Global regularity for the 2D anisotropic Boussinesq Equations with vertical dissipation

TL;DR

This paper proves the global regularity of solutions to the 2D anisotropic Boussinesq equations with only vertical dissipation by controlling derivatives through $L^r$ norms of the vertical velocity.

Contribution

It introduces a novel approach to establish global regularity using bounds on $L^r$ norms of vertical velocity and interpolation inequalities, addressing the challenge of anisotropic dissipation.

Findings

Global existence of classical solutions is proven.

The $L^r$ norms of vertical velocity grow at most like $\sqrt{r \, ext{log} \, r}$.

A new interpolation inequality links $\|v\\|_{L^\\infty}$ and $\|v\\|_{L^r}$.

Abstract

This paper establishes the global in time existence of classical solutions to the 2D anisotropic Boussinesq equations with vertical dissipation. When only the vertical dissipation is present, there is no direct control on the horizontal derivatives and the global regularity problem is very challenging. To solve this problem, we bound the derivatives in terms of the $L^\infty$-norm of the vertical velocity $v$ and prove that $\|v\|_{L^{r}}$ with $2\le r<\infty$ at any time does not grow faster than $\sqrt{r \log r}$ as $r$ increases. A delicate interpolation inequality connecting $\|v\|_{L^\infty}$ and $\|v\|_{L^r}$ then yields the desired global regularity.