Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces

TL;DR

This paper establishes global existence and uniqueness results for the Boussinesq system with partial viscosity in higher dimensions, using Lorentz and Besov space frameworks, despite the lack of smoothing effects on temperature.

Contribution

It provides new global existence and uniqueness results for the Boussinesq system with data in Lorentz and Besov spaces, including cases with negative regularity indices.

Findings

Global existence for small data in Lorentz spaces.

Uniqueness in Besov spaces with negative regularity.

Results applicable to higher dimensions (N ≥ 3).

Abstract

This paper is devoted to the study of the Cauchy problem for the Boussinesq system with partial viscosity in dimension $N\geq3.$ First we prove a global existence result for data in Lorentz spaces satisfying a smallness condition which is at the scaling of the equations. Second, we get a uniqueness result in Besov spaces with {\it negative} indices of regularity (despite the fact that there is no smoothing effect on the temperature). The proof relies on a priori estimates with loss of regularity for the nonstationary Stokes system with convection. As a corollary, we obtain a global existence and uniqueness result for small data in Lorentz spaces.