L r regularity for the stokes and navier-stokes problems

Abstract

The Lr regularity, 1<r<∞, of the weak solution of the Stokes problem has been analyzed by L. Cattabriga when the spatial domain Ω is of class C2. In this paper, Cattabriga's results are generalized for W2, ∞ domains. First, we prove a L2 regularity result by using appropriate difference quotients of the weak solution; for these, we obtain uniform estimates as a consequence of standard results concerning mixed problems. In order to obtain Lr regularity, we use the hydrodynamical potentials. We deduce Lr «a posteriori» estimates for the strong solution by arguing as D. Gilbarg and N. S. Trudinger have previously done to analyze the Dirichlet problem for the Laplace and Poisson equations. From these results, it is straightforward to demonstrate existence and uniqueness of the strong solution. Also, by applying the usual «boot-strap» argument, one deduces the Lr regularity of any weak solution to the Navier-Stokes problem.