On one multidimensional compressible nonlocal model of the dissipative QG equations

TL;DR

This paper investigates a multidimensional compressible nonlocal model related to dissipative quasi-geostrophic equations, establishing local and global existence, uniqueness, and decay properties of solutions across various parameter regimes.

Contribution

It introduces new results on existence, uniqueness, and decay of solutions for a novel nonlocal compressible QG model, covering sub-critical, critical, and super-critical cases.

Findings

Local existence and uniqueness of smooth solutions.

Global existence of solutions for sub-critical and critical cases.

Decay rates of solutions as time approaches infinity.

Abstract

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 $t\to\infty$.