Newtonian and Single Layer Potentials for the Stokes System with <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">∞</Emphasis></Superscript> Coefficients and the Exterior Dirichlet Problem

Mirela Kohr; Sergey E. Mikhailov; Wolfgang L. Wendland

doi:10.1007/978-3-030-02650-9_12

Analysis as a Life pp 237-260 | Cite as

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

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Mirela Kohr
Sergey E. Mikhailov
Wolfgang L. Wendland

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First Online: 31 January 2019

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Abstract

A mixed variational formulation of some problems in L²-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.

Keywords

Stokes system with L^∞ coefficients Newtonian and layer potentials Variational approach Inf-sup condition Sobolev spaces

Mathematics Subject Classification (2010)

Primary 35J25 35Q35 42B20 46E35 Secondary 76D 76M

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Notes

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.

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 Open image in new window , Open image in new window 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 Open image in new window

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 Open image in new window

and Open image in new window

be the operators defined by (3), and let Open image in new window

. 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 Open image in new window for any Open image in new window
(iii)

Open image in new window is an isomorphism and Open image in new window 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.

References

1.
M.S. Agranovich, Sobolev Spaces, Their Generalizations, and Elliptic Problems in Smooth and Lipschitz Domains (Springer, Heidelberg, 2015)CrossRefGoogle Scholar
2.
F. Alliot, C. Amrouche, The Stokes problem in ${\mathbb R}^n$: an approach in weighted Sobolev spaces. Math. Models Methods Appl. Sci. 9, 723–754 (1999)MathSciNetCrossRefGoogle Scholar
3.
F. Alliot, C. Amrouche, Weak solutions for the exterior Stokes problem in weighted Sobolev spaces. Math. Methods Appl. Sci. 23, 575–600 (2000)MathSciNetCrossRefGoogle Scholar
4.
C. Amrouche, M. Meslameni, Stokes problem with several types of boundary conditions in an exterior domain. Electron. J. Diff. Equ. 2013(196), 1–28 (2013)MathSciNetzbMATHGoogle Scholar
5.
C. Amrouche, H.H. Nguyen, L ^p-weighted theory for Navier-Stokes equations in exterior domains. Commun. Math. Anal. 8, 41–69 (2010)MathSciNetzbMATHGoogle Scholar
6.
I. Babus̆ka, The finite element method with Lagrangian multipliers. Numer. Math. 20, 179–192 (1973)MathSciNetCrossRefGoogle Scholar
7.
C. Băcuţă, M.E. Hassell, G.C. Hsiao, F-J. Sayas, Boundary integral solvers for an evolutionary exterior Stokes problem. SIAM J. Numer. Anal. 53, 1370–1392 (2015)MathSciNetCrossRefGoogle Scholar
8.
A. Barton, Layer potentials for general linear elliptic systems. Electron. J. Diff. Equ. 2017(309), 1–23 (2017)MathSciNetzbMATHGoogle Scholar
9.
K. Brewster, D. Mitrea, I. Mitrea, M. Mitrea, Extending Sobolev functions with partially vanishing traces from locally (𝜖, δ)-domains and applications to mixed boundary problems. J. Funct. Anal. 266, 4314–4421 (2014)MathSciNetCrossRefGoogle Scholar
10.
F. Brezzi, On the existence, uniqueness and approximation of saddle points problems arising from Lagrange multipliers. R.A.I.R.O. Anal. Numer. R2, 129–151 (1974)MathSciNetCrossRefGoogle Scholar
11.
F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 15 (Springer, New York, 1991)Google Scholar
12.
