Artificial boundary conditions for linearized stationary incompressible viscous flow around rotating and translating body

TL;DR

This paper develops and analyzes artificial boundary conditions for linearized and nonlinear stationary incompressible viscous flow around rotating and translating bodies, providing pressure estimates and error bounds for truncated computational domains.

Contribution

It introduces artificial boundary conditions for exterior flow problems and derives error estimates comparing truncated and full domain solutions.

Findings

Pressure pointwise estimates for both linearized and nonlinear flows.

Error bounds for solutions in truncated domains with artificial boundary conditions.

Comparison results between truncated and full domain solutions.

Abstract

We consider the linearized and nonlinear stationary incompressible flow around rotating and translating body in the exterior domain B with Lipschitz boundary. We derive the pointwise estimates for the pressure in both cases. Moreover, we consider the linearized problem in a truncation domain B_R of the exterior domain B under certain artificial boundary conditions on the truncating boundary and then compare this solution with the solution in the exterior domain B to get the truncation error estimate.

Equations

$$\begin{array}[]{crl}-\Delta u(z)+(\tau e_{1}-\varrho e_{1}\times z)\cdot\nabla u(z)+\varrho e_{1}\times u(z)\\ \qquad\qquad\qquad\qquad+\tau(u(z)\cdot\nabla)u(z)+\nabla\pi(z)=F(z)\\ {\rm div}\,u(z)=0\ \text{for}\ z\in\mathbb{R}^{3}\setminus\overline{\mathcal{D}}\end{array}$$

$$\begin{array}[]{crl}-\Delta u(z)+(\tau e_{1}-\varrho e_{1}\times z)\cdot\nabla u(z)+\varrho e_{1}\times u(z)+\nabla\pi(z)=F(z)\\ {\rm div}\,u(z)=0\ \text{for}\ z\in\mathbb{R}^{3}\setminus\overline{\mathcal{D}}\\ \end{array}$$

$$(Lu)(z):=$$

$$(L^{\ast }u)(z):=$$

$$K(z,t):=(4\pi t{)}^{-3/2}{e}^{-|z{|}^2/(4t)}(z\in {\mathbb{R}}^3,t\in (0,\infty )),$$

$$\Lambda (z,t):={\left(K(z,t){\delta }_{jk}+\partial z_j\partial z_k\left({\int }_{{\mathbb{R}}^3}(4\pi |z-y|{)}^{-1}K(y,t)dy\right)\right)}_{1\le j,k\le 3}$$

$$(z\in {\mathbb{R}}^3,t>0),$$

$$\displaystyle\Gamma(x,y,t):=\color[rgb]{0,0,0}\Lambda(x-\tau te_{1}-e^{-t\Omega}y,t)\color[rgb]{0,0,0}\cdot e^{-t\Omega},$$

$$\tilde{\Gamma }(x,y,t):=\Lambda (x+\tau t{e}_1-e^{t\Omega }y,t)\cdot e^{t\Omega }(x,y\in {\mathbb{R}}^3,t>0),$$

$$\mathcal{Z}(x,y):={\int }_0^{\infty }\Gamma (x,y,t)dt,\tilde{\mathcal{Z}}(x,y):={\int }_0^{\infty }\tilde{\Gamma }(x,y,t)dt,$$

$$(x,y\in {\mathbb{R}}^3,x\ne y).$$

$$\mathcal{R}(f)(x):={\int }_{{\mathbb{R}}^3}\mathcal{Z}(x,y)f(y)dy(x\in {\mathbb{R}}^3);$$

$$\displaystyle u|\mbox{$\mathcal{D}$}_{R}\in W^{1,p}(\mbox{$\mathcal{D}$}_{R})^{3},\quad\pi|\mbox{$\mathcal{D}$}_{R}\in L^{p}(\mbox{$\mathcal{D}$}_{R}),\quad u|\partial\mbox{$\mathcal{D}$}\in W^{2-1/p,\,p}(\partial\mbox{$\mathcal{D}$})^{3},$$

$$\displaystyle\hskip 30.00005pt\mbox{div}\,u|\mbox{$\mathcal{D}$}_{R}\in W^{1,p}(\mbox{$\mathcal{D}$}_{R}),\quad L(u)+\nabla\pi|\mbox{$\mathcal{D}$}_{R}\in L^{p}(\mbox{$\mathcal{D}$}_{R})^{3}$$

$$\mathcal{N}(h)(x):={\int }_{{\mathbb{R}}^3}-(4\pi |x-y|{)}^{-1}h(y)dy(x\in {\mathbb{R}}^3).$$

$$\mathcal{S}(h)(x):={\left({\int }_{{\mathbb{R}}^3}(4\pi |x-y{|}^3{)}^{-1}(x-y{)}_j\cdot h(y)dy\right)}_{1\le j\le 3}(x\in {\mathbb{R}}^3).$$

$$\mathcal{P}(h)(x):={\int }_{{\mathbb{R}}^3}(4\pi |x-y{|}^3{)}^{-1}((x-y)\cdot h(y))dy(x\in {\mathbb{R}}^3).$$

$$\mathcal{P}(h)\in W_{\mathrm{loc}}^{1,q}({\mathbb{R}}^3)\cap L^{(1/q-1/3{)}^{-1}}({\mathbb{R}}^3),$$

$${\left({\int }_{{\mathbb{R}}^3}{\left({\int }_{{\mathbb{R}}^3}|x-y{|}^{-2}|h(y)|dy\right)}^{(1/q-1/3{)}^{-1}}dx\right)}^{1/q-1/3}\le C\Vert h{\Vert }_q.$$

$${\int }_{\partial B_R}s(x{)}^{-B}d{o}_x\le C(S,B)\cdot R^{2-min\{1,B\}}\cdot \sigma (R)$$

$$\overline{\mathcal{D}}\cup \mathrm{supp}(\mathrm{div}u)\subset B_{S_1},u|B_S^c\in L^6(B_S^c{)}^3,\nabla u|B_S^c\in L^2(B_S^c{)}^9,$$

$$A+\min \{1,B\}\ge 3,|F(z)|\le \gamma |z{|}^{-A}s(z{)}^{-B}\text{for}z\in B_{S_1}^c.$$

$$|u(y)|\le C(|y|s(y){)}^{-1}{l}_{A,B}(y),$$

$$|\nabla u(y)|\leq C\,(|y|s(y))^{-3/2}\,s(y)^{\max{(0,7/2-A-B)}}\color[rgb]{0,0,0}\,l_{A,B}(y)\color[rgb]{0,0,0}$$

$$\left\{\begin{array}[]{crl}1&\text{if}\quad A+\min\{1,B\}>3\\ \max(1,\text{ln}(y))&\text{if}\quad A+\min\{1,B\}=3.\end{array}\right.$$

$$\displaystyle\int_{\overline{\mbox{$\mathcal{D}$}}^{c}}\bigl{[}\,\nabla u\cdot\nabla\varphi+\bigl{(}\,\tau\,\partial_{1}u-(\varrho\,e_{1}\times z)\cdot\nabla u+(\varrho\,e_{1}\times u)-F\,\bigr{)}\cdot\varphi\,\bigr{]}\,dz$$

$$\displaystyle=0\quad\mbox{for}\;\;\varphi\in C^{\infty}_{0}(\overline{\mbox{$\mathcal{D}$}}^{c})^{3}\;\;\mbox{with}\;\;{\rm div}\,\varphi=0.$$

