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

TL;DR

This paper investigates the ill-posedness of the 2D viscous shallow water equations in certain critical Besov spaces, demonstrating norm inflation at the critical case and in related spaces, highlighting limitations of well-posedness.

Contribution

It proves ill-posedness in the critical Besov space case $p=4$ and extends the ill-posedness results to a broader class of Besov spaces for the 2D viscous shallow water equations.

Findings

Ill-posedness for $p=4$ in the sense of norm inflation.

Ill-posedness in $ ext{Besov space}$ for $q eq 2$.

Limitations of well-posedness in critical Besov spaces.

Abstract

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 $q\neq 2$