Global strong solutions to the 3D incompressible Navier–Stokes equations with zero heat conduction

Abstract

This paper concerns the global regularity of strong solutions to the Cauchy problem of inhomogeneous incompressible Navier–Stokes fluids with vacuum, zero heat conduction, and density–temperature‐dependent viscosity. The existence of global‐in‐time strong solutions is proved when the initial energy is suitably small or the lower bound of the viscosity coefficient is large enough. Moreover, some exponential decay‐in‐time rates of strong solutions are obtained via time‐weighted a priori estimates.