Carleman estimates and Unique Continuation Property for 1-D viscous Camassa-Holm equation

Previous Article
On the strong-to-strong interaction case for doubly nonlocal Cahn-Hilliard equations
DCDS Home
This Issue
Next Article
The wind-driven ocean circulation: Applying dynamical systems theory to a climate problem

January 2017, 37(1): 169-188. doi: 10.3934/dcds.2017007

Carleman estimates and Unique Continuation Property for 1-D viscous Camassa-Holm equation

School of Mathematics and Statistics, and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, China

* Corresponding author: Peng Gao

Received March 2016 Revised September 2016 Published November 2016

This paper is devoted to studying the 1-D viscous Camassa-Holm equation on a bounded interval. We first deduce the existence and uniqueness of strong solution to the viscous Camassa-Holm equation by using Galerkin method. Then we establish an identity for a second order parabolic operator, by applying this identity we obtain two global Carleman estimates for the linear viscous Camassa-Holm operator. Based on these estimates, we obtain two types of Unique Continuation Property for the viscous Camassa-Holm equation.

Keywords: Viscous Camassa-Holm equation, strong solution, global Carleman estimate, Unique Continuation Property.

Mathematics Subject Classification: Primary:35K55;Secondary:35G2.

Citation: Peng Gao. Carleman estimates and Unique Continuation Property for 1-D viscous Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 169-188. doi: 10.3934/dcds.2017007

References:

