On the global regularity for anisotropic dissipative surface quasi-geostrophic equation

TL;DR

This paper proves the global regularity of solutions for a 2D anisotropic surface quasi-geostrophic equation with fractional dissipation, introducing new inequalities and a logarithmic Gronwall inequality to handle nonlocal operators.

Contribution

It develops novel anisotropic embedding and interpolation inequalities and a new logarithmic Gronwall inequality to establish global solutions for the anisotropic surface quasi-geostrophic equation.

Findings

Global existence of classical solutions under certain dissipation conditions

New anisotropic inequalities for fractional derivatives

A novel logarithmic Gronwall inequality for nonlocal operators

Abstract

In this paper, we consider the two-dimensional surface quasi-geostrophic equation with fractional horizontal dissipation and fractional vertical thermal diffusion. Global existence of classical solutions is established when the dissipation powers are restricted to a suitable range. Due to the nonlocality of these 1D fractional operators, some of the standard energy estimate techniques no longer apply, to overcome this difficulty, we establish several anisotropic embedding and interpolation inequalities involving fractional derivatives. In addition, in order to bypass the unavailability of the classical Gronwall inequality, we establish a new logarithmic type Gronwall inequality, which may be of independent interest and potential applications.