论文标题
$ b_ {p，1}^{1} \ cap c^{0,1} $中的Camassa-Holm方程不良
Ill-posedness for the Camassa-Holm equation in $B_{p,1}^{1}\cap C^{0,1}$
论文作者
论文摘要
在本文中，我们研究了真实线上Camassa-Holm方程的库奇问题。通过介绍新的初始数据构造，我们表明较小空间中的解决方案映射$ b_ {p，1}^{1} \ cap c^{0,1} $，$ p \ in（2，\ infty] $在原点不连续。 Camassa-Holm方程在$ W^{1，P} \ CAP C^{0,1} $中具有独特的本地解决方案，但是，对于某些初始数据，该解决方案是稳定的，并且可以在$ b_ {p，1}^{1}^{1}^{1}^{1}^{1}^{1}^{0,1} $中具有通胀。
In this paper, we study the Cauchy problem for the Camassa-Holm equation on the real line. By presenting a new construction of initial data, we show that the solution map in the smaller space $B_{p,1}^{1}\cap C^{0,1}$ with $p\in(2,\infty]$ is discontinuous at origin. More precisely, $u_0\in B_{p,1}^{1}\cap C^{0,1}$ can guarantee that the Camassa-Holm equation has a unique local solution in $W^{1,p}\cap C^{0,1}$, however, this solution is instable and can have an inflation in $B_{p,1}^{1}\cap C^{0,1}$ for certain initial data.
