Regularity of Weak Solutions to the Inhomogeneous Stationary Navier-Stokes Equations

One of the most intriguing issues in the mathematical theory of the stationary Navier-Stokes equations is the regularity of weak solutions. This problem has been deeply investigated for homogeneous fluids. In this paper, the regularity of the solutions in the case of not constant viscosity is analyzed. Precisely, it is proved that for a bounded domain Omega subset of R-2, a weak solution u is an element of W-1,W-q(Omega) is locally Holder continuous if q=2, and Holder continuous around x, if q is an element of(1,2) and vertical bar mu(x) - mu(0)vertical bar is suitably small, with mu(0) positive constant; an analogous result holds true for a bounded domain Omega subset of R-n (n>2) and weak solutions in W-1,W-n/2(Omega).