$$\displaystyle\int_{\overline{\mbox{$\mathcal{D}$}}^{c}}\bigl{[}\,\nabla u\cdot\nabla\varphi+\bigl{(}\,\tau\,\partial_{1}u-(\varrho\,e_{1}\times z)\cdot\nabla u+(\varrho\,e_{1}\times u)-F\,\bigr{)}\cdot\varphi$$

$$\displaystyle-\pi\,{\rm div}\,\varphi\,\bigr{]}\,dz=0\quad\mbox{for}\;\;\varphi\in C^{\infty}_{0}(\overline{\mbox{$\mathcal{D}$}}^{c})^{3}.$$

$$|F(z)|\le \gamma C(S_1,A)|z{|}^{-2}s(z{)}^{-A+2-B}\le \gamma C(S_1,A)|z{|}^{-2}s(z{)}^{-A+2-\min \{1,B\}}$$

$$\le \gamma C(S_1,A)|z{|}^{-2}s(z{)}^{-1}.$$

$${\int }_{B_{S_1}^c}|F(z){|}^{q}dz\le C{\int }_{S_1}^{\infty }{r}^{-2q}{\int }_{\partial B_r}s(z{)}^{-q}d{o}_{z}dr\le C{\int }_{S_1}^{\infty }{r}^{-2q+1}dr<\infty .$$

$$\displaystyle G(z):=F(z)-\bigl{(}\,\tau\,e_{1}-(\varrho\,e_{1}\times z)\,\bigr{)}\cdot\nabla u(z)-\bigl{(}\,\varrho\,e_{1}\times u(z)\,\bigr{)},\;\;z\in\overline{\mbox{$\mathcal{D}$}}^{c},$$

$$|\pi (x)+c_0|\le C|x{|}^{-2}\text{mbox}forx\in B_S^c.$$

$$\varphi |B_{S_1+\frac{1}{4}(S-S_1)}=0,\varphi |B_{S_1+\frac{3}{4}(S-S_1)}^c=1,$$

$$g_l(z):=$$

Full text

1 Univ. Littoral Côte d’Opale, Laboratoire de mathématiques

pures et appliquées Joseph Liouville

e-mail: Paul.Deuring@lmpa.univ-littoral.fr

2 Department of Technical Mathematics, Czech Technical University

3 Institute of Mathematics of the Academy of Sciences of the Czech Republic

e-mail: Stanislav.Kracmar@fs.cvut.cz

4 Institute of Mathematics of the Academy of Sciences of the Czech Republic

e-mail: matus@math.cas.cz

1 Introduction

We consider the systems of equations

[TABLE]

[TABLE]

where \mbox{\mathcal{D}}\subset\mathbb{R}^{3} is open and bounded, with Lipschitz boundary. Problems (1.1) and (1.2) together with some boundary conditions on $\partial\mathcal{D}$ constitute mathematical models (linear and non-linear, respectively) describing stationary flow of a viscous incompressible fluid around a rigid body which moves at a constant velocity and rotates at a constant angular velocity, where we consider that the rotation is parallel to the velocity at the infinity. For details concerning of deriving the model, see [11, 15]. The description and the analysis in the case when the rotation is not parallel to the velocity at infinity can be find in the following works, see [13, 17].

The aim of this paper is two folds:

First, we would like to derive the pointwise estimates for the pressure in the linear and also in the non-linear cases in order to complete the pointwise estimates for the velocity and its gradient from [8, 9] by the pointwise estimates of the pressure in order to get complete decay information of all parts $u,\,\pi$ of solutions to systems (1.1), (1.2). Let us mention that the decay of pressure was also investigated in the work of Galdi, Kyed [16] and in case of pure rotation see [12].

Second, to solve the linear system (1.2) in a truncation $\mathcal{D}_{R}:=B_{R}\backslash\overline{\mathcal{D}}$ of the exterior domain $\mathbb{R}^{3}\setminus\overline{\mathcal{D}}$ under certain artificial boundary conditions on the truncating boundary $\partial B_{R}$, and then compare this solution with the solution of (1.2) in the exterior domain, i.e. to get some sort of error estimates of the method of an artificial boundary condition. For this aim we use pointwise estimates of the velocity and of the pressure.

Mathematical analysis of the problem of the Navier-Stokes equations with artificial boundary condition was performed by many authors but without considering the rotation of body, see e.g. [1, 2, 4, 5]. The article can be seen as a first result in the case of motion of viscous fluids around rotating and translating body with artificial boundary condition.

The paper is organized as follows: In the rest of this section we introduce notation and give some auxiliary results. The next section 2 deals with pointwise estimates of the pressure of the linear system (1.2). In Section 3 we consider the linear system (1.2) with artificial boundary conditions. The error estimate of the velocity is derived comparing to the solution to the system given in the exterior domain. First let us introduce notation:

Definitions and notation related to the rotational system

Define $s(y):=1+|y|-y_{1}$ for $y\in\mathbb{R}^{3}$,

$\mathcal{D}_{R}:=B_{R}\setminus\overline{\mathcal{D}}$,

$B_{R}^{c}:=\mathbb{R}^{3}\setminus\overline{B_{R}}$,

where $B_{R}:=\{x\in\mathbb{R}^{3};|x|<R\},$ for $R>0$ such that $B_{R}\supset\overline{\mathcal{D}.}$

So, $\mathcal{D}_{R}$ is the truncation of the exterior domain $\overline{\mathcal{D}}^{c}:=\mathbb{R}^{3}\setminus\overline{\mathcal{D}}\,$ by the ball $B_{R}$. The boundary $\partial\mathcal{D}_{R}$ consists of parts $\partial\mathcal{D}$ and $\partial B_{R}$, the later we call the truncating boundary.

Fix $\tau\in(0,\infty)$, $\rho\in\mathbb{R}\setminus\{0\}$, and put $e_{1}:=(1,0,0),\;\Omega:=\rho\begin{pmatrix}0&0&0\\ 0&0&-1\\ 0&1&0\end{pmatrix}$,

so that \color[rgb]{0,0,0}\Omega\cdot z=\rho e_{1}\times z\color[rgb]{0,0,0}\ for $z\in\mathbb{R}^{3}$.

For $U\subset\mathbb{R}^{3}$ open, $u\in W^{2,1}_{{\rm loc}}(U)^{3}$, $z\in U$, put

[TABLE]

Put

[TABLE]

For $q\in(1,2)$, $f\in L^{q}(\mathbb{R}^{3})^{3}$, put

[TABLE]

see [7, Lemma 3.1]. We will use the space

$D^{1,2}_{0}(\overline{\mathcal{D}}^{c})^{3}:=\{v\in L^{6}(\overline{\mathcal{D}}^{c})^{3}\cap H^{1}_{loc}(\overline{\mathcal{D}}^{c})^{3}\,:\,\nabla v\in L^{2}(\overline{\mathcal{D}}^{c})^{9},\;v|\partial\mathcal{D}=0\}$ equipped with the norm $\|\nabla u\|_{2}$, where $v|\partial\mathcal{D}$ means the trace of $v$ on $\partial\mathcal{D}$.

For $p\in(1,\infty),$ define $M_{p}$ as the space of all pairs of functions $(u,\pi)$ such that u\in W^{2,p}_{loc}(\overline{\mbox{\mathcal{D}}}^{c})^{3},\;\pi\in W^{1,p}_{loc}(\overline{\mbox{\mathcal{D}}}^{c}),

[TABLE]

for some $R\in(0,\infty)$ with \overline{\mbox{\mathcal{D}}}\subset B_{R}.

