High Order Smoothness for Stochastic Navier-Stokes Equations with Transport and Stretching Noise on Bounded Domains

In the following $\mathscr{O}\subset\mathbb{R}^{2}$ will be a smooth bounded domain endowed with Euclidean norm and Lebesgue measure $\lambda$. We consider Banach Spaces as measure spaces equipped with their corresponding Borel $\sigma$-algebra. Let $(\mathcal{X},\mu)$ denote a general topological measure space, $(\mathcal{Y},\left\|\cdot\right\|_{\mathcal{Y}})$ and $(\mathcal{Z},\left\|\cdot\right\|_{\mathcal{Z}})$ be separable Banach Spaces, and $(\mathcal{U},\left\langle\cdot,\cdot\right\rangle_{\mathcal{U}})$, $(\mathcal{H},\left\langle\cdot,\cdot\right\rangle_{\mathcal{H}})$ be general separable Hilbert spaces. We introduce the following spaces of functions.

• •

  $L^{p}(\mathcal{X};\mathcal{Y})$ 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

  $$\left\|\phi\right\|_{L^{p}(\mathcal{X};\mathcal{Y})}^{p}:=\int_{\mathcal{X}}\left\|\phi(x)\right\|^{p}_{\mathcal{Y}}\mu(dx).$$

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

  $$\left\langle\phi,\psi\right\rangle_{L^{2}(\mathcal{X};\mathcal{Y})}=\int_{\mathcal{X}}\left\langle\phi(x),\psi(x)\right\rangle_{\mathcal{Y}}\mu(dx).$$

  In the case $\mathcal{X}=\mathscr{O}$ and $\mathcal{Y}=\mathbb{R}^{2}$ note that

  $$\left\|\phi\right\|_{L^{2}(\mathscr{O};\mathbb{R}^{2})}^{2}=\sum_{l=1}^{2}\left\|\phi^{l}\right\|^{2}_{L^{2}(\mathscr{O};\mathbb{R})},\qquad\phi=\left(\phi^{1},\dots,\phi^{2}\right),\quad\phi^{l}:\mathscr{O}\rightarrow\mathbb{R}.$$

  We denote $L^{p}(\mathscr{O};\mathbb{R}^{2})$ by simply $L^{p}$.

• •

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

  $$\left\|\phi\right\|_{L^{\infty}(\mathcal{X};\mathcal{Y})}:=\inf\{C\geq 0:\left\|\phi(x)\right\|_{Y}\leq C\textnormal{ for $\mu$-$a.e.$ }x\in\mathcal{X}\}.$$

  We denote $L^{\infty}(\mathscr{O};\mathbb{R}^{2})$ by simply $L^{\infty}$.

• •

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

• •

  $C^{m}(\mathscr{O};\mathbb{R})$ is the space of $m\in\mathbb{N}$ times continuously differentiable functions from $\mathscr{O}$ to $\mathbb{R}$, that is $\phi\in C^{m}(\mathscr{O};\mathbb{R})$ if and only if for every $2$ dimensional multi index $\alpha=\alpha_{1},\alpha_{2}$ with $\lvert\alpha\rvert\leq m$, $D^{\alpha}\phi\in C(\mathscr{O};\mathbb{R})$ where $D^{\alpha}$ is the corresponding classical derivative operator $\partial_{x_{1}}^{\alpha_{1}}\partial_{x_{2}}^{\alpha_{2}}$.

• •

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

• •

  $C^{m}_{0}(\mathscr{O};\mathbb{R})$ for $m\in\mathbb{N}$ or $m=\infty$ is the subspace of $C^{m}(\mathscr{O};\mathbb{R})$ of functions which have compact support.

• •

  $C^{m}(\mathscr{O};\mathbb{R}^{2}),C^{m}_{0}(\mathscr{O};\mathbb{R}^{2})$ for $m\in\mathbb{N}$ or $m=\infty$ is the space of functions from $\mathscr{O}$ to $\mathbb{R}^{2}$ whose component mappings each belong to $C^{m}(\mathscr{O};\mathbb{R}),C^{m}_{0}(\mathscr{O};\mathbb{R})$. These spaces are simply denoted by $C^{m},C^{m}_{0}$ respectively.

