Dissipative quasi-geostrophic equations in critical Sobolev spaces: smoothing effect and global well-posedness

TL;DR

This paper investigates the regularity, decay, and global existence of solutions to dissipative quasi-geostrophic equations in critical Sobolev spaces, demonstrating smoothing effects and extending results to critical and super-critical cases.

Contribution

It establishes higher regularity of solutions for arbitrary initial data and proves global existence for critical 2D quasi-geostrophic equations with periodic data.

Findings

Higher regularity of mild solutions in $H^{2-eta}$ spaces.

Global existence for critical 2D equations with periodic $ abla heta$ data.

Decay estimates for solutions over time.

Abstract

We study the critical and super-critical dissipative quasi-geostrophic equations in $\bR^2$ or $\bT^2$. Higher regularity of mild solutions with arbitrary initial data in $H^{2-\gamma}$ is proved. As a corollary, we obtain a global existence result for the critical 2D quasi-geostrophic equations with periodic $\dot H^1$ data. Some decay in time estimates are also provided.