Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 63, pp. 1-22.
Title: Quasi-geostrophic equations with initial data in Banach
spaces of local measures
Author: Sadek Gala (Univ. d'Evry Val d'Essonne, France)
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 $0< \gamma\leq 1$ 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
$1/2< \gamma\leq1$ 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.
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