论文标题
在某些关键空间中的二维粘性浅水方程的不适性问题
Ill-posedness issue for the 2D viscous shallow water equations in some critical Besov spaces
论文作者
论文摘要
我们在某个关键的besov空间中研究2D粘性浅水方程的库奇问题$ \ dot b^{\ frac {2} {p}} {p}} _ {p，1}（\ mathbb {r}^2） b^{\ frac {2} {p} -1} _ {p，q}（\ mathbb {r}^2）$。众所周知，该系统在本地适合大型初始数据，并且在全球范围内适用于$ \ dot b^{\ frac {2} {p} {p}}} _ {p，1}（\ mathbb {r}^2） b^{\ frac {2} {p} -1} _ {p，1}（\ Mathbb {r}^2）$ for $ p <4 $，在$ \ dot b^{\ frac {\ frac {2} {p}} {p}} {p}}} _ { b^{\ frac {2} {p} -1} _ {p，1}（\ mathbb {r}^2）$ for $ p> 4 $。在本文中，我们证明该系统在“正常通货膨胀”的意义上是针对关键案例$ p = 4 $的规定。此外，我们还表明，该系统以$ \ dot b^{\ frac {1} {2}}} _ {4,1}（\ Mathbb {r}^2）\ times \ diTs \ dot b^{ - \ frac {1} {2} {2} {2} {2} {2） $ q \ neq 2 $
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 $q\neq 2$
