Global Existence of Solutions to the 2D subcritical dissipative Quasi-Geostrophic equation and persistency of the initial regularity

TL;DR

This paper proves the global existence and regularity persistence of solutions to the 2D subcritical dissipative quasi-geostrophic equation for initial data in specific function spaces, extending understanding of solution behavior over time.

Contribution

It establishes the global existence and uniqueness of solutions for initial data in certain Besov and related spaces, and shows regularity persistence over time.

Findings

Global solutions exist for initial data in specified Besov spaces.

Solutions maintain initial regularity over time.

Unique solutions are obtained under given conditions.

Abstract

In this paper, we prove that if the initial data $\theta_0$ and its Riesz transforms ($\mathcal{R}_1(\theta_0)$ and $\mathcal{R}_2(\theta_0)$) belong to the space $(\overline{S(\mathbb{R}^2))}^{B_{\infty}^{1-2\alpha ,\infty}}$, where $\alpha \in ]1/2,1[$, then the 2D Quasi-Geostrophic equation with dissipation $\alpha$ has a unique global in time solution $\theta$. Moreover, we show that if in addition $\theta_0 \in X$ for some functional space $X$ such as Lebesgue, Sobolev and Besov's spaces then the solution $\theta$ belongs to the space $C([0,+\infty [,X).$