## A dispersive regularization for the modified camassa–holm equation

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 m0 in Radon measure space.

### Duke Scholars

## Published In

## DOI

## EISSN

## ISSN

## Publication Date

## Volume

## Issue

## Start / End Page

## Related Subject Headings

- Applied Mathematics
- 4904 Pure mathematics
- 4901 Applied mathematics
- 0102 Applied Mathematics
- 0101 Pure Mathematics

### Citation

*SIAM Journal on Mathematical Analysis*,

*50*(3), 2807–2838. https://doi.org/10.1137/17M1132756

*SIAM Journal on Mathematical Analysis*50, no. 3 (January 1, 2018): 2807–38. https://doi.org/10.1137/17M1132756.

*SIAM Journal on Mathematical Analysis*, vol. 50, no. 3, Jan. 2018, pp. 2807–38.

*Scopus*, doi:10.1137/17M1132756.

## Published In

## DOI

## EISSN

## ISSN

## Publication Date

## Volume

## Issue

## Start / End Page

## Related Subject Headings

- Applied Mathematics
- 4904 Pure mathematics
- 4901 Applied mathematics
- 0102 Applied Mathematics
- 0101 Pure Mathematics