Global regularity for a modified critical dissipative quasi-geostrophic equation

TL;DR

This paper proves the global existence and regularity of strong solutions for a modified critical dissipative quasi-geostrophic equation using Besov space techniques, extending understanding of such equations' behavior.

Contribution

It introduces a modification to the critical quasi-geostrophic equation that preserves scaling invariance and establishes global regularity results for strong solutions.

Findings

Global existence of strong solutions is proven.

Regularity of solutions is established for the modified system.

The approach uses Besov space techniques to analyze the equation.

Abstract

In this paper, we consider the modified quasi-geostrophic equation \begin{gather*} \del_t \theta + (u \cdot \grad) \theta + \kappa \Lambda^\alpha \theta = 0 u = \Lambda^{\alpha - 1} R^{\perp}\theta. \end{gather*} with $\kappa > 0$, $\alpha \in (0,1]$ and $\theta_0 \in \lp{2}(\R^2)$. We remark that the extra $\Lambda^{\alpha - 1}$ is introduced in order to make the scaling invariance of this system similar to the scaling invariance of the critical quasi-geostrophic equations. In this paper, we use Besov space techniques to prove global existence and regularity of strong solutions to this system.