• •

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

  $$\left\|\phi\right\|^{p}_{W^{m,p}(\mathscr{O},\mathbb{R})}:=\sum_{\lvert\alpha\rvert\leq m}\left\|D^{\alpha}\phi\right\|_{L^{p}(\mathscr{O};\mathbb{R})}^{p},$$

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

  $$\left\langle\phi,\psi\right\rangle_{W^{m,2}(\mathscr{O};\mathbb{R})}:=\sum_{\lvert\alpha\rvert\leq m}\left\langle D^{\alpha}\phi,D^{\alpha}\psi\right\rangle_{L^{2}(\mathscr{O};\mathbb{R})}.$$

• •

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

  $$\left\|\phi\right\|_{W^{m,\infty}(\mathscr{O},\mathbb{R})}:=\sup_{\lvert\alpha\rvert\leq m}\left\|D^{\alpha}\phi\right\|_{L^{\infty}(\mathscr{O};\mathbb{R}^{2})}.$$

• •

  $W^{s,p}(\mathscr{O};\mathbb{R})$ for $0<s<1$ and $1\leq p<\infty$ is the sub-class of functions $\phi\in L^{p}(\mathscr{O},\mathbb{R})$ such that

  $$\int_{\mathscr{O}\times\mathscr{O}}\frac{\lvert\phi(x)-\phi(y)\rvert^{p}}{\lvert x-y\rvert^{sp+2}}d\lambda(x,y)<\infty.$$

  This is a Banach space with respect to the norm

  $$\left\|\phi\right\|_{W^{s,p}(\mathscr{O};\mathbb{R})}^{p}=\left\|\phi\right\|_{L^{p}(U;\mathbb{R})}^{p}+\int_{\mathscr{O}\times\mathscr{O}}\frac{\lvert\phi(x)-\phi(y)\rvert^{p}}{\lvert x-y\rvert^{sp+2}}d\lambda(x,y).$$

  For $p=2$ this is a Hilbert space with inner product

  $$\left\langle\phi,\psi\right\rangle_{W^{s,2}(\mathscr{O};\mathbb{R})}=\left\langle\phi,\psi\right\rangle_{L^{2}(\mathscr{O};\mathbb{R})}+\int_{\mathscr{O}\times\mathscr{O}}\frac{\big{(}\phi(x)-\phi(y)\big{)}\big{(}\psi(x)-\psi(y)\big{)}}{\lvert x-y\rvert^{2s+2}}d\lambda(x,y).$$

• •

  $W^{s,p}(\mathscr{O};\mathbb{R})$ for $1\leq s<\infty,$ $s\not\in\mathbb{N}$ and $1\leq p<\infty$ is, using the notation $\left\lfloor{s}\right\rfloor$ to mean the integer part of $s$, the sub-class of $W^{\left\lfloor{s}\right\rfloor,p}(\mathscr{O};\mathbb{R})$ such that the distributional derivatives $D^{\alpha}\phi$ belong to $W^{s-\left\lfloor{s}\right\rfloor,p}(\mathscr{O};\mathbb{R})$ for every multi-index $\alpha$ such that $\lvert\alpha\rvert=\left\lfloor{s}\right\rfloor.$ This is a Banach space with norm

  $$\left\|\phi\right\|_{W^{s,p}(\mathscr{O};\mathbb{R})}^{p}=\left\|\phi\right\|_{W^{\left\lfloor{s}\right\rfloor,p}(\mathscr{O};\mathbb{R})}^{p}+\sum_{\lvert\alpha\rvert=\left\lfloor{s}\right\rfloor}\int_{\mathscr{O}\times\mathscr{O}}\frac{\lvert D^{\alpha}\phi(x)-D^{\alpha}\phi(y)\rvert^{p}}{\lvert x-y\rvert^{(s-\left\lfloor{s}\right\rfloor)p+2}}dxdy$$

