二维粘性浅水波系统在~Besov 空间中的~Cauchy 问题

In this paper we consider the Cauchy problem for 2D viscous shallow water system in Besov spaces. We firstly prove the local well-posedness of this problem in $B^s_{p,r}(mathbb{R}^2)$, $s> rac{2}{p}+1$ by using the Littlewood-Paley theory, the Bony decomposition and the theories of transport equations and transport diffusion equations. Then we give a blow-up criterion of solutions to the system in $B^s_{p,r}$, $s> rac{2}{p}+1$. Moreover, by this blow-up criterion, we can prove the global existence of the system with small enough initial data in $B^s_{p,r}(mathbb{R}^2)$, $pleq2$ and $s>1+ rac{2}{p}$. Our obtained results generalize and cover the recent results in cite{W}.