On Stationary Gevrey Solutions to the Gravitational Boussinesq System and Applications to Uniqueness

Abstract

The stationary version of the Boussinesq system with a general gravitational acceleration term is considered. Under suitable assumptions on this term, as well as on the external forces acting on each equation of this coupled system, we first establish the existence of weak solutions in the natural energy space $\dot{H}^1(\mathbb{R}^3)$. The uniqueness of these solutions is a challenging open problem. Within this framework, our first main contribution is to show that \emph{any} weak $\dot{H}^1$-solution exhibits an analytic smoothing effect in the Gevrey class. Our second main contribution is to show that the Gevrey class regularity can also be used to study the uniqueness problem, provided that these solutions satisfy a suitable low-frequency control. As a by-product, we also obtain new regularity results and a \emph{new Liouville-type result} for weak $\dot{H}^1$-solutions of the classical Navier--Stokes equations.