Quasi-geostrophic type equations with weak initial data

Abstract

We study the initial value problem for the quasi-geostrophic type equations $$ displaylines{ {partial heta over partial t}+ucdotablaheta + (-Delta)^{lambda}heta=0,quad hbox{on } {Bbb R}^nimes (0,infty), cr heta(x,0)=heta_0(x), quad xin {Bbb R}^n,, cr} $$ where $lambda$, ($0leq lambda leq 1$) is a fixed parameter and $u=(u_j)$ is divergence free and determined from $heta$ through the Riesz transform $u_j=pm {cal R}_{pi(j)}heta$, with $pi(j)$ a permutation of $1,2,cdots,n$. The initial data $heta_0$ is taken in the Sobolev space $dot{L}_{r,p}$ with negative indices. We prove local well-posedness when $$ {1 over2}<lambda le 1,quad 1<p<infty, quad {nover p}le 2lambda -1, quad r={nover p}-(2lambda-1) le 0,. $$ We also prove that the solution is global if $heta_0$ is sufficiently small.

Keywords

• Quasi-geostrophic equations
• Weak data
• Well-posedness.