We study the Cauchy problem of the 2D viscous shallow water equations in some critical Besov spaces $\dot B^{ \frac{ 2}{ p}}_{ p,1}(\mathbb{ R}^2)\times \dot B^{ \frac{ 2}{ p}-1}_{ p,q}(\mathbb{ R}^2)$. As is known, this system is locally well-posed for large initial data as well as globally well-posed for small initial data in $\dot B^{ \frac{ 2}{ p}}_{ p,1}(\mathbb{ R}^2)\times \dot B^{ \frac{ 2}{ p}-1}_{ p,1}(\mathbb{ R}^2)$ for $p<4$ and ill-posed in $\dot B^{ \frac{ 2}{ p}}_{ p,1}(\mathbb{ R}^2)\times \dot B^{ \frac{ 2}{ p}-1}_{ p,1}(\mathbb{ R}^2)$ for $p>4$. In this paper, we prove that this system is ill-posed for the critical case $p=4$ in the sense of "norm inflation". Furthermore, we also show that the system is ill-posed in $\dot B^{ \frac{ 1}{ 2}}_{ 4,1}(\mathbb{ R}^2)\times \dot B^{ -\frac{ 1}{ 2}}_{ 4,q}(\mathbb{ R}^2)$ for any $q\neq 2$
Categories: math.AP