Local strong solutions to the nonhomogeneous Bénard system with nonnegative density

Abstract

We study the Cauchy problem of the nonhomogeneous Bénard system in the whole two-dimensional (2D) space, where the density is allowed to vanish initially. We prove that there exists a unique local strong solution. To compensate for the lack of integrability of the velocity in the whole space, a careful space weight is imposed on the initial density, which cannot decay too slowly in the far field.