We write $C$ for generic constants. It should be clear from context which are the parameters these constants depend on. In order to lift possible ambiguities, we sometimes use the notation $C(\gamma_{1},\,...,\,\gamma_{n})$ in order to indicate that the constant in question depends in particular on $\gamma_{1},\,...,\,\gamma_{n}\in(0,\infty)$, for some $n\in\mathbb{N}$. But the relevant constant may depend on other parameters as well.

Auxiliary results to asymptotic behavior of the pressure

Lemma 1.1

(Weyl’s lemma).* Let $n\in\mathbb{N}$, $U\subset\mathbb{R}^{n}$ open, $u\in L^{1}_{{\rm loc}}(U)$ with $\int_{U}u\cdot\Delta l\,dx=0$ for $l\in C^{\infty}_{0}(U)$. Then $u\in C^{\infty}(U)$ and $\Delta u=0$.*

Proof: An elementary proof is given in [19, Appendix] $\Box$

For $q\in(1,3/2)$, $h\in L^{q}(\mathbb{R}^{3})$, put

[TABLE]

For $q\in(1,3)$, $h\in L^{q}(\mathbb{R}^{3})$, put

[TABLE]

For $q\in(1,3)$, $h\in L^{q}(\mathbb{R}^{3})^{3}$, put

[TABLE]

Note that $S(h)$ is a vector-valued function with $h$ being scalar, whereas $P(h)$ is a scalar function with $h$ being vector-valued.

Lemma 1.2

Let $q\in(1,3/2)$, $h\in L^{q}(\mathbb{R}^{3})$. Then $\mathcal{N}(h)\in W^{2,q}_{{\rm loc}}(\mathbb{R}^{3})\cap L^{(1/q-2/3)^{-1}}(\mathbb{R}^{3})$, $\Delta\mathcal{N}(h)=h$. If $h\in W^{1,q}(\mathbb{R}^{3})$, then $\partial_{l}\mathcal{N}(h)=\mathcal{N}(\partial_{l}h)$ $(1\leq l\leq 3)$.

Let $q\in(1,3)$, $h\in L^{q}(\mathbb{R}^{3})$. Then $\mathcal{S}(h)\in W^{1,q}_{{\rm loc}}(\mathbb{R}^{3})^{3},\ {\rm div}\,\mathcal{S}(h)=h.$ If $q\in(1,3/2)$, then $\nabla\mathcal{N}(h)=\mathcal{S}(h)$. If $h\in W^{1,q}(\mathbb{R}^{3})$, then $\mathcal{S}(h)\in W^{2,q}_{{\rm loc}}(\mathbb{R}^{3})^{3}$.

Let $q\in(1,3)$, $h\in L^{q}(\mathbb{R}^{3})^{3}$. Then

[TABLE]

Proof: The assertion of the Lemma 1.2 follows from well known Hardy-Littlewood-Sobolev inequality, Calderon-Zygmund inequality, and density arguments. $\Box$

Lemma 1.3

[18, Lemma 2.2]** Let $B\in\mathbb{R},\,S\in(0,\infty).$ Then

[TABLE]

for $R\in[S,\infty),$ with $\sigma(R):=1$ if $B\neq 1,$ and $\sigma(R)={\rm ln(1+R)}$ if $B=1$.

2 Decay estimates

In first part of this section we recall some known results from [7] and [9] about the decay of the velocity part of the solution of the system (1.2), and in order to get the full decay characterization of the solution we derive the decay of the pressure part of solution of (1.2). In the second part of this section we extend the result for the pressure to the non-linear case of (1.1).

Decay estimates in the linear case

Our starting point is a decay result from [9] for the velocity part $u$ of a solution to (1.2).

Theorem 2.1

([9, Theorem 3.12])* Suppose that $\mathcal{D}$ is $C^{2}$-bounded. Let $p\in(1,\infty)$, $(u,\pi)\in M_{p}$. Put $F=L(u)+\nabla\pi$. Suppose there are numbers $S_{1},S,\gamma\in(0,\infty)$, $A\in[2,\infty)$, $B\in\mathbb{R}$ such that $S_{1}<S$,*

[TABLE]

[TABLE]

Then

[TABLE]

[TABLE]

for $y\in B^{c}_{S}$, where function $l_{A,B}$ is given by

[TABLE]

The requirements $u|{B_{S}^{c}}\in L^{6}(B^{c}_{S})^{3},\;\nabla u|{B^{c}_{S}}\in L^{2}(B_{S}^{c})^{9}$ should be interpreted as decay conditions on $u$.

It may be deduced from Theorem 2.1 that inequalities (2.1) and (2.2) hold under assumptions weaker than those stated in that theorem. We specify this more general situation in the ensuing corollary, which in addition indicates some properties of $F$ that will be useful in the following.

Corollary 2.2

Let $p\in(1,\infty),\;\gamma,\,S_{1},\,S\in(0,\infty)$ with \overline{\mbox{\mathcal{D}}}\subset B_{S_{1}},\;S_{1}<S,\;A\in[2,\infty),\;B\in\mathbb{R} with $A+\min\{1,B\}\geq 3$. Let F:\overline{\mbox{\mathcal{D}}}^{c}\mapsto\mathbb{R}^{3} be measurable with F|\mbox{\mathcal{D}}_{S_{1}}\in L^{p}(\mbox{\mathcal{D}}_{S_{1}})^{3} and $|F(z)|\leq\gamma|z|^{-A}s(z)^{-B}\ \text{for}\ z\in B^{c}_{S_{1}}.$

Let u\in W^{1,p}_{loc}(\overline{\mbox{\mathcal{D}}}^{c})^{3} with $u|{B_{S}^{c}}\in L^{6}(B^{c}_{S})^{3}$, $\nabla u|{B^{c}_{S}}\in L^{2}(B_{S}^{c})^{9}$, ${\rm supp}({\rm div}\,u)\subset B_{S_{1}}$,

[TABLE]

Then inequalities (2.1) and (2.2) hold for $y\in B_{S}^{c}$.

Moreover F\in L^{q}(\overline{\mbox{\mathcal{D}}}^{c})^{3} for $q\in(1,p]$. If $p\geq 6/5,$ the function $F$ may be considered as a bounded linear functional on \mbox{\mathcal{D}}^{1,2}_{0}(\overline{\mbox{\mathcal{D}}}^{c})^{3}, in the usual sense.

Let \pi\in L^{p}_{loc}(\overline{\mbox{\mathcal{D}}}^{c}) with

[TABLE]

Fix some number $S_{0}\in(0,S_{1})$ with \overline{\mbox{\mathcal{D}}}\cup{\rm supp}({\rm div}\,u)\subset B_{S_{0}}. Then the relations $u|\overline{B_{S_{0}}}^{c}\in W^{2,p}_{loc}(\overline{B_{S_{0}}}^{c})^{3},\;\pi\in W^{1,p}_{loc}(\overline{B_{S_{0}}}^{c})$ and $L(u|\overline{B_{S_{0}}}^{c})+\nabla\pi=F|\overline{B_{S_{0}}}^{c}$ hold.

