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

Abstract

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 Ω⊂R2, a weak solution u∈W1,q(Ω) is locally Hölder continuous if q=2, and Hölder continuous around x, if q∈(1,2) and |μ(x)−μ0| is suitably small, with μ0 positive constant; an analogous result holds true for a bounded domain Ω⊂Rn(n>2) and weak solutions in W1,n/2(Ω).