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.