Proof: For $z\in B_{S_{1}}^{c}$, we have

[TABLE]

Thus for $q\in(1,\infty),$ with Lemma 1.3,

[TABLE]

It follows that F\in L^{q}(\overline{\mbox{\mathcal{D}}}^{c})^{3} for $q\in(1,p]$. According to [14, Theorem II.6.1], the inequality $\|v\|_{6}\leq C\,\|\nabla v\|_{2}$ holds for v\in\color[rgb]{0,0,0}D^{1,2}_{0}(\overline{\mbox{\mathcal{D}}}^{c})^{3}\color[rgb]{0,0,0}. Thus, if $p\geq 6/5$, hence F\in L^{6/5}(\overline{\mbox{\mathcal{D}}}^{c})^{3}, this function $F$ may be considered as a linear bounded functional on \mbox{\mathcal{D}}^{1,2}_{0}(\overline{\mbox{\mathcal{D}}}^{c})^{3}. The $L^{p}$-integrability of $F$ and the assumptions on $u$ imply that the function

[TABLE]

belongs to L^{p}_{loc}(\overline{\mbox{\mathcal{D}}}^{c})^{3}.

The choice of $S_{0}$ (see at the end of Corollary 2.2) means in particular that ${\rm div}\,(u|\overline{B_{S_{0}}}^{c})=0.$ This equation, (2.4), the relation G\in L^{p}_{loc}(\overline{\mbox{\mathcal{D}}}^{c})^{3} and interior regularity of solutions to the Stokes system (see [14, Theorem IV.4.1] for example) imply the claims in the last sentence of Corollary 2.2.

Put $S_{0}^{\prime}:=(S_{0}+S_{1})/2,\;A_{S_{0}^{\prime},R}:=B_{R}\backslash B_{S_{0}^{\prime}}$ for $R\in(S_{0}^{\prime},\infty).$ Then $u|A_{S_{0}^{\prime},R}\in W^{2,p}(A_{S_{0}^{\prime},R})^{3}$ and $\pi|A_{S_{0}^{\prime},R}\in W^{1,p}(A_{S_{0}^{\prime},R})^{3}$ for $R\in(S_{0}^{\prime},\infty),$ so $(u|A_{S_{0}^{\prime},R},\,\pi|A_{S_{0}^{\prime},R})\in M_{p}$, with $B_{S_{0}^{\prime}}$ in the role of $\mathcal{D}$. Note that $S_{0}<S_{o}^{\prime}<S_{1}<S.$ Thus the assumptions of Theorem 2.1 are satisfied with $\mathcal{D}$ replaced by $B_{S_{0}^{\prime}}$. As a consequence inequalities (2.1) and (2.2) hold. $\Box$

Remark 2.3

*Solutions as considered in Corollary 2.2 exist if, for example, Dirichlet boundary conditions are prescribed on $\partial\mathcal{D}$. In fact, as stated in [14, Theorem VIII.1.2], if $F$ is a bounded linear functional on the space $D^{1,2}_{0}(\overline{\mathcal{D}}^{c})^{3}$, and if $b\in H^{1/2}(\partial\overline{\mathcal{D}}^{c})^{3},$ then there is a function $u\in L^{6}(\overline{\mathcal{D}}^{c})^{3}\cap W^{1,1}_{loc}(\overline{\mathcal{D}}^{c})^{3}$ such that $\nabla u\in L^{2}(\overline{\mathcal{D}}^{c})^{9}$ and $u$ satisfies the equations (2.3) and ${\rm div}\,u=0$ (weak form of (1.2)), as well as the boundary conditions u|\partial\mbox{\mathcal{D}}=b.

Existence of a pressure $\pi\in L^{p}_{loc}(\overline{\mathcal{D}}^{c})$ with (2.4) holds according to [14, Lemma VIII.1.1].*

The main result of this section, dealing with the asymptotics of the pressure, is stated in

Theorem 2.4

*Let $p,\,\gamma,\,S_{1},\,S,\,A,\,B,\,F,\,u$ be given as in Corollary 2.2, but with the stronger assumptions $A=5/2,\;B\in(1/2,\,\infty)$ on $A$ and $B$.

Let \pi\in L^{p}_{loc}(\overline{\mbox{\mathcal{D}}}^{c}) such that (2.4) holds

Then there is $c_{0}\in\mathbb{R}$ such that*

[TABLE]

Proof:

By Corollary 2.2 we have $F\in L^{q}(\overline{\mathcal{D}}^{c})^{3}$ for $q\in(1,p]$. Fix some number $S_{0}\in(0,S_{1})$ with \overline{\mbox{\mathcal{D}}}\cup{\rm supp}({\rm div}\,u)\subset B_{S_{0}}. Then again by Corollary 2.2, the relations $u|\overline{B_{S_{0}}}^{c}\in W^{2,p}_{loc}(\overline{B_{S_{0}}}^{c})^{3},\;\pi|\overline{B_{S_{0}}}^{c}\in W^{1,p}_{loc}(\overline{B_{S_{0}}}^{c})$ and $L(u|\overline{B_{S_{0}}}^{c})+\nabla(\pi|\overline{B_{S_{0}}}^{c})=F|\overline{B_{S_{0}}}^{c}$ hold.

Note that $S_{0}<S_{1}<S$. Take $\phi\in C^{\infty}(\mathbb{R}^{3})$ with

[TABLE]

and put $\widetilde{u}:=\phi\cdot u$, $\widetilde{\pi}:=\phi\cdot\pi$, with $\widetilde{u},\widetilde{\pi}$ to be considered as functions in $\mathbb{R}^{3}$. By the choice of $\phi$ and the properties of $u$ and $\pi$, we get $\widetilde{u}\in W^{2,q}_{{\rm loc}}(\mathbb{R}^{3})^{3}$, $\widetilde{\pi}\in W^{1,q}_{{\rm loc}}(\mathbb{R}^{3})$ for $q\in[1,p]$, $\widetilde{u}|B_{S}^{c}=u|B_{S}^{c}\in L^{6}(\mathbb{R}^{3})^{3}$, $\nabla\widetilde{u}|B_{S}^{c}=\nabla u|B_{S}^{c}\in L^{2}(\mathbb{R}^{3})^{9}$. Put

[TABLE]

for $z\in\mathbb{R}^{3}$, $1\leq l\leq 3$, and set $\gamma:={\rm div}\,\widetilde{u}$. Then

[TABLE]

in particular $\mbox{ supp}(\gamma)\subset\overline{B_{S_{1}+3(S-S_{1})/4}}\setminus B_{S_{1}+(S-S_{1})/4}$, \color[rgb]{0,0,0}\gamma\in W^{2,q}(\mathbb{R}^{3}),\color[rgb]{0,0,0} $g+\phi\cdot F\in L^{q}(\mathbb{R}^{3})^{3}$ for $q\in(1,p]$. Let $x\in\mathbb{R}^{3}$, $\varepsilon>0$ with $\overline{B_{\varepsilon}(x)}\subset B_{S_{1}}$, where $B_{\varepsilon}(x)=\{y\in\mathbb{R}^{3};|y-x|<\varepsilon\}$. Since $\widetilde{u}|B_{S_{1}}=0$, $\widetilde{\pi}|B_{S_{1}}=0$, it follows from [9, Theorem 3.11] with $\mathcal{D}$ replaced by $B_{\varepsilon}(x)$ that

[TABLE]

Since this is true for any $x\in\mathbb{R}^{3}$, $\varepsilon>0$ with $\overline{B_{\varepsilon}(x)}\subset B_{S_{1}}$, it follows that

[TABLE]

But $\mathcal{S}(\gamma)\in W^{2,q}_{{\rm loc}}(\mathbb{R}^{3})^{3}$ for $q\in[1,\min\{3,p\})$ by Lemma 1.2, so from (2.8)

[TABLE]

This relation and [7, (3.11) and the

inequality

following (3.15)] imply

[TABLE]

