Partial boundary regularity for the Navier–Stokes equations in irregular domains

We write $f\lesssim g$ for two non-negative quantities $f$ and $g$ if there is a $c>0$ such that $f\leq\,cg$. Here $c$ is a generic constant which does not depend on the crucial quantities and can change from line to line. If necessary we specify particular dependencies. We write $f\approx g$ if $f\lesssim g$ and $g\lesssim f$. We do not distinguish in the notation for the function spaces between scalar- and vector-valued functions. However, vector-valued functions will usually be denoted in bold case.

Let $\mathcal{O}\subset\mathbb{R}^{m}$, $m\geq 1$, be open. Function spaces of continuous or $\alpha$-Hölder continuous functions, $\alpha\in(0,1)$, are denoted by $C(\overline{\mathcal{O}})$ or $C^{0,\alpha}(\overline{\mathcal{O}})$ respectively. Similarly, we write $C^{1}(\overline{\mathcal{O}})$ and $C^{1,\alpha}(\overline{\mathcal{O}})$. We denote as usual by $L^{p}(\mathcal{O})$ and $W^{k,p}(\mathcal{O})$ for $p\in[1,\infty]$ and $k\in\mathbb{N}$ Lebesgue and Sobolev spaces over $\mathcal{O}$. For a bounded domain $\mathcal{O}$ the space $L^{p}_{\perp}(\mathcal{O})$ denotes the subspace of $L^{p}(\mathcal{O})$ of functions with zero mean, that is $(f)_{\mathcal{O}}:=\mathop{\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\!\int}_{\mathcal{O}}f\,\mathrm{d}x:=\mathcal{L}^{m}(\mathcal{O})^{-1}\int_{\mathcal{O}}f\,\mathrm{d}x=0$. We denote by $W^{k,p}_{0}(\mathcal{O})$ the closure of the smooth and compactly supported functions in $W^{k,p}(\mathcal{O})$. This coincides with the functions vanishing $\mathcal{H}^{m-1}$ -a.e. on $\partial\mathcal{O}$ provided $\partial\mathcal{O}$ is sufficiently regular. We also denote by $W^{-k,p}(\mathcal{O})$ the dual of $W^{k,p}_{0}(\mathcal{O})$. Finally, we consider the subspace $W^{1,p}_{0,\operatorname{div}}(\mathcal{O})$ of divergence-free vector fields which is defined accordingly. We will use the shorthand notations $L^{p}_{x}$ and $W^{k,p}_{x}$ in the case of $3$-dimensional domains and $L^{p}_{y}$ and $W^{k,p}_{y}$ for $2$-dimensional sets.

For a separable Banach space $(X,\|\cdot\|_{X})$ we denote by $L^{p}(0,T;X)$ the set of (Bochner-) measurable functions $u:(0,T)\rightarrow X$ such that the mapping $t\mapsto\|u(t)\|_{X}\in L^{p}(0,T)$. The set $C([0,T];X)$ denotes the space of functions $u:[0,T]\rightarrow X$ which are continuous with respect to the norm topology on $(X,\|\cdot\|_{X})$. The space $W^{1,p}(0,T;X)$ consists of those functions from $L^{p}(0,T;X)$ for which the distributional time derivative belongs to $L^{p}(0,T;X)$ as well. We use the shorthand $L^{p}_{t}X$ for $L^{p}(0,T;X)$. For instance, we write $L^{p}_{t}W^{1,p}_{x}$ for $L^{p}(0,T;W^{1,p}(\mathcal{O}))$. Similarly, $W^{k,p}_{t}X$ stands for $W^{k,p}(0,T;X)$.

The space $C^{\alpha,\beta}([0,T]\times\overline{\mathcal{O}})$ with $\alpha,\beta\in(0,1]$ denotes the set of functions being $\alpha$-Hölder continuous in $t\in[0,T]$ and $\beta$-Hölder continuous in $x\in\overline{\mathcal{O}}$.

