Stability of solutions of stationary Boussinesq systems on weak-Morrey spaces

TL;DR

This paper proves the asymptotic stability of steady solutions for the Boussinesq systems in weak-Morrey spaces, establishing global existence and stability results using advanced functional analysis techniques.

Contribution

It introduces a novel approach to analyze Boussinesq systems in weak-Morrey spaces, proving stability and existence of solutions with new bilinear estimates and interpolation methods.

Findings

Global small mild solutions exist in weak-Morrey spaces.

Steady solutions are asymptotically stable in the considered framework.

The methods extend the analysis of Boussinesq systems to critical weak-Morrey spaces.

Abstract

In this paper we establish the asymptotic stability of steady solutions for the Boussinesq systems in the framework of Cartesian product of critical weak-Morrey spaces on $\mathbb{R}^n$, where $n \geqslant 3$. In our strategy, we first establish the continuity for the long time of the bilinear terms associated with the mild solutions of the Boussinesq systems, i.e., the bilinear estimates by using only the norm of the present spaces. As a direct consequence, we obtain the existence of global small mild solutions and asymptotic stability of steady solutions of the Boussinesq systems in the class of continuous functions from $[0, \infty)$ to the Cartesian product of critical weak-Morrey spaces. Our techniques consist interpolation of operators, duality, heat semigroup estimates , Holder and Young inequalities in block spaces (based on Lorentz spaces) that are preduals of Morrey-Lorentz spaces.