Let $\psi\in C^{\infty}_{0}(\mathbb{R}^{3})^{3}$. Due to (2.9), we may apply Fubini’s theorem, to obtain

[TABLE]

But for $a,b,x,y\in\mathbb{R}^{3}$ with $x\not=y$,

[TABLE]

hence with [7, Lemma 2.10],

[TABLE]

Therefore from (2.10)

[TABLE]

Since $\psi\in C^{\infty}_{0}(\mathbb{R}^{3})^{3}$, we may choose $x_{0}\in\mathbb{R}^{3}$, $\varepsilon>0$ such that

[TABLE]

Thus we get from [6, Theorem 4.3]

with $\mathcal{D},U,\omega,u,L$ replaced by $B_{\varepsilon}(x_{0})$, $\tau e_{1}$, $-\rho e_{1},\,\psi,\,L^{*}$, respectively,

and with $\pi=0$, that

[TABLE]

for $y\in\mathbb{R}^{3}\setminus\overline{B_{\varepsilon}(x_{0})}$. Since this is true for any $x_{0}\in\mathbb{R}^{3}$, $\varepsilon>0$ with $\overline{B_{\varepsilon}(x_{0})}\subset\mathbb{R}^{3}\setminus{\rm supp}(\psi)$, the preceding equation holds for any $y\in\mathbb{R}^{3}$. It follows from (2.11)

[TABLE]

Again recalling that $g+\phi F\in L^{q}(\mathbb{R}^{3})^{3}$ for $q\in(1,p]$, we get with Lemma 1.2 that

[TABLE]

Put $q_{0}:=\min\{6/5,p\}$, and note that $q_{0}\in(1,3/2)$, $q_{0}\leq p$.

Thus, by Hölder’s inequality and Lemma 1.2,

[TABLE]

As a consequence, we may apply Fubini’s theorem to deduce from (2.13) that

[TABLE]

From (2.12) and (2.14),

[TABLE]

Since this is true for any $\psi\in C^{\infty}_{0}(\mathbb{R}^{3})^{3}$, we have found that

[TABLE]

On the other hand, by (2.8) and (2.7)

[TABLE]

By subtracting this equation from (2.15), we get

[TABLE]

Next we consider the term {\rm div}\bigl{(}\,L\mathcal{S}(\gamma)\,\bigr{)}. Recall that

$q_{0}<3/2,\;q_{0}\leq p$ and $\gamma\in W^{2,q}(\mathbb{R}^{3})$ for $q\in[1,p]$ (see (2.7)),

so by Lemma 1.2

[TABLE]

Since $e_{1}\times\mathcal{S}(\gamma)=(0,\,-\mathcal{S}_{3}(\gamma),\,\mathcal{S}_{2}(\gamma))$ and because of the equation $\nabla\mathcal{N}(\gamma)=\mathcal{S}(\gamma)$ in (2.17), we may conclude that

[TABLE]

Moreover, for $z\in\mathbb{R}^{3}$, $1\leq j\leq 3$,

[TABLE]

Put $\varphi(z):=-z_{3}\partial_{2}\gamma(z)+z_{2}\partial_{3}\gamma(z)$ for $z\in\mathbb{R}^{3}.$ Then with (2.19), the equation ${\rm div}\mathcal{S}(\gamma)=\gamma$ in (2.17), and the second and third equation in (2.18),

[TABLE]

Let $\psi\in C^{\infty}_{0}(\mathbb{R}^{3})$. Then it follows that

[TABLE]

Obviously, again with (2.17),

[TABLE]

and similarly,

[TABLE]

Combining these equations, we get

[TABLE]

Now from (2.16)

[TABLE]

Since $\gamma\in W^{2,q}(\mathbb{R}^{3})$ for $q\in[1,p]$ and ${\rm supp}(\gamma)\subset B_{S}\backslash B_{S_{1}}$

due to (2.7)),

it follows that $\varphi+\Delta\gamma-\tau\partial_{1}\gamma\in L^{q}(\mathbb{R}^{3})$ for $q\in[1,p]$q, so we may consider $\mathcal{N}(\varphi+\Delta\gamma-\tau\partial_{1}\gamma)$. Lemma 1.2 yields

[TABLE]

Therefore from (2.20)

[TABLE]

Lemma 1.1 now yields

[TABLE]

Now we again apply Lemma 1.2. Since $g+\phi\cdot F\in L^{q_{0}}(\mathbb{R}^{3})^{3}$, we have

[TABLE]

Moreover $\varphi+\Delta\gamma-\tau\partial_{1}\gamma\in L^{q_{0}}(\mathbb{R}^{3})$, so

[TABLE]

Since $q_{0}\leq p,$ and in view of our remarks at the beginning of this proof we know that $u|\overline{B_{S_{0}}}^{c}\in W^{2,q_{0}}_{loc}(\overline{B_{S_{0}}}^{c}),\;\pi|\overline{B_{S_{0}}}^{c}\in W^{1,q_{0}}_{loc}(\overline{B_{S_{0}}}^{c}),\;L(u|\overline{B_{S_{0}}}^{c})+\nabla(\pi|\overline{B_{S_{0}}}^{c})=F|\overline{B_{S_{0}}}^{c},\;\mbox{div}(u|\overline{B_{S_{0}}}^{c})=0$ and $F\in L^{q_{0}}(\mathbb{R}^{3})^{3}$. By the choice of $u$ in Corollary 2.2, we have $u|B_{S}^{c}\in L^{6}(B_{S}^{c})^{3}$.

Therefore

[9, Theorem 2.1] yields there is $c_{0}\in\mathbb{R}$ such that

[TABLE]

But by (2.21),

[TABLE]

where ${\rm supp}(\phi-1)\subset B_{S}$ and ${\rm supp}(\phi)\subset B_{S_{1}}^{c}$. We may conclude that

[TABLE]

Let $\varepsilon\in(0,\infty)$, and let $(Q-c_{0})_{\varepsilon}$ be the usual Friedrich’s mollifier of $Q-c_{0}$ associated with $\varepsilon$.

Due to (2.21), (2.22) and by standard properties Friedrich’s mollifier, the function $(Q-c_{0})_{\varepsilon}$ is bounded and $\Delta(Q-c_{0})_{\varepsilon}=0$. Now Liouville’s theorem yields $(Q-c_{0})_{\varepsilon}=0$. Since this is true for any $\varepsilon>0$

and because $Q\in C^{\infty}(\mathbb{R}^{3}),$

we may conclude that $Q-c_{0}=0$, that is,

[TABLE]

hence

[TABLE]

where we used that ${\rm supp}(\phi-1)\subset B_{S}$ and $\phi|B_{S}^{c}=1$. Since ${\rm supp}(g)\subset\overline{B_{S_{1}+3(S-S_{1})/4}}$, we have

[TABLE]

Due to the assumptions $A=5/2,\;B\in(1/2,\,\infty)$ and because $\phi F|B_{S_{1}+(S-S_{1})/4}=0$ and $S_{1}<S_{1}+(S-S_{1})/4<S$, we get by [10, Theorem 3.2] or [18, Theorem 3.4] that

[TABLE]

Note that according to [18, Theorem 3.4 (iii)], a logarithmic factor should be added on the right-hand side of (2.25) in the case $A=5/2,\;B=1$. But this factor is superfluous. In fact, if the relation $|F(z)|\leq\gamma|z|^{-A}s(z)^{-B}\;(z\in B_{S_{1}}^{c})$ is valid with $A=5/2,\;B=1,$ it holds in the case $A=5/2,\;B=3/4,$ too. But then [18, Theorem 3.4 (i), (iii)] yields that (2.25) holds as it is, without additional factor.

