On the global regularity of the 2D Boussinesq equations with fractional dissipation

Abstract

In this paper, we consider the two-dimensional (2D) incompressible Boussinesq equations with fractional Laplacian dissipation and thermal diffusion. Attention is focused on the subcritical case when the velocity dissipation dominates. More precisely, we establish the global regularity result of the 2D Boussinesq equations in a new range of fractional powers of the Laplacian, namely 1−α<β<minα2,(3α−2)(α+2)10−7α,2−2α4α−3 with 0.783≈21−2178<α<1. Therefore, this result significantly improves the previous work [31] which obtained the global regularity result for 1−α<β<f(α) with 0.888≈6−64<α<1, where f(α)<1 is an explicit function.