On the global well-posedness of a class of Boussinesq- Navier-Stokes systems

TL;DR

This paper proves the global well-posedness of a class of 2D Boussinesq-Navier-Stokes systems with fractional dissipation for certain parameter ranges, even with rough initial data.

Contribution

It establishes the first global existence and uniqueness results for these systems under specific fractional dissipation conditions and initial data regularity.

Findings

Global well-posedness proven for certain fractional dissipation parameters.

Results hold for rough initial data within specified bounds.

Provides explicit bounds on parameters for well-posedness.

Abstract

In this paper we consider the following 2D Boussinesq-Navier-Stokes systems \partial_{t}u+u\cdot\nabla u+\nabla p+ |D|^{\alpha}u &= \theta e_{2} \partial_{t}\theta+u\cdot\nabla \theta+ |D|^{\beta}\theta &=0 \quad with $\textrm{div} u=0$ and $0<\beta<\alpha<1$. When $\frac{6-\sqrt{6}}{4}<\alpha< 1$, $1-\alpha<\beta\leq f(\alpha) $, where $f(\alpha)$ is an explicit function as a technical bound, we prove global well-posedness results for rough initial data.