Define $\zeta(x)=-x_{3}\gamma(x)$, $\tilde{\zeta}(x):=x_{2}\gamma(x)$ for $x\in\mathbb{R}^{3}$. Then ${\rm supp}(\zeta)\cup\,{\rm supp}(\tilde{\zeta})\subset B_{S}\setminus\overline{B_{S_{1}}}$,

[TABLE]

It follows with Lemma 1.2 that

[TABLE]

Similarly, since ${\rm supp}(\gamma)\subset B_{S}\setminus\overline{B_{S_{1}}}$, $\gamma\in W^{2,q}(\mathbb{R}^{3})$ for $q\in[1,p]$,

[TABLE]

Together

[TABLE]

Since supp$(\zeta)\cup{\rm supp}(\tilde{\zeta})\cup{\rm supp}(\gamma)\subset B_{S_{1}+3(S-S_{1})/4}$, we may conclude that

[TABLE]

Inequality (2.6) follows from (2.23)–(2.26). $\Box$

We remark that Theorem 2.4 remains valid if the assumptions on $A$ and $B$ are replaced by the conditions $A\geq 5/2,\;A+\min\{1,B\}>1$, which are weaker than those in Corollary 2.2. This observation is made precise by the ensuing corollary. Its proof is obvious, but this modified version of Theorem 2.4 still is interesting because its requirements on $A$ and $B$ are closer to the ones in Corollary 2.2 than those stated in Theorem 2.4.

Corollary 2.5

Let $p,\,\gamma,\,S_{1},\,S,\,A,\,B,\,F,\,u$ be given as in Corollary 2.2, but with the stronger assumptions $A\geq 5/2,\;A+\min\{1,B\}>3$ on $A$ and $B$. Let \pi\in L^{p}_{loc}(\overline{\mbox{\mathcal{D}}}^{c}) such that (2.4) holds. Then there is $c_{0}\in\mathbb{R}$ such that inequality (2.6) is valid.

Proof: Put $B^{\prime}:=A-5/2+\min\{1,B\}$. Since $A+\min\{1,B\}>3$, we have $B^{\prime}\in(1/2,\,\infty).$ Moreover, since $A\geq 5/2$, we find for

$z\in B_{S_{1}}^{c}$

that

[TABLE]

Thus the assumptions of Theorem 2.4 are satisfied with $B$ replaced by $B^{\prime}$ and with a modified parameter $\gamma$. This implies the conclusion of Theorem 2.4. $\Box$

Decay estimates in the non-linear case

Let us assume now the non-linear case, i.e. the system (1.1). First, recall the result about the decay properties of the velocity in this non-linear case:

Theorem 2.6

[8, Theorem 1.1]** Let $\gamma,\,S_{1}\in(0,\infty),\;p_{0}\in(1,\infty),\;A\in(2,\infty),\;B\in[0,\,3/2]$ with $\overline{\mathcal{D}}\subset B_{S_{1}},\,$ $A+\min\{B,1\}>3,\;A+B\geq 7/2$. Take $F:{\mathbb{R}}^{3}\mapsto\mathbb{R}^{3}$ measurable with $F|{B_{S_{1}}}\in L^{p_{0}}(B_{S_{1}})^{3}$,

[TABLE]

Let $u\in L^{6}(\overline{\mathcal{D}}^{c})^{3}\cap W^{1,1}_{loc}(\overline{\mathcal{D}}^{c})^{3},\pi\in L^{2}_{loc}(\overline{\mathcal{D}}^{c})$ with $\nabla u\in L^{2}(\overline{\mathcal{D}}^{c})^{9},\hbox{\rm div}\,u=0$ and

[TABLE]

for $\varphi\in C_{0}^{\infty}(\overline{\mathcal{D}}^{c})^{3}.$ Let $S\in(S_{1},\infty).$ Then

[TABLE]

Now, using Theorems 2.4 and 2.6, we are in the position to prove the result on the decay of the pressure in the non-linear case:

Theorem 2.7

Consider the situation in Theorem 2.6. Suppose in addition that $A\geq 5/2$. Then there is $c_{0}\in\mathbb{R}$ such that inequality (2.6) holds.

Proof: Observe that (u\cdot\nabla)u\in L^{3/2}(\overline{\mbox{\mathcal{D}}}^{c})^{3}. Thus, putting $p:=\min\{3/2,\,p_{0}\},\;\widetilde{F}:=F-\tau\,(u\cdot\nabla)u,$ we get $\widetilde{F}|\mathcal{D}_{S_{1}}\in L^{p}(\mathcal{D}_{S_{1}})^{3}$. Put $B^{\prime}:=\min\{5/2,\,A+B-5/2\}$. Since $A\geq 5/2$, we have

[TABLE]

On the other hand, by Theorem 2.6 with $(S_{1}+S)/2$ in the place of $S$,

[TABLE]

for $z\in B_{(S_{1}+S)/2}^{c}.$ In this way we get $|\widetilde{F}|\leq C\,|z|^{-5/2}\,s(z)^{-B^{\prime}}$ for $z\in B_{(S_{1}+S)/2}^{c}$.

We further note that $B^{\prime}\in(1/2,\,\infty)$. This is obvious in the case $B^{\prime}=5/2$. If $B^{\prime}<5/2$, we have $B^{\prime}=A+B-5/2$. Due to the assumption $A+\min\{1,B\}>3$ in Theorem 2.6, we thus get $B^{\prime}\in(1/2,\,\infty)$. (The requirement $A+B\geq 7/2$ in Theorem 2.6 even yields $B^{\prime}\geq 1$, but if this requirement is weakened in a suitable way, pointwise decay of $u$ and $\nabla u$ could still be proved. However, this point is not elaborated in [8], and therefore is not reflected in Theorem 2.6. But we still take account of it here by avoiding to use the assumption $A+B\geq 7/2$.)

We further have u\in W^{1,p}_{loc}(\overline{\mbox{\mathcal{D}}}^{c})^{3},\;\pi\in\color[rgb]{0,0,0}L^{p}_{loc}(\overline{\mbox{\mathcal{D}}}^{c})\color[rgb]{0,0,0}, and equation (2.4) holds with $F$ replaced by $\widetilde{F}$. Since in addition $u|B_{S}^{c}\in L^{6}(B_{S}^{c})^{3},\;\nabla u|B_{S}^{c}\in L^{2}(B_{S}^{c})^{9}$ and ${\rm div}\,u=0,$ we see that the assumptions of Theorem 2.4 are satisfied with $p$ as defined above and with $(S_{1}+S)/2,\;B^{\prime},\,\widetilde{F}$ in the role of $S_{1},\,B$ and $F$, respectively. Thus Theorem 2.4 implies the conclusion of Theorem 2.7. $\Box$

3 Formulation of the problem with artificial boundary conditions

Recall that we defined $\mathcal{D}_{R}=B_{R}\setminus\overline{\mathcal{D}}$. We introduce the subspace $W_{R}$ of $H^{1}(\mathcal{D}_{R})$ denoting

[TABLE]

where $v|\partial\mathcal{D}$ means the trace of $v$ on $\partial\mathcal{D}.$

Lemma 3.1

([3, Lemma 4.1])

The estimate*

[TABLE]

holds for $R\in(0,\infty)$ with $\overline{\mathcal{D}}\subset B_{R}$ and for $u\in W_{R}$.

