A DISPERSIVE REGULARIZATION FOR THE MODIFIED CAMASSA-HOLM EQUATION

作者：Gao, Yu^*; Li, Lei; Liu, Jian-Guo

来源：SIAM Journal on Mathematical Analysis, 2018, 50(3): 2807-2838.

DOI：10.1137/17M1132756

• peakon interaction
• dispersive limit
• nonuniqueness
• correct speed of singularity
• selection principle
• weak solutions

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 Npeakon 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.

• 出版日期2018
• 单位哈尔滨工业大学