Electron. J. Diff. Eqns., Vol. 2005(2005), No. 63, pp. 1-22.

Quasi-geostrophic equations with initial data in Banach spaces of local measures

Sadek Gala

Abstract:
This paper studies the well posedness of the initial value problem for the quasi-geostrophic type equations
$$\displaylines{ \partial_{t}\theta+u\nabla\theta+( -\Delta) ^{\gamma}\theta =0 \quad \hbox{on }\mathbb{R}^{d}\times] 0,+\infty[\cr \theta( x,0) =\theta_{0}(x), \quad x\in\mathbb{R}^{d} }$$
where 0 less than $\gamma\leq 1$ is a fixed parameter and the velocity field $u=(u_{1},u_{2},\dots,u_{d}) $ is divergence free; i.e., $\nabla u=0)$. The initial data $\theta_{0}$ is taken in Banach spaces of local measures (see text for the definition), such as Multipliers, Lorentz and Morrey-Campanato spaces. We will focus on the subcritical case 1/2 less than $\gamma\leq1$ and we analyse the well-posedness of the system in three basic spaces: $L^{d/r,\infty}$, $\dot {X}_{r}$ and $\dot {M}^{p,d/r}$, when the solution is global for sufficiently small initial data. Furtheremore, we prove that the solution is actually smooth. Mild solutions are obtained in several spaces with the right homogeneity to allow the existence of self-similar solutions.

After publication, the managing editors were informed that key results in this manuscript were ''borrowed" from the article below, without the proper citation. The reader should be aware of this claim and decide on her/his own about the originality of this article.
[1] Jose A. Carrillo, Lucas C. F. Ferreira; Self-similar solutions and large time asymptotics for the dissipative quasi-geostrophic equations, HYKE Preprints, 2005-10, http://www.hyke.org/preprint/index.php

Submitted April 24, 2005. Published June 15, 2005.
Math Subject Classifications: 35Q35, 35A07.
Key Words: Quasi-geostrophic equation; local spaces; mild solutions; self-similar solutions

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Sadek Gala
Université d'Evry Val d'Essonne
Département de Mathématiques
Bd F. Mitterrand. 91025 Evry Cedex. France
email: Sadek.Gala@maths.univ-evry.fr

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