Electron. J. Diff. Equ., Vol. 2011 (2011), No. 08, pp. 1-18.

Existence of global solutions to the 2-D subcritical dissipative quasi-geostrophic equation and persistency of the initial regularity

May Ramzi, Ezzeddine Zahrouni

Abstract:
In this article, 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
$$ (\overline{S(\mathbb{R}^2))} ^{B_{\infty }^{1-2\alpha ,\infty }}, \quad 1/2<\alpha <1, $$
then the 2-D 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 $C([0,+\infty [,X)$.

Submitted August 10, 2010. Published January 15, 2011.
Math Subject Classifications: 35Q35, 76D03
Key Words: Quasi-geostrophic equation; Besov Spaces

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May Ramzi
Département de Mathématiques
Faculté des Sciences de Bizerte, Tunisie
email: ramzi.may@fsb.rnu.tn
Ezzeddine Zahrouni
Département de Mathématiques
Faculté des Sciences de Monastir, Tunisie
email: Ezzeddine.Zahrouni@fsm.rnu.tn

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