We introduce an inner product $(\cdot,\,\cdot)^{(R)}$ in $W_{R}$ by defining

[TABLE]

The space $W_{R}$ equipped with this inner product is a Hilbert space. The norm generated by this scalar product $(\cdot,\cdot)^{(R)}$ is denoted by $|\cdot|^{(R)}$, that is

[TABLE]

We define the bilinear forms

[TABLE]

Lemma 3.2

Let $R\in(0,\infty)$ with $\overline{\mathcal{D}}\subset B_{R}$. Then

[TABLE]

for $u,w\in H^{1}(\mathcal{D}_{R})^{3}$.

Proof: The proof of Lemma 3.2 is based on use of Lemma 3.1.

The key observation in this section is stated in the following lemma, which is the basis of the theory presented in this section.

Lemma 3.3

Let $R\in(0,\infty)$ with $\overline{\mathcal{D}}\subset B_{R}$, and let $w\in W_{R}$. Then the equation $(|w|^{(R)})^{2}=a_{R}(w,w)+\delta_{R}(w,w)$ holds.

Proof: Using the definitions $a_{R}(\cdot,\cdot),\,\delta_{R}(\cdot,\cdot)$, we get

[TABLE]

We applied that

[TABLE]

As in [4], we obtain that the bilinear form $\beta_{R}$ is stable:

Theorem 3.4

([4, Corollary 4.3])* Let $R>0$ with $\overline{\mathcal{D}}\subset B_{R}$. Then*

[TABLE]

We note that functions from $W^{1,1}_{loc}(\overline{\mathcal{D}}^{c})$ with $L^{2}$-integrable gradient are $L^{2}$-integrable on truncated exterior domains:

Lemma 3.5** ([14, Lemma II.6.1])**

Let $w\in W^{1,1}_{loc}(\overline{\mathcal{D}}^{c})$ with $\nabla w\in L^{2}(\overline{\mathcal{D}}^{c})^{3}$, and let $R\in(0,\infty)$ with $\overline{\mathcal{D}}\subset B_{R}.$ Then $w{|\mathcal{D}_{R}}\in L^{2}(\mathcal{D}_{R}).$ In particular the trace of $w$ on $\partial\mathcal{D}$ is well defined.

The preceding lemma is implicitly used in the ensuing theorem, where we introduce an extension operator \mbox{\mathfrak{E}}:H^{1/2}(\partial\mathcal{D})^{3}\mapsto W^{1,1}_{loc}(\overline{\mathcal{D}}^{c})^{3} such that \mbox{div}\,\mbox{\mathfrak{E}}(b)=0.

Theorem 3.6** ([14, Exercise III.3.8])**

There is an operator $\mathfrak{E}$ from $H^{1/2}(\partial\mathcal{D})^{3}$ into $W^{1,1}_{loc}(\overline{\mathcal{D}}^{c})^{3}$ satisfying the relations \nabla\mbox{\mathfrak{E}}(b)\in L^{2}(\overline{\mathcal{D}}^{c})^{9},\;\mbox{\mathfrak{E}}(b)|{\partial\mathcal{D}}=b and {\rm div}\,\mbox{\mathfrak{E}}(b)=0 for $b\in H^{1/2}(\partial\mathcal{D})^{3}.$

In view of Lemma 3.2 and 3.3 and Theorem 3.6 and 3.4, the theory of mixed variational problems yields

Theorem 3.7

Let $S>0$ with $\overline{\mathcal{D}}\subset B_{S},\;R\in[2S,\infty),\;F\in L^{6/5}(\mathcal{D}_{R})^{3},\;b\in H^{1/2}(\partial\mathcal{D})^{3}.$ Then there is a uniquely determined pair of functions (\widetilde{V},P)=\bigl{(}\,\widetilde{V}(R,F,b),\,P(R,F,b)\,\bigr{)}\in W_{R}\times L^{2}(\mathcal{D}_{R}) such that

[TABLE]

where the operator $\mathfrak{E}$ was introduced in Theorem 3.6.

Let us interpret variational problem (3.1), (3.2) as a boundary value problem. Define the expression used in the boundary condition on the artificial boundary $\partial B_{R}:$

[TABLE]

for $x\in\partial B_{R},\;R\in(0,\infty)$ with \overline{\mbox{\mathcal{D}}}\subset B_{R},\;\color[rgb]{0,0,0}u\in W^{2,\,6/5}(\mathcal{D}_{R})^{3},\color[rgb]{0,0,0}\ \pi\in W^{1,\,6/5}(\mathcal{D}_{R}).

Lemma 3.8

Assume that $\mathcal{D}$ is $\mathcal{C}^{2}$-bounded. Let \color[rgb]{0,0,0}S\in(0,\infty)\color[rgb]{0,0,0} with \overline{\mathcal{D}}\subset B_{S},\;\color[rgb]{0,0,0}R\in[2S,\infty),\color[rgb]{0,0,0}\;F\in L^{6/5}(\mathcal{D}_{R})^{3} and $b\in W^{7/6,\,6/5}(\partial\mathcal{D})^{3}.$ Put $V:=\widetilde{V}(R,F,b)+\mathfrak{E}(b)|\mathcal{D}_{R}$, with $V(R,F,b)$ from Theorem 3.7 and \mbox{\mathfrak{E}}(b) from Theorem 3.6. Suppose that $V\in W^{2,6/5}(\mathcal{D}_{R})^{3}$ and $P=P(R,F,b)\in W^{1,\,6/5}(\mathcal{D}_{R})$, with $P(R,F,b)$ also introduced in Theorem 3.7. Then

[TABLE]

for $z\in\mathcal{D}_{R},$ and $V|\partial\mathcal{D}=b,\ \ \mathcal{L}_{R}(V,P)=0.$

The proof of Lemma 3.8 is obvious. This lemma means that a solution of variational problem (3.1), (3.2) may be considered as a weak solution of the modified Oseen system with rotation in $\mathcal{D}_{R}$, under the Dirichlet boundary condition on $\partial\mathcal{D}$ and under the artificial boundary condition $\mathcal{L}_{R}(V,P)=0$ on $\partial B_{R}$. The solution of (3.1), (3.2) will be now compared to the exterior modified Oseen flow introduced in Corollary 2.2:

Theorem 3.9

Suppose that $\mathcal{D}$ is $C^{2}$-bounded. Let $\gamma,\,S_{1}\in(0,\infty)$ with $\overline{\mathcal{D}}\subset B_{S_{1}},\;A\in[5/2,\,\infty),\;B\in\mathbb{R}$ with $\ A+\min\{1,B\}>3$. Let $F:\overline{\mathcal{D}}^{c}\mapsto\mathbb{R}^{3}$ be measurable with $F|\mathcal{D}_{S_{1}}\in L^{6/5}(\mathcal{D}_{S_{1}})^{3}$ and $|F(z)|\leq\gamma\,|z|^{-A}s(z)^{-B}\ \text{for}\ z\in B^{c}_{S_{1}}$.

Let b\in W^{7/6,\,6/5}(\partial\mbox{\mathcal{D}})^{3},\;u\in W^{1,1}_{loc}(\overline{\mathcal{D}}^{c})^{3}\cap L^{6}(\overline{\mathcal{D}}^{c})^{3} such that \nabla u\in L^{2}(\overline{\mathcal{D}}^{c})^{9},\;{\rm div}\,u=0,\;u|\partial\mbox{\mathcal{D}}=b and equation (2.3) is satisfied.