O. Chkadua, S.E. Mikhailov, D. Natroshvili, Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient, I: Equivalence and invertibility. J. Integr. Equ. Appl. 21, 499–543 (2009)MathSciNetzbMATHGoogle Scholar
13.
O. Chkadua, S.E. Mikhailov, D. Natroshvili, Localized boundary-domain singular integral equations based on harmonic parametrix for divergence-form elliptic PDEs with variable matrix coefficients. Integr. Equ. Oper. Theory. 76, 509–547 (2013)MathSciNetCrossRefGoogle Scholar
14.
O. Chkadua, S.E. Mikhailov, D. Natroshvili, Analysis of direct segregated boundary-domain integral equations for variable-coefficient mixed BVPs in exterior domains. Anal. Appl. 11(4), 1350006 (2013)MathSciNetCrossRefGoogle Scholar
15.
J. Choi, K-A. Lee, The Green function for the Stokes system with measurable coefficients. Commun. Pure Appl. Anal. 16, 1989–2022 (2017)MathSciNetCrossRefGoogle Scholar
16.
J. Choi, M. Yang, Fundamental solutions for stationary Stokes systems with measurable coefficients. J. Diff. Equ. 263, 3854–3893 (2017)MathSciNetCrossRefGoogle Scholar
17.
M. Costabel, Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19, 613–626 (1988)MathSciNetCrossRefGoogle Scholar
18.
R. Dautray, J. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Integral Equations and Numerical Methods, vol. 4 (Springer, Berlin, 1990)Google Scholar
19.
M. Dindos̆, M. Mitrea, The stationary Navier-Stokes system in nonsmooth manifolds: the Poisson problem in Lipschitz and C ¹ domains. Arch. Ration. Mech. Anal. 174, 1–47 (2004)Google Scholar
20.
A. Ern, J.L. Guermond, Theory and Practice of Finite Elements (Springer, New York, 2004)CrossRefGoogle Scholar
21.
E. Fabes, C. Kenig, G. Verchota, The Dirichlet problem for the Stokes system on Lipschitz domains. Duke Math. J. 57, 769–793 (1988)MathSciNetCrossRefGoogle Scholar
22.
E. Fabes, O. Mendez, M. Mitrea, Boundary layers on Sobolev-Besov spaces and Poisson’s equation for the Laplacian in Lipschitz domains. J. Funct. Anal. 159, 323–368 (1998)MathSciNetCrossRefGoogle Scholar
23.
G.P. Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Steady-State Problems, 2nd edn. (Springer, New York, 2011)Google Scholar
24.
G.N. Gatica, W.L. Wendland, Coupling of mixed finite elements and boundary elements for linear and nonlinear elliptic problems. Appl. Anal. 63, 39–75 (1996)MathSciNetCrossRefGoogle Scholar
25.
V. Girault, P.A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Springer Series in Comp. Math., vol. 5 (Springer, Berlin, 1986)CrossRefGoogle Scholar
26.
V. Girault, A. Sequeira, A well-posed problem for the exterior Stokes equations in two and three dimensions. Arch. Ration. Mech. Anal. 114, 313–333 (1991)MathSciNetCrossRefGoogle Scholar
27.
J. Giroire, Étude de quelques problèmes aux limites extérieurs et résolution par équations intégrales. Thése de Doctorat d’État, Université Pierre-et-Marie-Curie (Paris-VI), 1987Google Scholar
28.
B. Hanouzet, Espaces de Sobolev avec poids – application au probl$\grave {e}$me de Dirichlet dans un demi-espace. Rend. Sere. Mat. Univ. Padova. 46, 227–272 (1971)Google Scholar
29.
S. Hofmann, M. Mitrea, A.J. Morris, The method of layer potentials in L ^p and endpoint spaces for elliptic operators with L ^∞ coefficients. Proc. Lond. Math. Soc. 111, 681–716 (2015)MathSciNetCrossRefGoogle Scholar
30.
G.C. Hsiao, W.L. Wendland, Boundary Integral Equations (Springer, Heidelberg, 2008)CrossRefGoogle Scholar
31.
D.S. Jerison, C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130, 161–219 (1995)MathSciNetCrossRefGoogle Scholar
32.
M. Kohr, I. Pop, Viscous Incompressible Flow for Low Reynolds Numbers (WIT Press, Southampton, 2004)zbMATHGoogle Scholar
33.
