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.
"Local strong solutions to the nonhomogeneous Bénard system with nonnegative density." Rocky Mountain J. Math. 50 (4) 1497 - 1516, August 2020. https://doi.org/10.1216/rmj.2020.50.1497