Global well-posedness for the 2D Boussinesq Equations with Zero Viscosity

TL;DR

This paper establishes the global well-posedness of the 2D Boussinesq equations with zero viscosity and positive diffusivity in bounded domains, for rough initial data, using maximal regularity techniques.

Contribution

It proves the existence and uniqueness of solutions for rough initial data in the zero-viscosity 2D Boussinesq system, which was previously unresolved.

Findings

Global well-posedness for rough initial data

Applicability of maximal regularity for heat equations

Extension to bounded domains with zero viscosity

Abstract

We prove the global well-posedness of the two-dimensional Boussinesq equations with zero viscosity and positive diffusivity in bounded domains for rough initial data [ $u_{0}\in L^{2}$, $\text{curl}\,u_{0}\in L^{\infty}$ and $\theta_{0}\in B^{2-2/p}_{q,p}$ with $p\in (1,\infty)$, $q\in (2,\infty)$ ]. Our method is based on the maximal regularity for heat equation.