For $p\in[1,\infty)$ the fractional Sobolev space (Sobolev-Slobodeckij space) with differentiability $s>0$ with $s\notin\mathbb{N}$ will be denoted by $W^{s,p}(\mathcal{O})$. For $s>0$ we write $s=\lfloor s\rfloor+\{s\}$ with $\lfloor s\rfloor\in\mathbb{N}_{0}$ and $\{s\}\in(0,1)$. We denote by $W^{s,p}_{0}(\mathcal{O})$ the closure of the smooth and compactly supported functions in $W^{1,p}(\mathcal{O})$. For $s>\frac{1}{p}$ this coincides with the functions vanishing $\mathcal{H}^{m-1}$ -a.e. on $\partial\mathcal{O}$ provided $\partial\mathcal{O}$ is regular enough. We also denote by $W^{-s,p}(\mathcal{O})$ for $s>0$ the dual of $W^{s,p}_{0}(\mathcal{O})$. Similar to the case of unbroken differentiabilities above we use the shorthand notations $W^{s,p}_{x}$ and $W^{s,p}_{y}$.. We will denote by $B^{s}_{p,q}(\mathbb{R}^{m})$ the standard Besov spaces on $\mathbb{R}^{m}$ with differentiability $s>0$, integrability $p\in[1,\infty]$ and fine index $q\in[1,\infty]$. They can be defined (for instance) via Littlewood-Paley decomposition leading to the norm $\|\cdot\|_{B^{s}_{p,q}(\mathbb{R}^{m})}$. We refer to [21] and [33, 34] for an extensive picture. The Besov spaces $B^{s}_{p,q}(\mathcal{O})$ for a bounded domain $\mathcal{O}\subset\mathbb{R}^{m}$ are defined as the restriction of functions from $B^{s}_{p,q}(\mathbb{R}^{m})$, that is

If $s\notin\mathbb{N}$ and $p\in(1,\infty)$ we have $B^{s}_{p,p}(\mathcal{O})=W^{s,p}(\mathcal{O})$.

In accordance with [19, Chapter 14] the Sobolev multiplier norm is given by

where $p\in[1,\infty]$ and $s\geq 1$. The space $\mathcal{M}^{s,p}(\mathcal{O})$ of Sobolev multipliers is defined as those objects for which the $\mathcal{M}^{s,p}(\mathcal{O})$-norm is finite. For $\delta>0$ we denote by $\mathcal{M}^{s,p}(\mathcal{O})(\delta)$ the subset of functions from $\mathcal{M}^{s,p}(\mathcal{O})$ with $\mathcal{M}^{s,p}(\mathcal{O})$-norm not exceeding $\delta$. By mathematical induction with respect to $s$ one can prove for Lipschitz-continuous functions $\varphi$ that membership to $\mathcal{M}^{s,p}(\mathcal{O})$ in the sense of (5) implies that

The quantity (6) also serves as customary definition of the Sobolev multiplier norm in the literature but (5) is more suitable for our purposes.

Let us finally collect some some useful properties of Sobolev multipliers. By [19, Corollary 14.6.2] we have

provided that one of the following conditions holds:

• •

  $p(s-1)<m$ and $\phi\in B^{s}_{\varrho,p}(\mathbb{R}^{m})$ with $\varrho\in\big{[}\frac{pm}{p(s-1)-1},\infty\big{]}$;

• •

  $p(s-1)=m$ and $\phi\in B^{s}_{\varrho,p}(\mathbb{R}^{m})$ with $\varrho\in(p,\infty]$.

Note that the hidden constant in (7) depends on the $B^{s}_{\varrho,p}(\mathbb{R}^{m})$-norm of $\phi$. By [19, Corollary 4.3.8] it holds

for $p(s-1)>m$. Finally, we note the following rule about the composition with Sobolev multipliers which is a consequence of [19, Lemma 9.4.1]. For open sets $\mathcal{O}_{1},\mathcal{O}_{2}\subset\mathbb{R}^{m}$, $u\in W^{s,p}(\mathcal{O}_{2})$ and a Lipschitz continuous function ${\boldsymbol{\phi}}:\mathcal{O}_{1}\rightarrow\mathcal{O}_{2}$ with ${\boldsymbol{\phi}}\in\mathcal{M}^{s,p}(\mathcal{O}_{1})$ and Lipschitz continuous inverse ${\boldsymbol{\phi}}^{-1}:\mathcal{O}_{2}\rightarrow\mathcal{O}_{1}$ we have

with constant depending on ${\boldsymbol{\phi}}$.

