Variational approach for the Stokes and Navier–Stokes systems with nonsmooth coefficients in Lipschitz domains on compact Riemannian manifolds

Abstract

The purpose of this paper is to show well-posedness results for Dirichlet problems for the Stokes and Navier–Stokes systems with $$L^{\infty }$$ -variable coefficients in $$L^2$$ -based Sobolev spaces in Lipschitz domains on compact Riemannian manifolds. First, we refer to the Dirichlet problem for the nonsmooth coefficient Stokes system on Lipschitz domains in compact Riemannian manifolds and show its well-posedness by employing a variational approach that reduces the boundary value problem of Dirichlet type to a variational problem defined in terms of two bilinear continuous forms, one of them satisfying a coercivity condition and another one the inf-sup condition. We show also the equivalence between some transmission problems for the nonsmooth coefficient Stokes system in complementary Lipschitz domains on compact Riemannian manifolds and their mixed variational counterparts, and then their well-posedness in $$L^2$$ -based Sobolev spaces by using the remarkable Necas–Babuska–Brezzi technique (see Babuska in Numer Math 20:179–192, 1973; Brezzi in RAIRO Anal Numer R2:129–151, 1974; Necas in Rev Roum Math Pures Appl 9:47–69, 1964). As a consequence of these well-posedness results we define the layer potential operators for the nonsmooth coefficient Stokes system on Lipschitz surfaces in compact Riemannian manifolds, and provide their main mapping properties. These properties are used to construct explicitly the solution of the Dirichlet problem for the Stokes system. Further, we combine the well-posedness of the Dirichlet problem for the nonsmooth coefficient Stokes system with a fixed point theorem to show the existence of a weak solution to the Dirichlet problem for the nonsmooth variable coefficient Navier–Stokes system in $$L^2$$ -based Sobolev spaces in Lipschitz domains on compact Riemannian manifolds. The well developed potential theory for the smooth coefficient Stokes system on compact Riemannian manifolds (cf. Dindos and Mitrea in Arch Ration Mech Anal 174:1–47, 2004; Mitrea and Taylor in Math Ann 321:955–987, 2001) is also discussed in the context of the potential theory developed in this paper.