Newtonian and Single Layer Potentials for the Stokes System with L^∞ Coefficients and the Exterior Dirichlet Problem

Abstract

A mixed variational formulation of some problems in L ²-based Sobolev spaces is used to define the Newtonian and layer potentials for the Stokes system with L ^∞ coefficients on Lipschitz domains in ${\mathbb R}^3$. Then the solution of the exterior Dirichlet problem for the Stokes system with L ^∞ coefficients is presented in terms of these potentials and the inverse of the corresponding single layer operator.

Dedicated to Professor H. Begehr on the occasion of his 80th birthday

Appendix: Mixed Variational Formulations and Their Well-Posedness Property

Here we make a brief review of well-posedness results due to Babus̆ka [6] and Brezzi [10] for mixed variational formulations related to bounded bilinear forms in reflexive Banach spaces. We follow [20, Section 2.4], [11], and [25, §4].

Let X and ${\mathcal M}$ be reflexive Banach spaces, and let X ^∗ and ${\mathcal M}^*$ be their dual spaces. Let , be bounded bilinear forms. Then we consider the following abstract mixed variational formulation.

For f ∈ X ^∗ , $g\in {\mathcal M}^{*}$ given, find a pair $(u,p)\in X\times {\mathcal M}$ such that

$$\displaystyle \begin{aligned} \left\{\begin{array}{ll} a(u,v)+b(v,p)&=f(v), \ \ \forall \ v\in X,\\ b(u,q)&=g(q), \ \ \forall \ q\in {\mathcal M}. \end{array} \right. \end{aligned} $$

(1)

Let A : X → X ^∗ be the bounded linear operator defined by

$$\displaystyle \begin{aligned} \langle Av,w\rangle =a(v,w),\, \forall \, v,w\in X, \end{aligned} $$

(2)

where is the duality pairing of the dual spaces X ^∗ and X. We also use the notation 〈⋅, ⋅〉 for the duality pairing $_{{\mathcal M}^{*}}\langle \cdot ,\cdot \rangle _{\mathcal M}$. Let $B:X\to {\mathcal M}^{*}$ and $B^{*}:{\mathcal M}\to X^{*}$ be the bounded linear and transpose operators given by

$$\displaystyle \begin{aligned} &\langle Bv,q\rangle =b(v,q),\ \langle v,B^*q\rangle =\langle Bv,q\rangle ,\, \forall \, v\in X,\, \forall \, q\in {\mathcal M}. \end{aligned} $$

(3)

In addition, we consider the spaces

$$\displaystyle \begin{aligned} &V:=\mathrm{{Ker}}\, B=\left\{v\in X: b(v,q)=0,\ \forall \, q\in {\mathcal M}\right\}, \end{aligned} $$

(4)

$$\displaystyle \begin{aligned} &V^{\perp}:=\left\{T\in X^{*}: \langle T,v\rangle =0,\ \forall \, v\in V\right\}. \end{aligned} $$

(5)

Then the following well-posedness result holds (cf., e.g., [20, Theorem 2.34]).

Theorem 1

Let X and ${\mathcal M}$ be reflexive Banach spaces, f ∈ X ^∗ and $g\in {\mathcal M}^{*}$ , and $a(\cdot ,\cdot ):X\times X\to {\mathbb R}$ and $b(\cdot ,\cdot ):X\times {\mathcal M}\to {\mathbb R}$ be bounded bilinear forms. Let V  be the subspace of X defined by (4). Then the variational problem (1) is well-posed if and only if a(⋅, ⋅) satisfies the conditions

$$\displaystyle \begin{aligned} \left\{\begin{array}{lll} \exists \ \lambda >0\ \mathit{\mbox{ such that }} \ \displaystyle\inf _{u\in V\setminus \{0\}}\sup_{v\in V\setminus \{0\}}\frac{a(u,v)}{\|u\|{}_X\|v\|{}_X}\geq \lambda ,\\ \{v\in V: a(u,v)=0,\ \forall \ u\in V\}=\{0\}, \end{array}\right. \end{aligned} $$

(6)

and b(⋅, ⋅) satisfies the inf-sup (Ladyzhenskaya-Babus̆ka-Brezzi) condition,

$$\displaystyle \begin{aligned} \exists \, \beta >0\ \mathit{\mbox{ such that }} \ \inf _{q\in {\mathcal M}\setminus \{0\}}\sup_{v\in X\setminus \{0\}}\frac{b(v,q)}{\|v\|{}_X\|q\|{}_{\mathcal M}}\geq \beta . \end{aligned} $$

(7)

Moreover, there exists a constant C depending on β, λ and the norm of a(⋅, ⋅), such that the unique solution $(u,p)\in {X}\times {\mathcal M}$ of (1) satisfies the inequality

$$\displaystyle \begin{aligned} {\|u\|{}_{X}+\|p\|{}_{{\mathcal M}}\leq C\left(\|f\|{}_{X^{*}}+\|g\|{}_{{\mathcal M}^{*}}\right).} \end{aligned} $$

(8)

In addition, we have (see [20, Theorem A.56, Remark 2.7], [4, Theorem 2.7]).

Lemma 2

Let $X,{\mathcal M}$ be reflexive Banach spaces. Let $b(\cdot ,\cdot ):X\times {\mathcal M}\to {\mathbb R}$ be a bounded bilinear form. Let and be the operators defined by (3), and let . Then the following results are equivalent:

• (i) There exists a constant β > 0 such that b(⋅, ⋅) satisfies condition (7).

• (ii) $B:{X/V}\to {\mathcal M}^{*}$ is an isomorphism and for any

• (iii) is an isomorphism and for any $q\in {\mathcal M}.$

Remark 3

Let X be a reflexive Banach space and V  be a closed subspace of X. If a bounded bilinear form $a(\cdot ,\cdot ):V\times V\to {\mathbb R}$ is coercive on V , i.e., there exists a constant c _a > 0 such that

$$\displaystyle \begin{aligned} a(w,w)\geq c_a\|w\|{}_X^2,\ \forall \, w\in V, \end{aligned} $$

(9)

then the conditions (6) are satisfied as well (see, e.g., [20, Lemma 2.8]).

The next result known as the Babus̆ka-Brezzi theorem is the version of Theorem 1 for Hilbert spaces (see [6], [10, Theorems 0.1, 1.1, Corollary 1.2]).

Theorem 4

Let X and ${\mathcal M}$ be two real Hilbert spaces. Let $a(\cdot ,\cdot ):X\times X\to {\mathbb R}$ and $b(\cdot ,\cdot ):X\times {\mathcal M}\to {\mathbb R}$ be bounded bilinear forms. Let f ∈ X ^∗ and $g\in {\mathcal M}^{*}$ . Let V  be the subspace of X defined by (4). Assume that $a(\cdot ,\cdot ):V\times V\to {\mathbb R}$ is coercive and that $b(\cdot ,\cdot ):X\times {\mathcal M}\to {\mathbb R}$ satisfies the inf-sup condition (7). Then the variational problem (1) is well-posed.