97-1 Jiahong Wu

Quasi-geostrophic Type Equations with Initial Data in Morrey Spaces (36K, Latex) Jan 1, 97

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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.

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