Electron. J. Diff. Eqns., Vol. 1998(1998), No. 16, pp. 1-10.

Quasi-geostrophic type equations with weak initial data

Jiahong Wu

We study the initial value problem for the quasi-geostrophic type equations
$$ \displaylines{
{\partial \theta \over \partial t}+u\cdot\nabla\theta +
(-\Delta)^{\lambda}\theta=0,\quad \hbox{on } {\Bbb R}^n\times (0,\infty), \cr
\theta(x,0)=\theta_0(x), \quad x\in {\Bbb R}^n\,, \cr}
where $\lambda$, ( $0\leq \lambda \leq 1$) is a fixed parameter and $u=(u_j)$ is divergence free and determined from $\theta$ through the Riesz transform $u_j=\pm {\cal R}_{\pi(j)}\theta$, with $\pi(j)$ a permutation of $1,2,\cdots,n$. The initial data $\theta_0$ is taken in the Sobolev space $\dot{L}_{r,p}$ with negative indices. We prove local well-posedness when
{1 \over2}<\lambda \le 1,\quad 1 less than p less than \infty, 
\quad {n\over p}\le 2\lambda -1, \quad r={n\over p}-(2\lambda-1) \le 0\,.
We also prove that the solution is global if $\theta_0$ is sufficiently small.

Submitted November 26, 1996. Published June 12, 1998.
Math Subject Classification: 35K22, 35Q35, 76U05.
Key Words: Quasi-geostrophic equations, Weak data, Well-posedness.

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Jiahong Wu
School of Mathematics, Institute for Advanced Study
Princeton, NJ 08540. USA
e-mail: jiahong@math.utexas.edu

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