On one multidimensional compressible nonlocal model of the dissipative QG equations

In this paper we study the Cauchy problem for one multidimensional compressible nonlocal model of the dissipative quasi-geostrophic equations. First, we obtain the local existence and uniqueness of the smooth non-negative solution or the strong solution in time. Secondly, for the sub-critical and critical case $1\le \alpha \le 2$, we obtain the global existence and uniqueness results of the nonnegative smooth solution. Then, we prove the global existence of the weak solution for $0\le \alpha \le 2$ and $\nu \ge 0$. Finally, for the sub-critical case, we establish the global $H^1$ and $L^p,p>2,$ decay rate of the smooth solution as $t\to \mathrm{\infty }$.