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The modified camassa-holm equation in lagrangian coordinates

Publication ,  Journal Article
Gao, Y; Liu, JG
Published in: Discrete and Continuous Dynamical Systems - Series B
August 1, 2018

In this paper, we study the modified Camassa-Holm (mCH) equation in Lagrangian coordinates. For some initial data m0, we show that classical solutions to this equation blow up in finite time Tmax. Before Tmax, existence and uniqueness of classical solutions are established. Lifespan for classical solutions is obtained: Tmax ≥||m0||L∞ ||m0||L1 . And there is a unique solution 1 X(ξ, t) to the Lagrange dynamics which is a strictly monotonic function of ξ for any t ∈ [0, Tmax): Xξ(·, t) > 0. As t approaching Tmax, we prove that the classical solution m(·, t) in Eulerian coordinates has a unique limit m(·, Tmax) in Radon measure space and there is a point ξ0 such that Xξ(ξ0, Tmax) = 0 which means Tmax is an onset time of collisions of characteristics. We also show that in some cases peakons are formed at Tmax. After Tmax, we regularize the Lagrange dynamics to prove global existence of weak solutions m in Radon measure space.

Duke Scholars

Published In

Discrete and Continuous Dynamical Systems - Series B

DOI

ISSN

1531-3492

Publication Date

August 1, 2018

Volume

23

Issue

6

Start / End Page

2545 / 2592

Related Subject Headings

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

Citation

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Gao, Y., & Liu, J. G. (2018). The modified camassa-holm equation in lagrangian coordinates. Discrete and Continuous Dynamical Systems - Series B, 23(6), 2545–2592. https://doi.org/10.3934/dcdsb.2018067
Gao, Y., and J. G. Liu. “The modified camassa-holm equation in lagrangian coordinates.” Discrete and Continuous Dynamical Systems - Series B 23, no. 6 (August 1, 2018): 2545–92. https://doi.org/10.3934/dcdsb.2018067.
Gao Y, Liu JG. The modified camassa-holm equation in lagrangian coordinates. Discrete and Continuous Dynamical Systems - Series B. 2018 Aug 1;23(6):2545–92.
Gao, Y., and J. G. Liu. “The modified camassa-holm equation in lagrangian coordinates.” Discrete and Continuous Dynamical Systems - Series B, vol. 23, no. 6, Aug. 2018, pp. 2545–92. Scopus, doi:10.3934/dcdsb.2018067.
Gao Y, Liu JG. The modified camassa-holm equation in lagrangian coordinates. Discrete and Continuous Dynamical Systems - Series B. 2018 Aug 1;23(6):2545–2592.

Published In

Discrete and Continuous Dynamical Systems - Series B

DOI

ISSN

1531-3492

Publication Date

August 1, 2018

Volume

23

Issue

6

Start / End Page

2545 / 2592

Related Subject Headings

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