  and a Hilbert space in the case $p=2$, with inner product

  $$\left\langle\phi,\psi\right\rangle_{W^{s,2}(\mathscr{O};\mathbb{R})}=\left\langle\phi,\psi\right\rangle_{W^{\left\lfloor{s}\right\rfloor,2}(\mathscr{O};\mathbb{R})}+\int_{\mathscr{O}\times\mathscr{O}}\frac{\big{(}D^{\alpha}\phi(x)-D^{\alpha}\phi(y)\big{)}\big{(}D^{\alpha}\psi(x)-D^{\alpha}\psi(y)\big{)}}{\lvert x-y\rvert^{2(s-\left\lfloor{s}\right\rfloor)+2}}d\lambda(x,y).$$

• •

  $W^{s,p}(\mathscr{O};\mathbb{R}^{2})$ for $s>0$ and $1\leq p<\infty$ is the Banach space of functions $\phi:\mathscr{O}\rightarrow\mathbb{R}^{2}$ whose components $(\phi^{l})$ are each elements of the space $W^{s,p}(\mathscr{O};\mathbb{R}).$ We denote this space by simply $W^{s,p}$. The associated norm is

  $$\left\|\phi\right\|_{W^{s,p}}^{p}=\sum_{l=1}^{2}\left\|\phi^{l}\right\|^{p}_{W^{s,p}(\mathscr{O};\mathbb{R})}$$

  and similarly when $p=2$ this space is Hilbert with inner product

  $$\left\langle\phi,\psi\right\rangle_{W^{s,2}}=\sum_{l=1}^{2}\left\langle\phi^{l},\psi^{l}\right\rangle_{W^{s,2}(\mathscr{O},\mathbb{R})}.$$

• •

  $W^{m,\infty}(\mathscr{O};\mathbb{R}^{2})$ is the sub-class of $L^{\infty}(\mathscr{O},\mathbb{R}^{2})$ which has all weak derivatives up to order $m\in\mathbb{N}$ also of class $L^{\infty}(\mathscr{O},\mathbb{R}^{2})$. This is a Banach space with norm

  $$\left\|\phi\right\|_{W^{m,\infty}}:=\sup_{l\leq N}\left\|\phi^{l}\right\|_{W^{m,\infty}(\mathscr{O};\mathbb{R})}.$$

• •

  $W^{m,p}_{0}(\mathscr{O};\mathbb{R}),W^{m,p}_{0}(\mathscr{O};\mathbb{R}^{2})$ for $m\in\mathbb{N}$ and $1\leq p\leq\infty$ is the closure of $C^{\infty}_{0}(\mathscr{O};\mathbb{R}),C^{\infty}_{0}(\mathscr{O};\mathbb{R}^{2})$ in $W^{m,p}(\mathscr{O};\mathbb{R}),W^{m,p}(\mathscr{O};\mathbb{R}^{2})$.

• •

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

  $$\left\|F\right\|_{\mathscr{L}(\mathcal{Y};\mathcal{Z})}=\sup_{\left\|y\right\|_{\mathcal{Y}}=1}\left\|Fy\right\|_{\mathcal{Z}}$$

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

• •

  $\mathscr{L}^{2}(\mathcal{U};\mathcal{H})$ 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 $\mathcal{U}$,

  $$\sum_{i=1}^{\infty}\left\|Fe_{i}\right\|_{\mathcal{H}}^{2}<\infty.$$

  This is a Hilbert space with inner product

  $$\left\langle F,G\right\rangle_{\mathscr{L}^{2}(\mathcal{U};\mathcal{H})}=\sum_{i=1}^{\infty}\left\langle Fe_{i},Ge_{i}\right\rangle_{\mathcal{H}}$$

  which is independent of the choice of basis.

