This monograph provides a rigorous analysis of a wide range of stationary (steady state) boundary value problems for elliptic systems of Stokes and Navier-Stokes type, as encountered in fluid dynamics. Addressing Dirichlet, Neumann, Robin, mixed, and transmission problems in both the isotropic and anisotropic cases, it makes systematic use of the notion of relaxed ellipticity recently introduced by the authors. The problems are treated in Lipschitz domains in the Euclidean setting as well as in compact Riemannian manifolds and in manifolds with cylindrical ends (non-compact manifolds), with given data in a variety of spaces Lebesgue, standard or weighted Sobolev, Bessel potential, and Besov. A detailed and comprehensive study is provided of the main mathematical properties of boundary value problems related to the Navier-Stokes equations with variable coefficients, such as existence, uniqueness, and regularity of solutions. These are considered in bounded, periodic, and also unbounded domains, in the Euclidean setting as well as on manifolds (compact, or non-compact). The included results represent the authors contributions to the field of stationary Stokes, Navier-Stokes, and related equations, the main novelty being the analysis of the related boundary problems with anisotropic variable coefficients and on manifolds.
The book is aimed at researchers, graduate and advanced undergraduate mathematics students, physicists, and computational engineers interested in mathematical fluid mechanics, partial differential equations, and geometric analysis. The prerequisites include the basics of partial differential equations, the variational approach and function spaces; some sections need the fundamentals of integral equations, the theory of Riemannian manifolds, and fixed-point techniques.
Chapter
1. Introduction.- Part I. Boundary value problems for the
isotropic, constant-coefficient Brinkman and Navier-Stokes type systems in
bounded Lipschitz domains in Rn.
Chapter
2. Preliminaries.
Chapter
3. Layer
potentials for the constant-coefficient Brinkman system in bounded,
Lipschitz domains.
Chapter
4. Isotropic Navier-Stokes type models for flows
in bidisperse porous media.- Part II. Transmission and boundary value
problems for the anisotropic, L-coefficients Stokes and Navier-Stokes
systems in Lipschitz domains in Rn.
Chapter
5. Transmission problems for
Stokes and Navier-Stokes systems with strongly elliptic coefficients in Rn.-
Chapter
6. Layer potentials for the Stokes system with symmetric tensor
coefficient satisfying a relaxed ellipticity condition.
Chapter
7.
Dirichlet-transmission problems for Stokes and Navier-Stokes systems in
Lipschitz domains with internal interfaces.
Chapter
8.
Dirichlet-transmission problems for Stokes and Navier-Stokes systems on
bounded Lipschitz domains with transversal interfaces.
Chapter
9.
Mixed-transmission problems for Stokes and Navier-Stokes systems in bounded
Lipschitz domains with transversal interfaces.
Chapter
10. Periodic
solutions for anisotropic Stokes, Oseen, and Navier-Stokes systems.
Chapter
11. Anisotropic Navier-Stokes type models for flows in multidisperse porous
media.- Part III. Transmission and boundary value problems for Stokes and
Navier-Stokes type systems on compact Riemannian manifolds.
Chapter
12.
Generalized Brinkman operators with smooth coefficients: Fundamental
solutions and layer potentials.
Chapter
13. Transmission problems for Stokes
and Navier-Stokes type systems with smooth coefficients.
Chapter
14. Stokes
and Navier-Stokes systems with non-smooth coefficients in Lipschitz domains.-
Part IV. The generalized Stokes operator on manifolds with cylindrical ends.-
Chapter
15. The essentially translation invariant calculus on manifolds with
cylindrical ends.
Chapter
16. Invertibility of the generalized Stokes
operator and of its layer potential operators on manifolds with straight
cylindrical ends.
Mirela Kohr is a Full Professor in the Department of Mathematics at Babe-Bolyai University, Cluj-Napoca, Romania. She has an extensive list of publications on various areas of fluid mechanics, PDEs (especially Navier-Stokes), complex analysis, and geometric analysis. She has been the principal investigator on six research grants, an invited speaker to many conferences, and has made several research visits to the University of Toronto, and various universities in Italy, Germany, the UK, Japan and France. Sergey E. Mikhailov is a Full Professor in Applied Mathematics and Analysis at Brunel University, London, UK. He has numerous publications on the analysis and numerics of boundary integral equations, partial differential (especially Navier-Stokes) equations and theoretical solid mechanics. He graduated from the Moscow Institute of Physics and Technology and worked in Moscow before moving to the University of Stuttgart, Germany, in 1993, as a Humboldt Fellow, and then to the UK. Victor Nistor is a Full Professor in the Department of Mathematics at Lorraine University, Metz, France. He has an extensive list of publications covering various areas of partial differential equations, geometric analysis, applied mathematics, and operator algebras. He was an NSF Young Investigator and a Sloan Fellow. Wolfgang L. Wendland studied Mathematics and Fluid Mechanics at TU Berlin. He is Professor Emeritus Dr.-Ing. at University of Stuttgart, where he was a Full Professor of Applied Mathematics (19862005). He was a Full Professor of Mathematics at TH Darmstadt (19701986), Uni del Chair Prof. at the University of Delaware, USA (19731974) and J.G. Herder Professor at Babe-Bolyai University in Cluj, Romania (2005 and 2007). He has numerous publications on analysis and numerical methods for PDEs and boundary integral equations.
