Global well-posedeness and large time asymptotic behavior of 2D density-dependent Boussinesq equations of Korteweg type with vacuum

Abstract

In this paper, we are devoted to study the Cauchy problem of the incompressible density-dependent Boussinesq equations of Korteweg type with vacuum. We establish some key spatial weighted estimates and a priori decay-in-time rate of the strong solutions by using the energy method. Furthermore, we prove that there is a unique global strong solution for the 2D Cauchy problem when the spatial weighted norm of the initial density is suitably small. Finally, we also obtain the large time decay rates for the gradients of velocity, temperature and pressure.