## Quasi-geostrophic type equations with weak initial data

### Summary

Summary: 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 $${1 \over2}<\lambda \le 1,\quad 1 less than p less than \infty, \quad {n\over p}\le 2\lambda -1, \quad r={n\over p}-(2\lambda-1) \le 0\,.$$ We also prove that the solution is global if $\theta_0$ is sufficiently small.

### Mathematics Subject Classification

35K22, 35Q35, 76U05

### Keywords/Phrases

quasi-geostrophic equations, weak data, well-posedness