M. Kohr, M. Lanza de Cristoforis, W.L. Wendland, Nonlinear Neumann-transmission problems for Stokes and Brinkman equations on Euclidean Lipschitz domains. Potential Anal. 38, 1123–1171 (2013)MathSciNetCrossRefGoogle Scholar
34.
M. Kohr, M. Lanza de Cristoforis, S.E. Mikhailov, W.L. Wendland, Integral potential method for transmission problem with Lipschitz interface in ${\mathbb R}^3$ for the Stokes and Darcy-Forchheimer-Brinkman PDE systems. Z. Angew. Math. Phys. 67(5), 1–30, 116 (2016)Google Scholar
35.
M. Kohr, M. Lanza de Cristoforis, W.L. Wendland, On the Robin-transmission boundary value problems for the nonlinear Darcy-Forchheimer-Brinkman and Navier-Stokes systems. J. Math. Fluid Mech. 18, 293–329 (2016)MathSciNetCrossRefGoogle Scholar
36.
M. Kohr, S.E. Mikhailov, W.L. Wendland, Transmission problems for the Navier-Stokes and Darcy-Forchheimer-Brinkman systems in Lipschitz domains on compact Riemannian manifolds. J. Math. Fluid Mech. 19, 203–238 (2017)MathSciNetCrossRefGoogle Scholar
37.
J. Lang, O. Méndez, Potential techniques and regularity of boundary value problems in exterior non-smooth domains: regularity in exterior domains. Potential Anal. 24, 385–406 (2006)MathSciNetCrossRefGoogle Scholar
38.
W. McLean, Strongly Elliptic Systems and Boundary Integral Equations (Cambridge University Press, Cambridge, 2000)zbMATHGoogle Scholar
39.
D. Medková, Transmission problem for the Brinkman system. Complex Var. Elliptic Equ. 59, 1664–1678 (2014)MathSciNetCrossRefGoogle Scholar
40.
S.E. Mikhailov, Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains. J. Math. Anal. Appl. 378, 324–342 (2011)MathSciNetCrossRefGoogle Scholar
41.
S.E. Mikhailov, Solution regularity and co-normal derivatives for elliptic systems with non-smooth coefficients on Lipschitz domains. J. Math. Anal. Appl. 400, 48–67 (2013)MathSciNetCrossRefGoogle Scholar
42.
M. Mitrea, M. Wright, Boundary value problems for the Stokes system in arbitrary Lipschitz domains. Astérisque. 344, viii+241 pp. (2012)Google Scholar
43.
M. Mitrea, S. Monniaux, M. Wright, The Stokes operator with Neumann boundary conditions in Lipschitz domains. J. Math. Sci. (New York). 176(3), 409–457 (2011)MathSciNetCrossRefGoogle Scholar
44.
J.-C. Nédélec, Approximation des Équations Intégrales en Mécanique et en Physique. Cours de DEA, 1977Google Scholar
45.
D.A. Nield, A. Bejan, Convection in Porous Media, 3rd edn. (Springer, New York, 2013)CrossRefGoogle Scholar
46.
F-J. Sayas, V. Selgas, Variational views of Stokeslets and stresslets. SEMA J 63, 65–90 (2014)MathSciNetCrossRefGoogle Scholar
47.
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators (North-Holland, Amsterdam, 1978)zbMATHGoogle Scholar
48.
W. Varnhorn, The Stokes Equations (Akademie Verlag, Berlin, 1994)zbMATHGoogle Scholar

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Authors and Affiliations

Mirela Kohr
- 1
Sergey E. Mikhailov
- 2
Wolfgang L. Wendland
- 3
Email author

1.Faculty of Mathematics and Computer ScienceBabeş-Bolyai UniversityCluj-NapocaRomania
2.Department of MathematicsBrunel University LondonUxbridgeUK
3.Institut für Angewandte Analysis und Numerische SimulationUniversität StuttgartStuttgartGermany

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

Abstract

Keywords

Mathematics Subject Classification (2010)

Notes

Acknowledgements

References

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Cite chapter

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

Abstract

Keywords

Mathematics Subject Classification (2010)

Notes

Acknowledgements

References

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Cite chapter

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