[1]	L. Baudouin and J. P. Puel, Uniqueness and stability in an inverse problem for the Schrödinger equation, Inverse Problems, 18 (2002), 1537-1554. doi: 10.1088/0266-5611/18/6/307. Google Scholar
[2]	M. Bellassoued, Logarithmic stability in the dynamical inverse problem for the Schrödinger equation by arbitrary boundary observation, Journal de mathematiques pures et appliquees, 91 (2009), 233-255. doi: 10.1016/j.matpur.2008.06.002. Google Scholar
[3]	R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661. Google Scholar
[4]	T. Carleman, Sur un probléme d'unicité pour les systémes d'équations aux derivées partielles á deux variables independentes, Ark. Mat. Astr.Fys., 2B (1939), 1-9. Google Scholar
[5]	E. Cerpa, A. Mercado and A. F. Pazoto, On the boundary control of a parabolic system coupling KS-KdV and Heat equations, Sci. Ser. A Math. Sci., 22 (2012), 55-74. Google Scholar
[6]	M. Chen and P. Gao, A New Unique Continuation Property for the Korteweg de-Vries Equation, Bull. Aust. Math. Soc., 90 (2014), 90-98. doi: 10.1017/S000497271300110X. Google Scholar
[7]	P. N. da Silva, Unique Continuation for the Kawahara Equation, TEMA Tend. Mat. Apl. Comput., 8 (2007), 463-473. doi: 10.5540/tema.2007.08.03.0463. Google Scholar
[8]	M. Eller, I. Lasiecka and R. Triggiani, Unique continuation for over-determined Kirchoff plate equations and related thermoelastic systems, J. Inverse Ill-Posed Probl., 9 (2001), 103-148. doi: 10.1515/jiip.2001.9.2.103. Google Scholar
[9]	L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998. Google Scholar
[10]	C. Foias, D. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence in: Advances in Nonlinear Mathematics and Science, Phys. D, 152/153 (2001), 505-519. doi: 10.1016/S0167-2789(01)00191-9. Google Scholar
[11]	Y. Fu and B. Guo, Time periodic solution of the viscous Camassa-Holm equation, J. Math. Anal. Appl., 313 (2006), 311-321. doi: 10.1016/j.jmaa.2005.08.073. Google Scholar
[12]	P. Gao, Carleman estimate and unique continuation property for the linear stochastic Korteweg-de Vries equation, Bull. Austral. Math. Soc., 90 (2014), 283-294. doi: 10.1017/S0004972714000276. Google Scholar
[13]	P. Gao, Insensitizing controls for the Cahn-Hilliard type equation, Electron. J. Qual. Theory Differ. Equ., 35 (2014), 1-22. Google Scholar
[14]	P. Gao, A new global Carleman estimate for the one-dimensional Kuramoto-Sivashinsky equation and applications to exact controllability to the trajectories and an inverse problem, Nonlinear Analysis: Theory, Methods & Applications, 117 (2015), 133-147. doi: 10.1016/j.na.2015.01.015. Google Scholar
[15]	P. Gao, M. Chen and Y. Li, Observability estimates and null controllability for forward and backward linear stochastic Kuramoto-Sivashinsky equations, SIAM J. Control Optim., 53 (2015), 475-500. doi: 10.1137/130943820. Google Scholar
[16]	P. Gao, A new global Carleman estimate for Cahn-Hilliard type equation and its applications, J. Differential Equations, 260 (2016), 427-444. doi: 10.1016/j.jde.2015.08.053. Google Scholar
[17]	P. Gao, Local exact controllability to the trajectories of the Swift-Hohenberg equation, Nonlinear Analysis: Theory, Methods & Applications, 139 (2016), 169-195. doi: 10.1016/j.na.2016.02.023. Google Scholar
[18]	P. Gao, Global Carleman estimates for linear stochastic Kawahara equation and their applications, Mathematics of Control, Signals, and Systems, 28 (2016), 1-22. doi: 10.1007/s00498-016-0173-6. Google Scholar
[19]	O. Glass, Controllability and asymptotic stabilization of the Camassa-Holm equation, Journal of Differential Equations, 245 (2008), 1584-1615. doi: 10.1016/j.jde.2008.06.016. Google Scholar
[20]	O. Glass and S. Guerrero, Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit, Asymptot. Anal., 60 (2008), 61-100. Google Scholar
[21]	O. Glass and S. Guerrero, On the controllability of the fifth-order Korteweg-de Vries equation, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 26 (2009), 2181-2209. doi: 10.1016/j.anihpc.2009.01.010. Google Scholar
