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 $1\leα\leq 2$, we obtain the global existence and uniqueness results of the nonnegative smooth solution. Then, we prove the global existence of the weak solution for $0\le α\le 2$ and $ν\ge 0$. Finally, for the sub-critical case, we establish the global $H^1$ and $L^p, p>2,$ decay rate of the smooth solution as $t\to\infty$.

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