We next give the probabilistic set up. Let $(\Omega,\mathcal{F},(\mathcal{F}_{t}),\mathbbm{P})$ 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 (e.g. [51], Definition 3.2.36) that $\mathcal{W}$ admits the representation $\mathcal{W}_{t}=\sum_{i=1}^{\infty}e_{i}W^{i}_{t}$ as a limit in $L^{2}(\Omega;\mathfrak{U}^{\prime})$ whereby the $(W^{i})$ are a collection of i.i.d. standard real valued Brownian Motions and $\mathfrak{U}^{\prime}$ is an enlargement of the Hilbert Space $\mathfrak{U}$ such that the embedding $J:\mathfrak{U}\rightarrow\mathfrak{U}^{\prime}$ is Hilbert-Schmidt and $\mathcal{W}$ is a $JJ^{*}-$cylindrical Brownian Motion over $\mathfrak{U}^{\prime}$. Given a process $F:[0,T]\times\Omega\rightarrow\mathscr{L}^{2}(\mathfrak{U};\mathscr{H})$ 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

$$\int_{0}^{t}F_{s}d\mathcal{W}_{s}:=\sum_{i=1}^{\infty}\int_{0}^{t}F_{s}(e_{i})dW^{i}_{s},$$

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 $\mathbbm{P}-a.e.$ $\omega$ via the traditional localisation procedure. In this case the stochastic integral is a local martingale in $\mathscr{H}$. ³³3A complete, direct construction of this integral, a treatment of its properties and the fundamentals of stochastic calculus in infinite dimensions can be found in [58] Section 2.

We now recap the classical functional framework for the study of the deterministic Navier-Stokes Equation. We formally define the operator $\mathcal{L}$ as well as the divergence-free and Navier boundary conditions. The nonlinear operator $\mathcal{L}$ is defined for sufficiently regular functions $f,g:\mathscr{O}\rightarrow\mathbb{R}^{2}$ by $\mathcal{L}_{f}g:=\sum_{j=1}^{2}f^{j}\partial_{j}g.$ 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 $\partial_{j}g$ by its component mappings $(\partial_{l}g)^{l}:=\partial_{j}g^{l}$. For any $m\geq 1$, the mapping $\mathcal{L}:W^{m+1,2}\rightarrow W^{m,2}$ defined by $f\mapsto\mathcal{L}_{f}f$ is continuous (see [33] Lemma 1.2). Some more technical properties of the operator are given at the end of this subsection. For the divergence-free condition we mean a function $f$ such that the property

$$\textnormal{div}f:=\sum_{j=1}^{2}\partial_{j}f^{j}=0$$

holds. We require this property and the boundary condition to hold for our solution $u$ at all times, understood in their traditional weak senses: that is for weak derivatives $\partial_{j}$ so $\sum_{j=1}^{2}\partial_{j}f^{j}=0$ holds as an identity in $L^{2}(\mathscr{O};\mathbb{R})$ whilst the boundary condition is understood in terms of trace. To be precise we first define the restriction mapping on functions $f\in W^{1,2}(\mathscr{O};\mathbb{R})\cap C(\bar{\mathscr{O}};\mathbb{R})$ by the restriction of $f$ to the boundary $\partial\mathscr{O}$, which is then shown to be a bounded operator into $W^{\frac{1}{2},2}(\partial\mathscr{O};\mathbb{R})$ (see Lemma 1.6 and more classical sources of e.g. [21]). By the density of $C(\bar{\mathscr{O}};\mathbb{R})$ then the trace operator is well defined on the whole of $W^{1,2}(\mathscr{O};\mathbb{R})$ as a continuous linear extension of the restriction mapping, and furthermore on $W^{1,2}(\mathscr{O};\mathbb{R}^{2})$ by the trace of the components. The rate of strain tensor $D$ appearing in (3) is a mapping $D:W^{1,2}\rightarrow L^{2}(\mathscr{O};\mathbb{R}^{2\times 2})$ defined by

$$f\mapsto\begin{bmatrix}\partial_{1}f^{1}&\frac{1}{2}\left(\partial_{1}f^{2}+\partial_{2}f^{1}\right)\\ \frac{1}{2}\left(\partial_{2}f^{1}+\partial_{1}f^{2}\right)&\partial_{2}f^{2}\end{bmatrix}$$

