Existence and Decay of Global Strong Solution to 3D Density-Dependent Boussinesq Equations with Vacuum

Abstract

This paper is concerned with the initial boundary problem for the three-dimensional density-dependent Boussinesq equations with vacuum. We obtain the existence of the global strong solution under the initial density in the norm $L^∞$ is small enough without any smallness condition of $u$ and $θ.$ Furthermore, the exponential decay rates of the solution and their derivatives in some norm was established. In addition, we show that the solution and their derivatives are monotonically decreasing with respect to time $t$ on $[0, T].$