doi: 10.3934/dcdsb.2021229
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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:
[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. GuoX. LiuX. 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. YangY. 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

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. GuoX. LiuX. 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. YangY. 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

[1]

Kapil Kumar Choudhary, Rajiv Kumar, Rajesh Kumar. Global classical and weak solutions of the prion proliferation model in the presence of chaperone in a banach space. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021039

[2]

Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure & Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35

[3]

Hongjun Yu. Global classical solutions to the Boltzmann equation with external force. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1647-1668. doi: 10.3934/cpaa.2009.8.1647

[4]

Fucai Li, Yue Li. Global weak solutions for a kinetic-fluid model with local alignment force in a bounded domain. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021122

[5]

Bingyuan Huang, Shijin Ding, Huanyao Wen. Local classical solutions of compressible Navier-Stokes-Smoluchowski equations with vacuum. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1717-1752. doi: 10.3934/dcdss.2016072

[6]

Yuming Chu, Yihang Hao, Xiangao Liu. Global weak solutions to a general liquid crystals system. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2681-2710. doi: 10.3934/dcds.2013.33.2681

[7]

Zhenhua Guo, Mina Jiang, Zhian Wang, Gao-Feng Zheng. Global weak solutions to the Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 883-906. doi: 10.3934/dcds.2008.21.883

[8]

Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085

[9]

Xiangdong Zhao. Global boundedness of classical solutions to a logistic chemotaxis system with singular sensitivity. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 5095-5100. doi: 10.3934/dcdsb.2020334

[10]

Md. Rabiul Haque, Takayoshi Ogawa, Ryuichi Sato. Existence of weak solutions to a convection–diffusion equation in a uniformly local lebesgue space. Communications on Pure & Applied Analysis, 2020, 19 (2) : 677-697. doi: 10.3934/cpaa.2020031

[11]

Monica Marras, Nicola Pintus, Giuseppe Viglialoro. On the lifespan of classical solutions to a non-local porous medium problem with nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - S, 2020, 13 (7) : 2033-2045. doi: 10.3934/dcdss.2020156

[12]

Shijin Ding, Boling Guo, Junyu Lin, Ming Zeng. Global existence of weak solutions for Landau-Lifshitz-Maxwell equations. Discrete & Continuous Dynamical Systems, 2007, 17 (4) : 867-890. doi: 10.3934/dcds.2007.17.867

[13]

Hafedh Bousbih. Global weak solutions for a coupled chemotaxis non-Newtonian fluid. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 907-929. doi: 10.3934/dcdsb.2018212

[14]

Shihui Zhu. Existence and uniqueness of global weak solutions of the Camassa-Holm equation with a forcing. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 5201-5221. doi: 10.3934/dcds.2016026

[15]

Jonatan Lenells. Weak geodesic flow and global solutions of the Hunter-Saxton equation. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 643-656. doi: 10.3934/dcds.2007.18.643

[16]

Bo Su and Martin Burger. Global weak solutions of non-isothermal front propagation problem. Electronic Research Announcements, 2007, 13: 46-52.

[17]

Zhen Lei, Yi Zhou. BKM's criterion and global weak solutions for magnetohydrodynamics with zero viscosity. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 575-583. doi: 10.3934/dcds.2009.25.575

[18]

Bernard Ducomet, Eduard Feireisl, Hana Petzeltová, Ivan Straškraba. Global in time weak solutions for compressible barotropic self-gravitating fluids. Discrete & Continuous Dynamical Systems, 2004, 11 (1) : 113-130. doi: 10.3934/dcds.2004.11.113

[19]

Chien-Hong Cho, Marcus Wunsch. Global weak solutions to the generalized Proudman-Johnson equation. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1387-1396. doi: 10.3934/cpaa.2012.11.1387

[20]

Kristian Moring, Christoph Scheven, Sebastian Schwarzacher, Thomas Singer. Global higher integrability of weak solutions of porous medium systems. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1697-1745. doi: 10.3934/cpaa.2020069

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (29)
  • HTML views (29)
  • Cited by (0)

Other articles
by authors

[Back to Top]