Global well-posedness of the 2D Boussinesq equations with vertical dissipation

TL;DR

This paper proves the global well-posedness of 2D Boussinesq equations with only vertical dissipation for minimal initial data regularity, using advanced inequalities to establish global solutions.

Contribution

It extends previous results by allowing initial data in a lower regularity space, ensuring global solutions with minimal assumptions.

Findings

Global well-posedness for minimal initial data regularity

Development of logarithmic Sobolev embedding inequalities

Establishment of a logarithmic Gronwall inequality

Abstract

We prove the global well-posedness of the two-dimensional Boussinesq equations with only vertical dissipation. The initial data $(u_0,\theta_0)$ are required to be only in the space $X=\{f\in L^2(\mathbb R^2)\,|\,\partial_xf\in L^2(\mathbb R^2)\}$, and thus our result generalizes that in [C. Cao, J. Wu, Global regularity for the two-dimensional anisotropic Boussinesq equations with vertical dissipation, Arch. Rational Mech. Anal., Vol. 208 (2013), 985-1004], where the initial data are assumed to be in $H^2(\mathbb R^2)$. The assumption on the initial data is at the minimal level that is required to guarantee the uniqueness of the solutions. A logarithmic type limiting Sobolev embedding inequality for the $L^\infty(\mathbb R^2)$ norm, in terms of anisotropic Sobolev norms, and a logarithmic type Gronwall inequality are established to obtain the global in time a priori estimates, which guarantee the local solution to be a global one.