In this section we present the necessary framework to parametrise the boundary of the underlying domain $\Omega\subset\mathbb{R}^{3}$ by local maps of a certain regularity. This yields, in particular, a rigorous definition of a $\mathcal{M}^{s,p}$-boundary. We follow the presentation from [2].

We assume that $\partial\Omega$ can be covered by a finite number of open sets $\mathcal{U}^{1},\dots,\mathcal{U}^{\ell}$ for some $\ell\in\mathbb{N}$, such that the following holds. For each $j\in\{1,\dots,\ell\}$ there is a reference point $y^{j}\in\mathbb{R}^{3}$ and a local coordinate system $\{e^{j}_{1},e^{j}_{2},e_{3}^{j}\}$ (which we assume to be orthonormal and set $\mathcal{Q}_{j}=(e_{1}^{j}|e_{2}^{j}|e_{3}^{j})\in\mathbb{R}^{3\times 3}$), a function $\varphi_{j}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ and $r_{j}>0$ with the following properties:

1. (A1)

   There is $h_{j}>0$ such that

   $$\mathcal{U}^{j}=\{x=\mathcal{Q}_{j}z+y^{j}\in\mathbb{R}^{3}:\,z=(z^{\prime},z_{3})\in\mathbb{R}^{3},\,|z^{\prime}|<r_{j},\,|z_{3}-\varphi_{j}(z^{\prime})|<h_{j}\}.$$

2. (A2)

   For $x\in\mathcal{U}^{j}$ we have with $z=\mathcal{Q}_{j}^{t}(x-y^{j})$

   • •

     $x\in\partial\Omega$ if and only if $z_{3}=\varphi_{j}(z^{\prime})$;

   • •

     $x\in\Omega$ if and only if $0<z_{3}-\varphi_{j}(z^{\prime})<h_{j}$;

   • •

     $x\notin\Omega$ if and only if $0>z_{3}-\varphi_{j}(z^{\prime})>-h_{j}$.

3. (A3)

   We have that

   $$\partial\Omega\subset\bigcup_{j=1}^{\ell}\mathcal{U}^{j}.$$

In other words, for any $x_{0}\in\partial\Omega$ there is a neighborhood $U$ of $x_{0}$ and a function $\phi:\mathbb{R}^{2}\rightarrow\mathbb{R}$ such that after translation and rotation¹¹1By translation via $y_{j}$ and rotation via $\mathcal{Q}_{j}$ we can assume that $x_{0}=0$ and that the outer normal at $x_{0}$ is pointing in the negative $x_{3}$-direction.

The regularity of $\partial\Omega$ will be described by means of local coordinates as just described.

Clearly, we can define similarly a $\mathcal{M}^{s,p}(\delta)$-boundary for some $\delta>0$ by requiring that $\varphi_{1},\dots,\varphi_{\ell}\in\mathcal{M}^{s,p}(\mathbb{R}^{2})(\delta)$. Analogous definitions apply for various other function spaces such as $B^{s}_{\varrho,p}$ for $s>0$ and $\varrho,p\in[1,\infty]$ or $C^{1,\alpha}$ for $\alpha\in(0,1]$. Of particular importance for us is also a Lipschitz boundary, where $\varphi_{1},\dots,\varphi_{\ell}\in W^{1,\infty}(\mathbb{R}^{2})$. We say that the Lipschitz constant of $\partial\Omega$, denoted by $\mathrm{Lip}(\partial\Omega)$, is (smaller or) equal to some number $L>0$ provided the Lipschitz constants of $\varphi_{1},\dots,\varphi_{\ell}$ are not exceeding $L$.

