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doi: 10.3934/dcdsb.2021229
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Global weak solutions to the generalized mCH equation via characteristics

Fanqin Zeng 1, , Yu Gao 2, and  Xiaoping Xue 3,,

1. 

School of Mathematics, Harbin Institute of Technology, Harbin 150001, China
2. 

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China
3. 

School of Mathematics, Harbin Institute of Technology, Harbin 150001, China

* Corresponding author

Received  February 2021 Early access September 2021

In this paper, we study the generalized modified Camassa-Holm (gmCH) equation via characteristics. We first change the gmCH equation for unknowns $ (u,m) $ into its Lagrangian dynamics for characteristics $ X(\xi,t) $, where $ \xi\in\mathbb{R} $ is the Lagrangian label. When $ X_\xi(\xi,t)>0 $, we use the solutions to the Lagrangian dynamics to recover the classical solutions with $ m(\cdot,t)\in C_0^k(\mathbb{R}) $ ($ k\in\mathbb{N},\; \; k\geq1 $) to the gmCH equation. The classical solutions $ (u,m) $ to the gmCH equation will blow up if $ \inf_{\xi\in\mathbb{R}}X_\xi(\cdot,T_{\max}) = 0 $ for some $ T_{\max}>0 $. After the blow-up time $ T_{\max} $, we use a double mollification method to mollify the Lagrangian dynamics and construct global weak solutions (with $ m $ in space-time Radon measure space) to the gmCH equation by some space-time BV compactness arguments.

Keywords: Lagrangian dynamics, local classical solutions, global weak solutions, double mollification method.
Mathematics Subject Classification: Primary: 35G25, 35L05; Secondary: 35D30.
Citation: Fanqin Zeng, Yu Gao, Xiaoping Xue. Global weak solutions to the generalized mCH equation via characteristics. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021229
References:
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show all references

References:
[1]

S. C. Anco and E. Recio, A general family of multi-peakon equations and their properties, J. Phys. A, 52 (2019), 125203. doi: 10.1088/1751-8121/ab03dd.  Google Scholar

[2]

A. Bressan, Hyperbolic Systems of Conservation Laws: The One-Dimensional Cauchy Problem, Oxford University Press on Demand, 2000.  Google Scholar

[3]

A. S. Fokas, The Korteweg-de Vries equation and beyond, Acta Appl. Math., 39 (1995), 295-305.  doi: 10.1007/BF00994638.  Google Scholar

[4]

B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa-Holm equation, Phys. D, 95 (1996), 229-243.  doi: 10.1016/0167-2789(96)00048-6.  Google Scholar

[5]

Y. Gao and H. Liu, Global $N$-peakon weak solutions to a family of nonlinear equations, J. Differential Equations, 271 (2021), 343-355.  doi: 10.1016/j.jde.2020.08.042.  Google Scholar

[6]

Y. Gao and J.-G. Liu, The modified Camassa-Holm equation in Lagarange coordinates, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 2545-2592.  doi: 10.3934/dcdsb.2018067.  Google Scholar

[7]

Z. Guo, X. Liu, X. Liu and C. Qu, Stability of peakons for the generalized modified Camassa-Holm equation, J. Differential Equations, 266 (2019), 7749-7779.  doi: 10.1016/j.jde.2018.12.014.  Google Scholar

[8]

X. Liu, Orbital stability of peakons for a modified Camassa-Holm equation with higher-order nonlinearity, Discrete Contin. Dyn. Syst., 38 (2018), 5505-5521.  doi: 10.3934/dcds.2018242.  Google Scholar

[9]

X. Liu, Stability in the energy space of the sum of N peakons for a modified Camassa-Holm equation with higher-order nonlinearity, J. Math. Phys., 59 (2018), 121505. doi: 10.1063/1.5034143.  Google Scholar

[10] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, 2002.   Google Scholar
[11]

P. J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900-1906.  doi: 10.1103/PhysRevE.53.1900.  Google Scholar

[12]

Z. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons, J. Math. Phys., 47 (2006), 112701. doi: 10.1063/1.2365758.  Google Scholar

[13]

M. Yang, Y. Li and Y. Zhao, On the Cauchy problem of generalized Fokas-Olver-Resenau-Qiao equation, Appl. Anal., 97 (2018), 2246-2268.  doi: 10.1080/00036811.2017.1359565.  Google Scholar

[14]

S. Yang, Blow-up phenomena for the generalized FORQ/MCH equation, Z. Angew. Math. Phys., 71 (2020), Paper No. 20, 13 pp. doi: 10.1007/s00033-019-1241-9.  Google Scholar

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