Global strong solution for viscous incompressible heat conducting Navier–Stokes flows with density-dependent viscosity

Abstract

We study an initial boundary value problem for the nonhomogeneous heat conducting fluids with non-negative density. First of all, we show that for the initial density allowing vacuum, the strong solution exists globally if the gradient of viscosity satisfies [Formula: see text]. Then, under certain smallness condition, we prove that there exists a unique global strong solution to the 2D viscous nonhomogeneous heat conducting Navier–Stokes flows with variable viscosity. Our method relies upon the delicate energy estimates and regularity properties of Stokes system and elliptic equation.