On the solvability of the compressible Navier–Stokes system in bounded domains

Abstract

This paper is dedicated to the well-posedness issue for the barotropic Navier–Stokes system with homogeneous Dirichlet boundary conditions in bounded domains of . We aim at considering data in as large a class as possible. Our main result is that if the initial density is bounded away from zero and belongs to some W1,r with r > N, if the initial velocity is in the Besov space (and satisfies a suitable boundary condition), and if the body force is in for some p > 1 then the system has a unique local solution. Our regularity assumptions are consistent with a dimensional analysis which shows that critical data would correspond to r = N and p = 1, and improve an old result by Solonnikov (1980 J. Sov. Math. 14 1120–32).