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

TL;DR

This paper proves the global regularity of smooth solutions to the 2D Boussinesq equations with fractional Laplacian dissipation, extending the range of fractional powers for which well-posedness holds.

Contribution

It provides an elementary proof of global regularity for the 2D Boussinesq equations with a broader range of fractional Laplacian powers, improving previous results.

Findings

Established global regularity for fractional powers in a new range

Simplified proof based on nonlinear lower bounds

Extended the applicability of previous well-posedness results

Abstract

As a continuation of the previous work [40], in this paper we focus on the Cauchy problem of the two-dimensional (2D) incompressible Boussinesq equations with fractional Laplacian dissipation. We give an elementary proof of the global regularity of the smooth solutions of the 2D Boussinesq equations with a new range of fractional powers of the Laplacian. The argument is based on the nonlinear lower bounds for the fractional Laplacian established in [12]. Consequently, this result significantly improves the recent works [12, 38, 40].