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

Ill-posedness issue for the 2D viscous shallow water equations in some critical Besov spaces