Electronic Journal of Differential Equations,
Vol. 1998(1998), No. 16, pp. 1-10.

Title: Quasi-geostrophic type equations with weak initial data 

Author: Jiahong Wu (Institute for Advanced Study, Princeton, USA)

Abstract: 
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 
$$ 
{1 \over2}<\lambda \le 1,\quad 1<p<\infty, 
\quad {n\over p}\le 2\lambda -1, \quad r={n\over p}-(2\lambda-1) \le 0\,.
$$
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.