论文标题
具有非零背景的修改后的Camassa-Holm方程的全球解决方案的存在
Existence of global solutions for the modified Camassa-Holm equation with a nonzero background
论文作者
论文摘要
本文中的考虑是存在具有非零背景初始值的修饰Camassa-Holm（MCH）方程的全局解决方案。 MCH方程是完全可集成的，可以被视为浅水波单向传播的模型。通过应用cauchy投影操作员的应用应用逆散射变换，在加权Sobolev空间$ h^{2，1}（\ \ mathbb {r}）\ cap h^{1，2}（1，2}（1，2，2，2，2，2，2，2，1，2}（1，1，2}（r MathBB））中Riemann-Hilbert（RH）问题与MCH方程相关的问题。一种关键的技术是在Sobolev空间中得出解决方案的界限$ W^{1，\ infty}（\ Mathbb {r}），然后$然后重建一个新的RH问题，用于Cauchy Reflection Acefflection系数。全局解决方案的规律性是通过对相应RH问题的解决方案的精制估计参数来实现的。
Consideration in the present paper is the existence of global solutions for the modified Camassa-Holm (mCH) equation with a nonzero background initial value. The mCH equation is completely integrable and can be considered as a model for the unidirectional propagation of shallow-water waves. By applying the inverse scattering transform with an application of the Cauchy projection operator, the existence of a unique global solution to the mCH equation in the line with a nonzero background initial value is established in the weighted Sobolev space $ H^{2, 1} (\mathbb{R})\cap H^{1, 2} (\mathbb{R})$ based on the representation of a Riemann-Hilbert (RH) problem associated with the Cauchy problem to the mCH equation. A crucial technique used is to derive the boundedness of the solution in the Sobolev space $ W^{1,\infty}(\mathbb{R}),$ then reconstruct a new RH problem for the Cauchy projection operator of reflection coefficients. The regularity of the global solution is achieved by the refined estimate arguments on those solutions of the corresponding RH problem.
