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

 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.

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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