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

TL;DR

This paper proves that stationary solutions to the gravitational Boussinesq system become analytically smooth in the Gevrey class and explores how this regularity can aid in establishing uniqueness, also deriving new results for Navier-Stokes solutions.

Contribution

It demonstrates Gevrey class regularity for weak solutions of the stationary Boussinesq system and applies this to address the open problem of solution uniqueness.

Findings

Weak solutions exhibit Gevrey class regularity.

Gevrey regularity can be used to study uniqueness under low-frequency control.

New regularity and Liouville-type results for Navier-Stokes solutions.

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.