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
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Notes
- 1.
The trace operators defined on Sobolev spaces of vector fields on Ω± or \({\mathbb R}^3\) are also denoted by γ ± and γ, respectively.
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Acknowledgements
The research has been supported by the grant EP/M013545/1: “Mathematical Analysis of Boundary-Domain Integral Equations for Nonlinear PDEs” from the EPSRC, UK. Part of this work was done in April/May 2018, when M. Kohr visited the Department of Mathematics of the University of Toronto. She is grateful to the members of this department for their hospitality.
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Appendix: Mixed Variational Formulations and Their Well-Posedness Property
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
Let A : X → X ∗ be the bounded linear operator defined by
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
In addition, we consider the spaces
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
and b(⋅, ⋅) satisfies the inf-sup (Ladyzhenskaya-Babus̆ka-Brezzi) condition,
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
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
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.
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Kohr, M., Mikhailov, S.E., Wendland, W.L. (2019). Newtonian and Single Layer Potentials for the Stokes System with L∞ Coefficients and the Exterior Dirichlet Problem. In: Rogosin, S., Çelebi, A. (eds) Analysis as a Life. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-02650-9_12
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