Existence of global solutions to the modified Camassa-Holm equation on the line

Yiling Yang, Gaozhan Li, Engui Fan

Published 2022-07-26Version 1

We address the existence of global solutions to the Cauchy problem of the modified Camassa-Holm (mCH) equation on the line \begin{align} &m_{t}+\left(m\left(u^{2}-u_{x}^{2}\right)\right)_{x}=0, \quad m=u-u_{xx}, \nonumber \\ &u(x,t=0)=u_0(x), \nonumber \end{align} where initial data $( u_0(x), m_0(x)-1) \in H^{4}(\mathbb{R} )\times H^{2,1}(\mathbb{R} )$.Our main technical tool is the inverse scattering transform method based on the representation of a Riemann-Hilbert (RH) problem associated with the above Cauchy problem. The existence and uniqueness of the RH problem is shown via a generalization of Zhou Vanishing Lemma. Based on Cauchy integral projection of reflection coefficients and solutions of the time evolution RH problem, the reconstruction formula is directly used to obtain a unique global solution $( u(x,t), m(x,t) -1) \in C ([0, +\infty); W^{2,\infty}(\mathbb{R} ) \times W^{2,\infty}(\mathbb{R} ) )$ to the Cauchy problem for the mCH equation. Further by establishing a new RH problem and its more careful estimates, we obtain regularity of the solution and a unique global solution $( u(x,t), m(x,t) -1) \in C ([0, +\infty); H^{4}(\mathbb{R} ) \times H^{2,1}(\mathbb{R} ) )$ to the Cauchy problem for the mCH equation.