In order to describe the behaviour of functions defined in $\Omega$ close to the boundary we need to extend the functions $\varphi_{1},\dots,\varphi_{\ell}$ from (A1)–(A3) to the half space $\mathbb{H}:={\{{\xi=(\xi^{\prime},\xi_{3})\,:\,\xi_{3}>0}\}}$. Hence we are confronted with the task of extending a function $\phi\,:\,\mathbb{R}^{2}\to\mathbb{R}$ to a mapping $\Phi\,:\,\mathbb{H}\to\mathbb{R}^{3}$ that maps the 0-neighborhood in $\mathbb{H}$ to the $x_{0}$-neighborhood in $\Omega$. The mapping $(\xi^{\prime},0)\mapsto(\xi^{\prime},\phi(\xi^{\prime}))$ locally maps the boundary of $\mathbb{H}$ to the one of $\partial\Omega$. We extend this mapping using the extension operator of Maz’ya and Shaposhnikova [19, Section 9.4.3]. Let $\zeta\in C^{\infty}_{c}(B_{1}(0^{\prime}))$ with $\zeta\geq 0$ and $\int_{\mathbb{R}^{2}}\zeta(x^{\prime})\,\mathrm{d}x^{\prime}=1$. Let $\zeta_{t}(x^{\prime}):=t^{-2}\zeta(x^{\prime}/t)$ denote the induced family of mollifiers. We define the extension operator

where $\phi:\mathbb{R}^{2}\to\mathbb{R}$ is a Lipschitz function with Lipschitz constant $L$. Then the estimate

follows from [19, Theorem 8.7.2]. Moreover, [19, Theorem 8.7.1] yields

It is shown in [19, Lemma 9.4.5] that (for sufficiently large $N$, i.e., $N\geq c(\zeta)L+1$) the mapping

is for every $z^{\prime}\in\mathbb{R}^{2}$ one to one and the inverse is Lipschitz with gradient bounded by $(N-L)^{-1}$. Now, we define the mapping ${\boldsymbol{\Phi}}\,:\,\mathbb{H}\to\mathbb{R}^{3}$ as a rescaled version of the latter one by setting

Thus, ${\boldsymbol{\Phi}}$ is one-to-one (for sufficiently large $N=N(L)$) and we can define its inverse ${\boldsymbol{\Psi}}:={\boldsymbol{\Phi}}^{-1}$. The Jacobi matrix of the mapping ${\boldsymbol{\Phi}}$ satisfies

Since ${\lvert{\partial_{\xi_{3}}\mathcal{T}\phi}\rvert}\leq L$, we have

using that $N$ is large compared to $L$. Finally, we note the implication

which holds, for instance, if ${\boldsymbol{\Phi}}$ is Lipschitz continuous, cf. [19, Lemma 9.4.2].

We start with a definition of boundary suitable weak solutions adapting the notation from [26]. These solutions satisfy a local form of the energy inequality in the neighborhood of boundary points. For that purpose we fix two numbers $r_{\ast}\in(1,2)$ and $s_{\ast}\in(1,\frac{3}{2})$ such that $\frac{1}{r_{\ast}}+\frac{3}{2s_{\ast}}\geq 2$. The choice comes from the fact that the convective term $(\nabla{\bf u}){\bf u}$ of a weak solution to (1) belongs to $L^{r_{\ast}}_{t}L^{s_{\ast}}_{x}$. Later on we will choose $r_{\ast}=5/3$ and, accordingly, $s_{\ast}=15/14$.

The next theorem shows that, under suitable assumptions on $\partial\Omega$, there is a solution to (1) which is a boundary suitable weak solution around every boundary point.

Denoting by $\mathcal{H}^{s}_{\mathrm{para}}$ the $s$-dimensional parabolic Hausdorff measure (and using its definition based on covering with parabolic cubes) it is standard to deduce the following result concerning the size of the singular set from Theorem 2.6.

Our result in Theorem 2.7 is in terms of the size of the singular set weaker than the result in [26] for more regular domains. It is shown there that the dimension of the singular set is one rather than $5/3$. We comment on this gap in more detail in Section 6.1. It is unlcear if this is an intrinsic feature of irregular domains or a drawback of our method.