Navier-Stokes equations on Lipschitz domains in Riemannian manifolds

Abstract

The Navier-Stokes equations are a system of nonlinear evolution equations modeling the flow of a viscous, incompressible fluid. One ingredient in the analysis of this system is the stationary, linear system known as the Stokes system, a boundary value problem (BVP) that will be described in detail in the next section. Layer potential methods in smoothly bounded domains in Euclidean space have proven useful in the analysis of the Stokes system, starting with work of Odqvist and Lichtenstein, and including work of Solonnikov and many others. See the discussion in Chapter III of [10] and in [17], for the case of flow in regions with smooth boundary. A treatment based on the modern language of pseudodifferential operators can be found in [18]. In 1988, E. Fabes, C. Kenig and G. Verchota [6], extended this classical layer potential approach to cover Lipschitz domains in Euclidean space. In [6] the main result concerning the (constant coefficient) Stokes system on Lipschitz domains with connected boundary in Euclidean space, is the treatment of the L-Dirichlet boundary value problem (and its regular version). To achieve this, the authors solve certain auxiliary Neumann type problems and then exploit the duality between these and the original BVP’s at the level of boundary integral operators. P. Deuring and W. von Wahl [4] made use of the analysis in [6] to demonstrate the short-time existence of solutions to the Navier-Stokes equations in bounded Lipschitz domains in threedimensional Euclidean space. It was necessary in [4] to include the hypothesis that the boundary be connected. The hypothesis that the boundary be connected pervaded much work on the application of layer potentials to analysis on Lipschitz domains. It was certainly natural to speculate that this restriction was an artifact of the methods used and not ∗Partly supported by NSF grant DMS-9870018. †Partially supported by NSF grant DMS-9877077. 1991Mathematics Subject Classification. Primary 35Q30, 76D05, 35J25; Secondary 42B20, 45E05.