Global Regularity for the Two-dimensional Boussinesq Equations Without Diffusivity in Bounded Domains

TL;DR

This paper proves the global regularity of solutions to the 2D Boussinesq equations without diffusivity in bounded domains, even for initial data with minimal fractional smoothness.

Contribution

It establishes the first global regularity results for rough initial data in bounded domains for these equations without diffusivity.

Findings

Global regularity for rough initial data with fractional derivatives

Well-posedness of 2D Boussinesq equations without diffusivity

Applicable to bounded domains with minimal smoothness assumptions

Abstract

We address the well-posedness for the two-dimensional Boussinesq equations with zero diffusivity in bounded domains. We prove global in time regularity for rough initial data: both the initial velocity and temperature have $\epsilon$ fractional derivatives in $L^{q}$ for some $q>2$ and $\epsilon>0$ arbitrarily small.