  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 $1\le\alpha\leq 2$, 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 $t\to\infty$.
