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

Abstract

In this paper, we study the sub-critical dissipative quasi-geostrophic equations $({\mathbf{S}}_{\alpha })$. We prove that there exists a unique local-in-time solution for any large initial data ${\theta }_0$ in the space ${\mathcal{X}}^{1-2\alpha }({\mathbb{R}}^2)$ defined by (\ref{ec}). Moreover we show that $({\mathbf{S}}_{\alpha })$ has a global solution in time if the norms of the initial data in ${\mathcal{X}}^{1-2\alpha }({\mathbb{R}}^2)$ are bounded by $1/4$. Also, we prove a blow-up criterion of the non global solution of $({\mathbf{S}}_{\alpha })$.

arXiv comment16 pages