论文标题
全球保守的解决方案的存在和独特性，用于弱化的camassa-holm方程，随时间加权$ h^1（\ mathbb {r}）$ space
Existence and uniqueness of the globally conservative solutions for a weakly dissipative Camassa-Holm equation in time weighted $H^1(\mathbb{R})$ space
论文作者
论文摘要
在本文中，我们证明了cauchy问题的全球薄弱解决方案的存在和独特性在及时加权$ h^1 $空间的弱耗散的Camassa-Holm方程中。首先，我们通过引入一些新变量来得出等效的半线性系统，并在时间加权$ h^1 $空间中介绍该方程式的全球保守解决方案。其次，我们表明，Pearmon解决方案是$ H^1中的保守弱解决方案。$最后，鉴于保守的解决方案，我们引入了一套针对该特定解决方案量身定制的辅助变量，并证明这些变量满足具有独特解决方案的特定半线性系统。反过来，我们在原始变量中获得了保守解决方案的唯一性。
In this paper, we prove that the existence and uniqueness of globally weak solutions to the Cauchy problem for the weakly dissipative Camassa-Holm equation in time weighted $H^1$ space. First, we derive an equivalent semi-linear system by introducing some new variables, and present the globally conservative solutions of this equation in time weighted $H^1$ space. Second, we show that the peakon solutions are conservative weak solutions in $H^1.$ Finally, given a conservative solution, we introduce a set of auxiliary variables tailored to this particular solution, and prove that these variables satisfy a particular semilinear system having unique solutions. In turn, we get the uniqueness of the conservative solution in the original variables.