or in component form, $(Df)^{k,l}:=\frac{1}{2}\left(\partial_{k}f^{l}+\partial_{l}f^{k}\right)$. Note that if $f\in W^{2,2}$ then $Df\in W^{1,2}(\mathscr{O};\mathbb{R}^{2\times 2})$ so the trace of its components is well defined and henceforth we understand the boundary condition

$$2(Df)\mathbf{n}\cdot\iota+\alpha f\cdot\iota=0$$

on $\partial\mathscr{O}$ to be in this trace sense. The same is true for $f\cdot\mathbf{n}=0$. We look to impose these conditions by incorporating them into the function spaces where our solution takes value, and will always assume that $\alpha\in C^{2}(\partial\mathscr{O};\mathbb{R})$ so as to match the regularity required in [8].

We note that the Poincaré Inequality holds for the component mappings of functions in $\bar{W}^{1,2}_{\sigma}$. The inequality (see e.g. [59] Theorem 1.9) holds for the component mapping $f^{j}$ of $f\in\bar{W}^{1,2}_{\sigma}$ if

$$\int_{\mathscr{O}}f^{j}d\lambda=0,$$

and observe that via an approximation with the set $C^{\infty}_{0,\sigma}$ which is dense in $L^{2}_{\sigma}$, one can conclude that for all $g\in L^{2}_{\sigma}$ and $\psi\in W^{1,2}(\mathscr{O};\mathbb{R})$,

$$\left\langle g,\nabla\psi\right\rangle=0$$

(this is the statement of [59] Lemma 2.11). Moreover by choosing $\phi$ as the function $\phi(x^{1},x^{2}):=x^{j}$, then

$$\int_{\mathscr{O}}f^{j}d\lambda=\left\langle f,\nabla\phi\right\rangle=0$$

so the Poincaré Inequality holds, and as such we equip $\bar{W}^{1,2}_{\sigma}$ with the inner product

$$\left\langle f,g\right\rangle_{1}:=\sum_{j=1}^{2}\left\langle\partial_{j}f,\partial_{j}g\right\rangle$$

which is equivalent to the full $W^{1,2}$ one. We introduce the Leray Projector $\mathcal{P}$ as the orthogonal projection in $L^{2}$ onto $L^{2}_{\sigma}$. It is well known (see e.g. [65] Remark 1.6.) that for any $m\in\mathbb{N}$, $\mathcal{P}$ is continuous as a mapping $\mathcal{P}:W^{m,2}\rightarrow W^{m,2}$. Through $\mathcal{P}$ we define the Stokes Operator $A:W^{2,2}\rightarrow L^{2}_{\sigma}$ by $A:=-\mathcal{P}\Delta$. 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 $m\in\{0\}\cup\mathbb{N}$, $A:W^{m+2,2}\rightarrow W^{m,2}$ is continuous. We also note that $A=A\mathcal{P}$ as the Laplacian leaves the complement space of $L^{2}_{\sigma}$ invariant, see e.g. [34] Subsection 2.2 and Lemma 2.7, and by iterating this property then for $\beta\in\mathbb{N}$ we have that $\mathcal{P}(-\Delta)^{\beta}=A^{\beta}$.

One can see that we are building a framework to parallel that of the classic no-slip case, though a significant difference comes in the presence of a boundary integral for Green’s type identities. What we achieve now is the following, recalling $\kappa\in C^{2}(\partial\mathscr{O};\mathbb{R})$ to be the curvature of $\partial\mathscr{O}$.

Due to this boundary integral we will make great use of Lebesgue spaces and fractional Sobolev spaces defined on the boundary $\partial\mathscr{O}$. The spaces $L^{p}(\partial\mathscr{O};\mathbb{R}^{2})$ can be defined precisely as in Subsection 1.1 for $\partial\mathscr{O}$ equipped with its surface measure, and in fact the same is true for $W^{s,2}(\partial\mathscr{O};\mathbb{R})$ with $0<s<1$ and hence $W^{s,2}(\partial\mathscr{O};\mathbb{R}^{2})$ in the same manner. Indeed this definition is given in [36] pp.20, where it is shown to be equivalent to the often used definition locally as the space $W^{s,2}(\mathbb{R};\mathbb{R})$ via a coordinate transformation and partition of unity. The stability under a change of variables ensures results of Hölder’s Inequality and (one dimensional) Sobolev Embeddings hold on the boundary for these spaces. The relation to the trace of functions in $\mathscr{O}$ is stated now.