[22]	D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81. doi: 10.1006/aima.1998.1721. Google Scholar
[23]	D. D. Holm and E. S. Titi, Computational models of turbulence: The lans-$α$ model and the role of global analysis, SIAM News, 38 (2005), 1-5. Google Scholar
[24]	J. U. Kim, On the Stochastic Wave Equation with Nonlinear Damping, Appl. Math. Optim., 58 (2008), 29-67. doi: 10.1007/s00245-007-9029-2. Google Scholar
[25]	K. H. Kwek, H. Gao, W. Zhang and C. Qu, An initial boundary value problem of Camassa-Holm equation, J. Math. Phys., 41 (2000), 8279-8285. doi: 10.1063/1.1288498. Google Scholar
[26]	N. A. Larkin, Modified KdV equation with a source term in a bounded domain, Math. Meth. Appl. Sci., 29 (2006), 751-765. doi: 10.1002/mma.704. Google Scholar
[27]	I. Lasiecka, R. Triggiani and P. F. Yao, Carleman estimates for a plate equation on a Riemann manifold with energy level terms, Analysis and Applications, 10 (2003), 199-236. doi: 10.1007/978-1-4757-3741-7_15. Google Scholar
[28]	W. K. Lim, Global well-posedness for the viscous Camassa-Holm equation, J. Math. Anal. Appl., 326 (2007), 432-442. doi: 10.1016/j.jmaa.2006.01.095. Google Scholar
[29]	J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer-Verlag, NewYork-Heidelberg, 1972. Google Scholar
[30]	S. Liu and R. Triggiani, Global uniqueness in determining electric potentials for a system of strongly coupled Schrödinger equations with magnetic potential terms, J. Inverse Ill-Posed Probl., 19 (2011), 223-254. doi: 10.1515/JIIP.2011.030. Google Scholar
[31]	J. E. Marsden and S. Shkoller, Global well-posedness for the Lagrangian averaged Navier-Stokes (LANS-$ α$) equations on bounded domains, Topological Methods in the Physical Sciences, London, 2000, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 359 (2001), 1449-1468. doi: 10.1098/rsta.2001.0852. Google Scholar
[32]	A. Mercado, A. Osses and L. Rosier, Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights, Inverse Problems, 24 (2008), 015017, 18pp. doi: 10.1088/0266-5611/24/1/015017. Google Scholar
[33]	M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, 2nd edn, Texts in Applied Mathematics, Vol. 13, Springer-Verlag, New York, 2004. Google Scholar
[34]	L. Rosier and B.-Y. Zhang, Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain, SIAM Journal on Control and Optimization, 45 (2006), 927-956. doi: 10.1137/050631409. Google Scholar
[35]	L. Rosier and B.-Y. Zhang, Unique continuation property and control for the Benjamin-Bona-Mahony equation on a periodic domain, Journal of Differential Equations, 254 (2013), 141-178. doi: 10.1016/j.jde.2012.08.014. Google Scholar
[36]	J. Simon, Compact sets in the space $ L^{p}(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360. Google Scholar
[37]	L. Tian, C. Shen and D. Ding, Optimal control of the viscous Camassa-Holm equation, Nonlinear Analysis: Real World Applications, 10 (2009), 519-530. doi: 10.1016/j.nonrwa.2007.10.016. Google Scholar
[38]	R. Triggiani and P. F. Yao, Inverse/observability estimates for Schrödinger equations with variable coefficients, Control and Cybernetics, 28 (1999), 627-664. Google Scholar
[39]	M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 123013, 75pp. doi: 10.1088/0266-5611/25/12/123013. Google Scholar
[40]	G. Yuan and M. Yamamoto, Lipschitz stability in inverse problems for a Kirchhoff plate equation, Asymptotic Analysis, 53 (2007), 29-60. Google Scholar
[41]	X. Zhang, Exact controllability of semilinear plate equations, Asymptotic Analysis, 27 (2001), 95-125. Google Scholar
[42]	X. Zhang and E. Zuazua, A sharp observability inequality for Kirchhoff plate systems with potentials, Comput. Appl. Math., 25 (2006), 353-373. doi: 10.1590/S0101-82052006000200013. Google Scholar
[43]	Z. C. Zhou, Observability estimate and null controllability for one-dimensional fourth order parabolic equation, Taiwanese J. Math., 16 (2012), 1991-2017. Google Scholar

