Jiahong Wu
Quasi-geostrophic Type Equations with Initial Data in Morrey Spaces
(36K, Latex)

ABSTRACT.  This paper studies the well-posedness of the initial 
value problem for the quasi-geostrophic type
equations
$$
\frac{\partial \theta}{\partial t}+u\cdot\nabla\theta +
(-\Delta)^{\gamma}\theta=0,\quad \mbox{on}\quad {\Bbb R}^n
\times (0,\infty),
$$
$$
\theta(x,0)=\theta_0(x), \quad x\in {\Bbb R}^n
$$
where $\gamma(0\le \gamma\le 1)$ is a fixed parameter and
$u=(u_j)$ is divergence free and
determined from $\theta$ through the Riesz transform
$u_j=\pm {\cal R}_{\pi(j)}\theta$ ($\pi(j)$ being a permutation of
$j$, $j=1,2,\cdots,n)$.
The initial data $\theta_0$ is taken in certain Morrey spaces ${\cal M}
_{p,\lambda}({\Bbb R}^n)$ (see text for the definition).
The local well-posedness is proved for
$$
\frac{1}{2}<\gamma \le 1, \quad 1<p<\infty,\quad \lambda=n-(2\gamma-1)p\ge 0
$$
and the solution is global for sufficiently small data.
Furthermore, the solution is shown to be smooth.