sum_{i=1}^\infty \langle \mathcal{G}_i(t,\boldsymbol{\phi}) - \mathcal{G}_i(t,\boldsymbol{\psi}), \boldsymbol{\phi}-\boldsymbol{\psi} \rangle ^2_X \leq {} c_{t} \tilde{K}_2(\boldsymbol{\phi},\boldsymbol{\psi}) \lVert \boldsymbol{\phi}-\boldsymbol{\psi} \rVert _X^4 \end{aligned} $$

(90)

Assumption 5.18

For every $\phi \in V$ , it holds that

$$\displaystyle \begin{aligned} {} 2\langle \mathcal{A}(t,\boldsymbol{\phi}), \boldsymbol{\phi} \rangle _U + \sum_{i=1}^\infty\lVert \mathcal{G}_i(t,\boldsymbol{\phi}) \rVert _U^2 &\leq c_tK(\boldsymbol{\phi}) - \kappa\lVert \boldsymbol{\phi} \rVert _H^2, \end{aligned} $$

(91)

$$\displaystyle \begin{aligned}{} \sum_{i=1}^\infty \langle \mathcal{G}_i(t,\boldsymbol{\phi}), \boldsymbol{\phi} \rangle ^2_U &\leq c_tK(\boldsymbol{\phi}). \end{aligned} $$

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Remark 6

Note that Assumption 5.18 is stronger than Assumption 5.10, as we are bounding the same terms but we are not afforded a control in the H norm of $\phi$ in addition to its U norm. Thus in applying Theorem 5.22 it is sufficient to only demonstrate Assumption 5.18.

Analagously to Susbection 5.3, we state the relevant definitions and the resulting theorem in this context (again proved in [32]). Definition 5.19 is stated for an $\mathcal {F}_0-$ measurable ).

Definition 5.19

A pair $(\boldsymbol {\Psi },\tau )$ where $\tau$ is a $\mathbb {P}-a.s.$ positive stopping time and $\boldsymbol {\Psi }$ is a process such that for $\mathbb {P}-a.e.$ $\omega$, $\boldsymbol {\Psi }_{\cdot }(\omega ) \in C\left ([0,T];U\right )$ and $\boldsymbol {\Psi }_{\cdot }(\omega ){\mathbf {1}}_{\cdot \leq \tau (\omega )} \in L^2\left ([0,T];H\right )$ for all $T>0$ with $\boldsymbol {\Psi }_{\cdot }{\mathbf {1}}_{\cdot \leq \tau }$ progressively measurable in H, is said to be a U-valued local strong solution of the Eq. (69) if the identity

$$\displaystyle \begin{aligned} {} \boldsymbol{\Psi}_{t} = \boldsymbol{\Psi}_0 + \int_0^{t\wedge \tau} \mathcal{A}(s,\boldsymbol{\Psi}_s)ds + \int_0^{t \wedge \tau}\mathcal{G} (s,\boldsymbol{\Psi}_s) d\mathcal{W}_s \end{aligned} $$

(93)

holds $\mathbb {P}-a.s.$ in X for all $t \geq 0$.

Definition 5.20

A pair $(\boldsymbol {\Psi },\Theta )$ such that there exists a sequence of stopping times $(\theta _j)$ which are $\mathbb {P}-a.s.$ monotone increasing and convergent to $\Theta$, whereby $(\boldsymbol {\Psi }_{\cdot \wedge \theta _j},\theta _j)$ is a $U-$valued local strong solution of the Eq. (69) for each j, is said to be an $H-$valued maximal strong solution of the Eq. (69) if for any other pair $(\boldsymbol {\Phi },\Gamma )$ with this property then $\Theta \leq \Gamma$ $\mathbb {P}-a.s.$ implies $\Theta = \Gamma$ $\mathbb {P}-a.s.$.

Definition 5.21

A $U-$valued maximal strong solution $(\boldsymbol {\Psi },\Theta )$ of the Eq. (69) is said to be unique if for any other such solution $(\boldsymbol {\Phi },\Gamma )$, then $\Theta = \Gamma$ $\mathbb {P}-a.s.$ and for all $t \in [0,\Theta )$,

$$\displaystyle \begin{aligned} \mathbb{P}\left(\left\{\omega \in \Omega: \boldsymbol{\Psi}_{t}(\omega) = \boldsymbol{\Phi}_{t}(\omega) \right\} \right) = 1. \end{aligned}$$

Theorem 5.22

Suppose that Assumptions 5.6–5.11 and 5.16–5.18 are satisfied in this framework. Then for any given $\mathcal {F}_0-$ measurable ), there exists a unique $U-$valued maximal strong solution $(\boldsymbol {\Psi },\Theta )$ of the Eq.(69). Moreover at $\mathbb {P}-a.e.$ $\omega$ for which $\Theta (\omega )<\infty$, we have that

$$\displaystyle \begin{aligned} {}\sup_{r \in [0,\Theta(\omega))}\lVert \boldsymbol{\Psi}_r(\omega) \rVert _U^2 + \int_0^{\Theta(\omega)}\lVert \boldsymbol{\Psi}_r(\omega) \rVert _H^2dr = \infty.\end{aligned} $$

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Proof

See [32] Theorem 4.9. □