A Dispersive Regularization for the Modified Camassa--Holm Equation

Abstract

In this paper, we present a dispersive regularization approach to construct a global $N$-peakon weak solution to the modified Camassa--Holm equation (mCH) in one dimension. In particular, we perform a double mollification for the system of ODEs describing trajectories of $N$-peakon solutions and obtain $N$ smoothed peakons without collisions. Though the smoothed peakons do not give a solution to the mCH equation, the weak consistency allows us to take the smoothing parameter to zero and the limiting function is a global $N$-peakon weak solution. The trajectories of the peakons in the constructed solution are globally Lipschitz continuous and do not cross each other. When $N=2$, the solution is a sticky peakon weak solution. At last, using the $N$-peakon solutions and through a mean field limit process, we obtain global weak solutions for general initial data $m_0$ in Radon measure space.