show all references

References:

[1]	L. Baudouin and J. P. Puel, Uniqueness and stability in an inverse problem for the Schrödinger equation, Inverse Problems, 18 (2002), 1537-1554. doi: 10.1088/0266-5611/18/6/307. Google Scholar
[2]	M. Bellassoued, Logarithmic stability in the dynamical inverse problem for the Schrödinger equation by arbitrary boundary observation, Journal de mathematiques pures et appliquees, 91 (2009), 233-255. doi: 10.1016/j.matpur.2008.06.002. Google Scholar
[3]	R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661. Google Scholar
[4]	T. Carleman, Sur un probléme d'unicité pour les systémes d'équations aux derivées partielles á deux variables independentes, Ark. Mat. Astr.Fys., 2B (1939), 1-9. Google Scholar
[5]	E. Cerpa, A. Mercado and A. F. Pazoto, On the boundary control of a parabolic system coupling KS-KdV and Heat equations, Sci. Ser. A Math. Sci., 22 (2012), 55-74. Google Scholar
[6]	M. Chen and P. Gao, A New Unique Continuation Property for the Korteweg de-Vries Equation, Bull. Aust. Math. Soc., 90 (2014), 90-98. doi: 10.1017/S000497271300110X. Google Scholar
[7]	P. N. da Silva, Unique Continuation for the Kawahara Equation, TEMA Tend. Mat. Apl. Comput., 8 (2007), 463-473. doi: 10.5540/tema.2007.08.03.0463. Google Scholar
[8]	M. Eller, I. Lasiecka and R. Triggiani, Unique continuation for over-determined Kirchoff plate equations and related thermoelastic systems, J. Inverse Ill-Posed Probl., 9 (2001), 103-148. doi: 10.1515/jiip.2001.9.2.103. Google Scholar
[9]	L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998. Google Scholar
[10]	C. Foias, D. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence in: Advances in Nonlinear Mathematics and Science, Phys. D, 152/153 (2001), 505-519. doi: 10.1016/S0167-2789(01)00191-9. Google Scholar
[11]	Y. Fu and B. Guo, Time periodic solution of the viscous Camassa-Holm equation, J. Math. Anal. Appl., 313 (2006), 311-321. doi: 10.1016/j.jmaa.2005.08.073. Google Scholar
[12]	P. Gao, Carleman estimate and unique continuation property for the linear stochastic Korteweg-de Vries equation, Bull. Austral. Math. Soc., 90 (2014), 283-294. doi: 10.1017/S0004972714000276. Google Scholar
[13]	P. Gao, Insensitizing controls for the Cahn-Hilliard type equation, Electron. J. Qual. Theory Differ. Equ., 35 (2014), 1-22. Google Scholar
[14]	P. Gao, A new global Carleman estimate for the one-dimensional Kuramoto-Sivashinsky equation and applications to exact controllability to the trajectories and an inverse problem, Nonlinear Analysis: Theory, Methods & Applications, 117 (2015), 133-147. doi: 10.1016/j.na.2015.01.015. Google Scholar
[15]	P. Gao, M. Chen and Y. Li, Observability estimates and null controllability for forward and backward linear stochastic Kuramoto-Sivashinsky equations, SIAM J. Control Optim., 53 (2015), 475-500. doi: 10.1137/130943820. Google Scholar
[16]	P. Gao, A new global Carleman estimate for Cahn-Hilliard type equation and its applications, J. Differential Equations, 260 (2016), 427-444. doi: 10.1016/j.jde.2015.08.053. Google Scholar
[17]	P. Gao, Local exact controllability to the trajectories of the Swift-Hohenberg equation, Nonlinear Analysis: Theory, Methods & Applications, 139 (2016), 169-195. doi: 10.1016/j.na.2016.02.023. Google Scholar
[18]	P. Gao, Global Carleman estimates for linear stochastic Kawahara equation and their applications, Mathematics of Control, Signals, and Systems, 28 (2016), 1-22. doi: 10.1007/s00498-016-0173-6. Google Scholar
[19]	O. Glass, Controllability and asymptotic stabilization of the Camassa-Holm equation, Journal of Differential Equations, 245 (2008), 1584-1615. doi: 10.1016/j.jde.2008.06.016. Google Scholar
[20]	O. Glass and S. Guerrero, Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit, Asymptot. Anal., 60 (2008), 61-100. Google Scholar
[21]	O. Glass and S. Guerrero, On the controllability of the fifth-order Korteweg-de Vries equation, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 26 (2009), 2181-2209. doi: 10.1016/j.anihpc.2009.01.010. Google Scholar