We shall make use of this result for the characterisation of the fractional Sobolev spaces as interpolation spaces. In fact the renowned book of Adams [1] defines these spaces in this way (7.57, pp.250), and their equivalence is well understood (see for example [64] pp.83). A proof of this equivalence relies on a norm preserving extension operator for the space as defined by interpolation (which follows from the classical integer valued case, see [1] Theorem 5.24 and [60] chapter 6), the equivalence of the interpolation space on $\mathbb{R}^{2}$ with that defined by Fourier transformations (see [1] 7.63 pp.252), the further equivalence of this space with our definition on $\mathbb{R}^{2}$ ([17] Proposition 4.17), and finally a norm preserving extension for this fractional Sobolev space ([18] Theorem 5.4). Therefore there exists a constant $c$ such that for $f\in W^{1,2}$ and $0<s<1$, we have that

$$\left\|f\right\|_{W^{s,2}}\leq c\left\|f\right\|^{1-s}\left\|f\right\|_{W^{1,2}}^{s}.$$

(7)

In particular for $s=\frac{1}{2}$, if the result of Lemma 1.6 were true in this limiting case (that is, an embedding of $W^{\frac{1}{2},2}(\mathscr{O};\mathbb{R}^{2})$ into $L^{2}(\partial\mathscr{O};\mathbb{R}^{2})$) then combined with (7) we would obtain

$$\left\|f\right\|_{L^{2}(\partial\mathscr{O};\mathbb{R}^{2})}^{2}\leq c\left\|f\right\|\left\|f\right\|_{W^{1,2}}.$$

(8)

This inequality is in fact true and is classical in the study of our problem (see for example [50] pp.130, [44] equation (2.5)) though some additional machinery is required to prove it. In short the result can be achieved by showing that the trace operator is a continuous linear operator from an appropriate Besov space which similarly interpolates between $L^{2}$ and $W^{1,2}$ (see [1] Theorem 7.43 and Remark 7.45 for the trace embedding, and the Besov Spaces subchapter for the interpolation). Moving on we introduce the finite dimensional projections $(\bar{\mathcal{P}}_{n})$, where $\bar{\mathcal{P}}_{n}$ is the orthogonal projection onto $\bar{V}_{n}:=\textnormal{span}\{\bar{a}_{1},\dots,\bar{a}_{n}\}$ in $L^{2}$ (note that $\bar{V}_{n}$ is a Hilbert Space equipped with any $W^{k,2}$). That is, $\bar{\mathcal{P}}_{n}$ is given by

$$\bar{\mathcal{P}}_{n}:f\mapsto\sum_{k=1}^{n}\left\langle f,\bar{a}_{k}\right\rangle\bar{a}_{k}.$$

It will also be of use to us to consider the vorticity in this context, which we do by introducing the operator $\textnormal{curl}:W^{1,2}\rightarrow L^{2}(\mathscr{O};\mathbb{R})$ by

$$\textnormal{curl}:f\rightarrow\partial_{1}f^{2}-\partial_{2}f^{1}.$$

A significant property in the study of vorticity is the following.

The curl is intrinsically related to the Navier boundary conditions through $\kappa$, by the relation proven as Lemma 2.1 in [8] which we state here.

Moreover we can make the condition (4) discussed in the introduction precise.

Returning to the nonlinear term, we shall frequently understand the $L^{2}$ inner product as a duality pairing between $L^{\frac{4}{3}}$ and $L^{4}$ as justified in the following.

This subsection concludes with a symmetry result for the trilinear form defined by the nonlinear term which is classical when zero trace is assumed, but perhaps not in lieu of this assumption.