 971 Jiahong Wu
 Quasigeostrophic Type Equations with Initial Data in Morrey Spaces
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Jan 1, 97

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Abstract. This paper studies the wellposedness of the initial
value problem for the quasigeostrophic 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 wellposedness is proved for
$$
\frac{1}{2}<\gamma \le 1, \quad 1<p<\infty,\quad \lambda=n(2\gamma1)p\ge 0
$$
and the solution is global for sufficiently small data.
Furthermore, the solution is shown to be smooth.
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