96-671 Jiahong Wu
Quasi-geostrophic type equations with weak initial data (23K, Latex) Dec 16, 96
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Abstract. We study the initial value problem for the quasi-geostrophic type equations $$\frac{\partial \theta}{\partial t}+u\cdot\nabla\theta + (-\Delta)^{\lambda}\theta=0,\quad \mbox{on}\quad {\Bbb R}^n \times (0,\infty),$$ $$\theta(x,0)=\theta_0(x), \quad x\in {\Bbb R}^n$$ where $\lambda$ 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 the Sobolev space of negative indices $\dot{L}_{r,p}$ (see the text for the definition). We prove local well-posedness if $\frac{1}{2}<\lambda \le 1$ and $$1<p<\infty, \quad \frac{n}{p}\le 2\lambda -1, \quad r=\frac{n}{p }-(2\lambda-1) (\le 0)$$ The solution is global if $\theta_0$ is sufficiently small.

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