Global strong solutions to the 3D inhomogeneous heat-conducting incompressible fluids

This paper concerns the global regularity to an initial boundary value problem for the three-dimensional (3D) inhomogeneous heat-conducting fluids with density and absolute temperature-dependent viscosity. Let be the initial velocity. Through some time-weighted a priori estimates, we establish global existence of strong solutions for positive initial density under assumption that is small. For the case when initial vacuum is allowed, we show that strong solutions globally exist provided small. It is worth pointing out that the initial temperature can be arbitrarily large.