On global solutions to the inhomogeneous, incompressible Navier–Stokes equations with temperature-dependent coefficients

Abstract

In this paper, we study the initial-boundary value problem for the full inhomogeneous, incompressible Navier–Stokes equations with temperature-dependent viscosity and heat conductivity coefficients. The viscosity coefficient may be degenerate in the sense that it may vanish in the region of absolutely zero temperature. Our main result is to prove the global existence of large weak solutions to such a system. The proof is based on a three-level approximate scheme, the Galerkin method, De Giorgi’s method, and appropriate compactness arguments.