[22]	D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81. doi: 10.1006/aima.1998.1721. Google Scholar
[23]	D. D. Holm and E. S. Titi, Computational models of turbulence: The lans-$α$ model and the role of global analysis, SIAM News, 38 (2005), 1-5. Google Scholar
[24]	J. U. Kim, On the Stochastic Wave Equation with Nonlinear Damping, Appl. Math. Optim., 58 (2008), 29-67. doi: 10.1007/s00245-007-9029-2. Google Scholar
[25]	K. H. Kwek, H. Gao, W. Zhang and C. Qu, An initial boundary value problem of Camassa-Holm equation, J. Math. Phys., 41 (2000), 8279-8285. doi: 10.1063/1.1288498. Google Scholar
[26]	N. A. Larkin, Modified KdV equation with a source term in a bounded domain, Math. Meth. Appl. Sci., 29 (2006), 751-765. doi: 10.1002/mma.704. Google Scholar
[27]	I. Lasiecka, R. Triggiani and P. F. Yao, Carleman estimates for a plate equation on a Riemann manifold with energy level terms, Analysis and Applications, 10 (2003), 199-236. doi: 10.1007/978-1-4757-3741-7_15. Google Scholar
[28]	W. K. Lim, Global well-posedness for the viscous Camassa-Holm equation, J. Math. Anal. Appl., 326 (2007), 432-442. doi: 10.1016/j.jmaa.2006.01.095. Google Scholar
[29]	J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer-Verlag, NewYork-Heidelberg, 1972. Google Scholar
[30]	S. Liu and R. Triggiani, Global uniqueness in determining electric potentials for a system of strongly coupled Schrödinger equations with magnetic potential terms, J. Inverse Ill-Posed Probl., 19 (2011), 223-254. doi: 10.1515/JIIP.2011.030. Google Scholar
[31]	J. E. Marsden and S. Shkoller, Global well-posedness for the Lagrangian averaged Navier-Stokes (LANS-$ α$) equations on bounded domains, Topological Methods in the Physical Sciences, London, 2000, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 359 (2001), 1449-1468. doi: 10.1098/rsta.2001.0852. Google Scholar
[32]	A. Mercado, A. Osses and L. Rosier, Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights, Inverse Problems, 24 (2008), 015017, 18pp. doi: 10.1088/0266-5611/24/1/015017. Google Scholar
[33]	M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, 2nd edn, Texts in Applied Mathematics, Vol. 13, Springer-Verlag, New York, 2004. Google Scholar
[34]	L. Rosier and B.-Y. Zhang, Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain, SIAM Journal on Control and Optimization, 45 (2006), 927-956. doi: 10.1137/050631409. Google Scholar
[35]	L. Rosier and B.-Y. Zhang, Unique continuation property and control for the Benjamin-Bona-Mahony equation on a periodic domain, Journal of Differential Equations, 254 (2013), 141-178. doi: 10.1016/j.jde.2012.08.014. Google Scholar
[36]	J. Simon, Compact sets in the space $ L^{p}(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360. Google Scholar
[37]	L. Tian, C. Shen and D. Ding, Optimal control of the viscous Camassa-Holm equation, Nonlinear Analysis: Real World Applications, 10 (2009), 519-530. doi: 10.1016/j.nonrwa.2007.10.016. Google Scholar
[38]	R. Triggiani and P. F. Yao, Inverse/observability estimates for Schrödinger equations with variable coefficients, Control and Cybernetics, 28 (1999), 627-664. Google Scholar
[39]	M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 123013, 75pp. doi: 10.1088/0266-5611/25/12/123013. Google Scholar
[40]	G. Yuan and M. Yamamoto, Lipschitz stability in inverse problems for a Kirchhoff plate equation, Asymptotic Analysis, 53 (2007), 29-60. Google Scholar
[41]	X. Zhang, Exact controllability of semilinear plate equations, Asymptotic Analysis, 27 (2001), 95-125. Google Scholar
[42]	X. Zhang and E. Zuazua, A sharp observability inequality for Kirchhoff plate systems with potentials, Comput. Appl. Math., 25 (2006), 353-373. doi: 10.1590/S0101-82052006000200013. Google Scholar
[43]	Z. C. Zhou, Observability estimate and null controllability for one-dimensional fourth order parabolic equation, Taiwanese J. Math., 16 (2012), 1991-2017. Google Scholar