For $R\in[2S_{1},\,\infty),$ put V_{R}:=\widetilde{V}(R,F,b)+\mbox{\mathfrak{E}}(b), with \mbox{\mathfrak{E}}(b) from Theorem 3.6, and $\widetilde{V}(R,F,b)$ from Theorem 3.7. Then

[TABLE]

We note that since W^{2,\,6/5}(\mbox{\mathcal{D}})\subset H^{1}(\mbox{\mathcal{D}}) by a Sobolev inequality, we have W^{7/6,\,6/5}(\partial\mbox{\mathcal{D}})\subset H^{1/2}(\partial\mbox{\mathcal{D}}), as follows with the usual

lifting and trace properties.

As a consequence, b\in H^{1/2}(\partial\mbox{\mathcal{D}})^{3}, so the term \mbox{\mathfrak{E}}(b) is well defined. We further remark that by Corollary 2.2 with $p=6/5$, the function $F$ may be considered as a bounded linear functional on \mbox{\mathcal{D}}^{1,2}_{0}(\overline{\mbox{\mathcal{D}}}^{c})^{3}. Therefore, as explained in Remark 2.3, a function $u$ with properties as stated in Theorem 3.9 does in fact exist.

Proof of Theorem 3.9: All conditions in Corollary 2.2 are verified if $\gamma,\,S_{1},\,A,\,B,\,F,\,u$ are given as in Theorem 3.9, and if $p=6/5$ and $S=2\,S_{1}.$ Note in this respect that the conditions on $u$ in Theorem 3.9 obviously imply u\in W^{1,\,6/5}_{loc}(\overline{\mbox{\mathcal{D}}}^{c})^{3}. Corollary 2.2 now yields that F\in L^{6/5}(\overline{\mbox{\mathcal{D}}}^{c})^{3} and that the function $u$ satisfies inequalities (2.1) and (2.2) with $S=2\,S_{1}$.

On the other hand, since u\in W^{1,\,6/5}_{loc}(\overline{\mbox{\mathcal{D}}}^{c})^{3}, the function $G$ already considered in the proof of Corollary 2.2 (see (2.5)) belongs to L^{6/5}_{loc}(\overline{\mbox{\mathcal{D}}}^{c})^{3}. Therefore, by interior regularity of solutions to the Stokes system (see [14, Theorem IV.4.1]), we may deduce from the equations (2.3) and ${\rm div}\,u=0$ that u\in W^{2,\,6/5}_{loc}(\overline{\mbox{\mathcal{D}}}^{c})^{3} and that there is \pi\in W^{1,\,6/5}_{loc}(\overline{\mbox{\mathcal{D}}}^{c})^{3} with $L(u)+\nabla\pi=F$. In particular the pair $(u,\pi)$ verifies (2.4). In view of our assumptions on $A$ and $B$, we thus see that the requirements in Corollary 2.5 are fulfilled for $\gamma,\,S_{1},\,A,\,B,\,F,\,u$ as in Theorem 3.9 and for $p=6/5$ and $S=2\,S_{1}$. As a consequence, Corollary 2.5 yields that there is $c_{0}\in\mathbb{R}$ such that (2.6) holds with $S=2\,S_{1}$.

Take $R\in[2\,S_{1},\,\infty).$ Since u\in W^{2,6/5}_{loc}(\overline{\mbox{\mathcal{D}}}^{c})^{3}, we have \color[rgb]{0,0,0}u|\partial B_{R}\in W^{7/5,\,6/5}(\partial B_{R})^{3}\color[rgb]{0,0,0}. Combining this relation with the assumption \color[rgb]{0,0,0}b\in W^{7/5,\,6/5}(\partial\mbox{\mathcal{D}})^{3}\color[rgb]{0,0,0} and the boundary condition u|\partial\mbox{\mathcal{D}}=b, we get u|\partial\mbox{\mathcal{D}}_{R}\in W^{7/5,\,6/5}(\partial\mbox{\mathcal{D}}_{R})^{3}. Moreover our requirements on $u$ yield that u|\mbox{\mathcal{D}}_{R}\in W^{1,\,6/5}(\mbox{\mathcal{D}}_{R})^{3}. Since F\in L^{6/5}(\overline{\mbox{\mathcal{D}}}^{c})^{3}, as already mentioned, we get G|\mbox{\mathcal{D}}_{R}\in L^{6/5}(\mbox{\mathcal{D}}_{R})^{3}, with $G$ from (2.5). Recalling that $\mathcal{D}$ is supposed to be $C^{2}$-bounded, we may now apply the result in [14, Lemma IV.6.1] on boundary regularity of solutions to the Stokes system. This reference yields that u|\mbox{\mathcal{D}}_{R}\in W^{2,\,6/5}(\mbox{\mathcal{D}}_{R})^{3},\;\pi|\mbox{\mathcal{D}}_{R}\in W^{1,\,6/5}(\mbox{\mathcal{D}}_{R}) and that the pair $(u,\pi)$ solves (1.2).

Let $P_{R}:=P(R,F,b)$ be given as in Theorem 3.7, and put $w:=u-V_{R},\;\kappa:=\pi-P_{R}\,$, and let $\,g\in W_{R}$. Note that by Theorem 3.7, we have

[TABLE]

Thus

[TABLE]

[TABLE]

Since the pair $(u,\pi)$ solves (1.2), we now get

[TABLE]

Let $c\in\mathbb{R}$ be an arbitrary constant. For $g:=w$ we get with Lemma 3.3 that

[TABLE]

because by the assumptions on $u$ and Theorem 3.6 and 3.7,

[TABLE]

where $n$ denotes the outward unit normal to $\mathcal{D}.$

Let $c_{0}$ be the constant introduced above as part of estimate (2.6). Because

[TABLE]

we get from (3.5)

[TABLE]

The last step is estimation: $\|\mathcal{L}_{R}(u,\pi+c_{0})\|_{2}\leq C\,\cdot R^{-1}$.

We start by observing that

[TABLE]

As explained above, inequalities (2.1), (2.2) and (2.6) are valid with $S=2\,S_{1}$. According to (2.1) and (2.6), we have $|u(x)|\leq C\,(|x|s(x))^{-1},$ and $|\pi(x)+c_{0}|\leq C\,|x|^{-2}$ for $x\in B_{2\cdot S_{1}}^{c}$. Inequality (2.2) yields |\nabla u(x)|\leq C\,|x|^{\color[rgb]{0,0,0}-3/2\color[rgb]{0,0,0}}s(x)^{-B^{\prime}} for $x$ as before, with $B^{\prime}:=3/2-\max\{0,\,7/2-A-B\}$. If $B\geq 1$, we recall that $A\geq 5/2$, getting $A+B\geq 7/2,$ hence $B^{\prime}=3/2$. On the other hand, if $B<1$, then \color[rgb]{0,0,0}\min\color[rgb]{0,0,0}\{1,B\}=B, so that the assumption A+\color[rgb]{0,0,0}\min\color[rgb]{0,0,0}\{1,B\}>3 becomes $A+B>3$, hence $B^{\prime}>1$. Thus we get in any case that $B^{\prime}>1>1/2$. In view of these observations, and with Lemma 1.3, we obtain

[TABLE]

This completes the proof of Theorem 3.9. $\Box$

Acknowledgements: * The works of S.K. and Š. N. were supported by Grant No. 16-03230S of GAČR in the framework of RVO 67985840, S.K. is supported by RVO 12000. Final version was supported by Grant No. 19-04243S of GAČR.*