Global existence of the two-dimensional QGE with sub-critical dissipation

TL;DR

This paper proves the existence and uniqueness of solutions for the two-dimensional sub-critical dissipative quasi-geostrophic equations, establishing conditions for global existence and criteria for blow-up.

Contribution

It provides new results on global existence, uniqueness, and blow-up criteria for the 2D sub-critical quasi-geostrophic equations with large initial data.

Findings

Existence of unique local-in-time solutions for large initial data.

Global solutions exist if initial data norms are sufficiently small.

A blow-up criterion for non-global solutions is established.

Abstract

In this paper, we study the sub-critical dissipative quasi-geostrophic equations $({\bf S}_\alpha)$. We prove that there exists a unique local-in-time solution for any large initial data $\theta_0$ in the space ${\bf{\mathcal X}}^{1-2\alpha}(\mathbb R^2)$ defined by (\ref{ec}). Moreover we show that $({\bf S}_\alpha)$ has a global solution in time if the norms of the initial data in ${\bf{\mathcal X}}^{1-2\alpha}(\mathbb R^2)$ are bounded by $1/4$. Also, we prove a blow-up criterion of the non global solution of $({\bf S}_\alpha)$.