[1]	Milena Stanislavova, Atanas Stefanov. Attractors for the viscous Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 159-186. doi: 10.3934/dcds.2007.18.159
[2]	Zhenhua Guo, Mina Jiang, Zhian Wang, Gao-Feng Zheng. Global weak solutions to the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 883-906. doi: 10.3934/dcds.2008.21.883
[3]	Qiaoyi Hu, Zhijun Qiao. Persistence properties and unique continuation for a dispersionless two-component Camassa-Holm system with peakon and weak kink solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2613-2625. doi: 10.3934/dcds.2016.36.2613
[4]	Shaojie Yang, Tianzhou Xu. Symmetry analysis, persistence properties and unique continuation for the cross-coupled Camassa-Holm system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 329-341. doi: 10.3934/dcds.2018016
[5]	Shouming Zhou, Chunlai Mu. Global conservative and dissipative solutions of the generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1713-1739. doi: 10.3934/dcds.2013.33.1713
[6]	Shihui Zhu. Existence and uniqueness of global weak solutions of the Camassa-Holm equation with a forcing. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5201-5221. doi: 10.3934/dcds.2016026
[7]	Laurent Bourgeois. Quantification of the unique continuation property for the heat equation. Mathematical Control & Related Fields, 2017, 7 (3) : 347-367. doi: 10.3934/mcrf.2017012
[8]	Yongsheng Mi, Boling Guo, Chunlai Mu. On an $N$-Component Camassa-Holm equation with peakons. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1575-1601. doi: 10.3934/dcds.2017065
[9]	Helge Holden, Xavier Raynaud. Dissipative solutions for the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1047-1112. doi: 10.3934/dcds.2009.24.1047
[10]	Defu Chen, Yongsheng Li, Wei Yan. On the Cauchy problem for a generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 871-889. doi: 10.3934/dcds.2015.35.871
[11]	Yu Gao, Jian-Guo Liu. The modified Camassa-Holm equation in Lagrangian coordinates. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2545-2592. doi: 10.3934/dcdsb.2018067
[12]	Giuseppe Maria Coclite, Lorenzo Di Ruvo. A note on the convergence of the solution of the high order Camassa-Holm equation to the entropy ones of a scalar conservation law. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1247-1282. doi: 10.3934/dcds.2017052
[13]	Peng Gao. Global Carleman estimate for the Kawahara equation and its applications. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1853-1874. doi: 10.3934/cpaa.2018088
[14]	Stephen C. Anco, Elena Recio, María L. Gandarias, María S. Bruzón. A nonlinear generalization of the Camassa-Holm equation with peakon solutions. Conference Publications, 2015, 2015 (special) : 29-37. doi: 10.3934/proc.2015.0029
[15]	Li Yang, Zeng Rong, Shouming Zhou, Chunlai Mu. Uniqueness of conservative solutions to the generalized Camassa-Holm equation via characteristics. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5205-5220. doi: 10.3934/dcds.2018230
[16]	Yongsheng Mi, Chunlai Mu. On a three-Component Camassa-Holm equation with peakons. Kinetic & Related Models, 2014, 7 (2) : 305-339. doi: 10.3934/krm.2014.7.305
[17]	Feng Wang, Fengquan Li, Zhijun Qiao. On the Cauchy problem for a higher-order μ-Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4163-4187. doi: 10.3934/dcds.2018181
[18]	Danping Ding, Lixin Tian, Gang Xu. The study on solutions to Camassa-Holm equation with weak dissipation. Communications on Pure & Applied Analysis, 2006, 5 (3) : 483-492. doi: 10.3934/cpaa.2006.5.483
[19]	Priscila Leal da Silva, Igor Leite Freire. An equation unifying both Camassa-Holm and Novikov equations. Conference Publications, 2015, 2015 (special) : 304-311. doi: 10.3934/proc.2015.0304
[20]	Stephen Anco, Daniel Kraus. Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4449-4465. doi: 10.3934/dcds.2018194

American Institute of Mathematical Sciences