On the degenerate boussinesq equations on surfaces

Abstract

In this paper we study the non-degenerate and partially degenerate Boussinesq equations on a closed surface $ \Sigma $. When $ \Sigma $ has intrinsic curvature of finite Lipschitz norm, we prove the existence of global strong solutions to the Cauchy problem of the Boussinesq equations with full or partial dissipations. The issues of uniqueness and singular limits (vanishing viscosity/vanishing thermal diffusivity) are also addressed. In addition, we establish a breakdown criterion for the strong solutions for the case of zero viscosity and zero thermal diffusivity. These appear to be among the first results for Boussinesq systems on Riemannian manifolds.