A note on the convergence of the solutions of the Camassa-Holm equation to the entropy ones of a scalar conservation law

Giuseppe Maria Coclite; Lorenzo di  Ruvo

doi:10.3934/dcds.2016.36.2981

Previous Article
A degenerate edge bifurcation in the 1D linearized nonlinear Schrödinger equation
DCDS Home
This Issue
Next Article
Moving recurrent properties for the doubling map on the unit interval

June 2016, 36(6): 2981-2990. doi: 10.3934/dcds.2016.36.2981

A note on the convergence of the solutions of the Camassa-Holm equation to the entropy ones of a scalar conservation law

Giuseppe Maria Coclite ^1, and Lorenzo di Ruvo ^2,

1.	Department of Mathematics, University of Bari, Via E. Orabona 4, I--70125 Bari
2.	Department of Mathematics, University of Bari, via E. Orabona 4, 70125 Bari, Italy

Received July 2015 Revised October 2015 Published December 2015

Abstract
Related Papers

We consider a shallow water equation of Camassa-Holm type, which contains nonlinear dispersive effects. We prove that as the diffusion parameter tends to zero, the solution of the dispersive equation converges to the unique entropy solution of a scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the $L^p$ setting.

Keywords: compensated compactness, Camassa-Holm equation, entropy condition., Singular limit.

Mathematics Subject Classification: Primary: 35G25, 35L65; Secondary: 35L0.

Citation: Giuseppe Maria Coclite, Lorenzo di Ruvo. A note on the convergence of the solutions of the Camassa-Holm equation to the entropy ones of a scalar conservation law. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 2981-2990. doi: 10.3934/dcds.2016.36.2981

References:

[1]	A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z. Google Scholar
[2]	A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Analysis and Applications, 5 (2007), 1-27. doi: 10.1142/S0219530507000857. Google Scholar
[3]	R. Camassa and D. D. Holm, An integrable shallow water equation with peaked soliton, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661. Google Scholar
[4]	G. M. Coclite and L. di Ruvo, A singular limit problem for conservation laws related to the Rosenau equation,, submitted., (). Google Scholar
[5]	G. M. Coclite, F. Gargano and V. Sciacca, Analytic solutions and Singularity formation for the Peakon b-Family equations, Acta Appl. Math., 122 (2012), 419-434. doi: 10.1007/s10440-012-9753-8. Google Scholar
[6]	G. M. Coclite, H. Holden and K. H. Karlsen, Wellposedness of solutions of a parabolic-elliptic system, Discrete Cont. Dynam. Syst., 13 (2005), 659-682. doi: 10.3934/dcds.2005.13.659. Google Scholar
[7]	G. M. Coclite, H. Holden and K. H. Karlsen, Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. Math. Anal., 37 (2005), 1044-1069. doi: 10.1137/040616711. Google Scholar
[8]	G. M. Coclite, H. Holden and K. H. Karlsen, Well-posedness of higher-order Camassa-Holm equations, J. Differential Equations, 246 (2009), 929-963. doi: 10.1016/j.jde.2008.04.014. Google Scholar
[9]	G. M. Coclite, K. H. Karlsen and N. H. Risebro, A convergent finite difference scheme for the Camassa-Holm equation with general $H^1$ initial data, SIAM J. Numer. Anal., 46 (2008), 1554-1579. doi: 10.1137/060673242. Google Scholar
[10]	G. M. Coclite, K. H. Karlsen and N. H. Risebro, An explicit finite difference scheme for the Camassa-Holm equation, Adv. Differ. Equ., 13 (2008), 681-732. Google Scholar
[11]	G. M. Coclite and K. H. Karlsen, A singular limit problem for conservation laws related to the Camassa-Holm shallow water equation, Comm. Partial Differential Equations, 31 (2006), 1253-1272. doi: 10.1080/03605300600781600. Google Scholar
[12]	G. M. Coclite and K. H. Karlsen, A note on the Camassa-Holm equation, J. Differential Equations, 259 (2015), 2158-2166. doi: 10.1016/j.jde.2015.03.020. Google Scholar
[13]	A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362. doi: 10.5802/aif.1757. Google Scholar
[14]	A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5. Google Scholar
[15]	A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307. doi: 10.1093/imamat/hxs033. Google Scholar
[16]	A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 303-328. Google Scholar
[17]	A. Constantin and J. Escher, Global weak solutions for a shallow water equation, Indiana Univ. Math. J., 47 (1998), 1527-1545. doi: 10.1512/iumj.1998.47.1466. Google Scholar
[18]	A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431. doi: 10.1090/S0273-0979-07-01159-7. Google Scholar
[19]	A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12. Google Scholar
[20]	A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586. Google Scholar
[21]	A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2. Google Scholar
[22]	A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61. doi: 10.1007/s002200050801. Google Scholar
[23]	C. De Lellis, F. Otto and M. Westdickenberg, Minimal entropy conditions for Burgers equation, Quart. Appl. Math., 62 (2004), 687-700. Google Scholar
[24]	H.-H. Dai, Exact travelling-wave solutions of an integrable equation arising in hyperelastic rods, Wave Motion, 28 (1998), 367-381. doi: 10.1016/S0165-2125(98)00014-6. Google Scholar
[25]	H.-H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech., 127 (1998), 193-207. doi: 10.1007/BF01170373. Google Scholar
[26]	H. H. Dai and Y. Huo, Solitary shock waves and other travelling waves in a general compressible hyperelastic rod, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 456 (2000), 331-363. doi: 10.1098/rspa.2000.0520. Google Scholar
[27]	A. A. Himonas and G. Misiolek, The Cauchy problem for an integrable shallow-water equation, Differential Integral Equations, 14 (2001), 821-831. Google Scholar
[28]	H. Holden and X. Raynaud, Global conservative solutions of the generalized hyperelastic-rod wave equation, J. Differential Equations, 233 (2007), 448-484. doi: 10.1016/j.jde.2006.09.007. Google Scholar
[29]	H. Holden and X. Raynaud, Global conservative solutions of the Camassa-Holm equation-a Lagrangian point of view, Comm. Partial Differential Equations, 32 (2007), 1511-1549. doi: 10.1080/03605300601088674. Google Scholar
[30]	H. Holden and X. Raynaud, Dissipative solutions for the Camassa-Holm equation, Discrete Contin. Dyn. Syst., 24 (2009), 1047-1112. doi: 10.3934/dcds.2009.24.1047. Google Scholar
[31]	S. Hwang, Singular limit problem of the Camassa-Holm type equation, J. Differential Equations, 235 (2007), 74-84. doi: 10.1016/j.jde.2006.12.011. Google Scholar
[32]	S. Hwang and A. E. Tzavaras, Kinetic decomposition of approximate solutions to conservation laws: Application to relaxation and diffusion-dispersion approximations, Comm. Partial Differential Equations, 27 (2002), 1229-1254. doi: 10.1081/PDE-120004900. Google Scholar
[33]	D. Ionescu-Kruse, Variational derivation of the Camassa-Holm shallow water equation, J. Nonlinear Math. Phys., 14 (2007), 303-312. doi: 10.2991/jnmp.2007.14.3.1. Google Scholar
[34]	R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82. doi: 10.1017/S0022112001007224. Google Scholar
[35]	P. G. LeFloch and R. Natalini, Conservation laws with vanishing nonlinear diffusion and dispersion, Nonlinear Anal. Ser. A: Theory Methods, 36 (1999), 212-230. doi: 10.1016/S0362-546X(98)00012-1. Google Scholar
[36]	A. Y. Li and P. J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63. doi: 10.1006/jdeq.1999.3683. Google Scholar
[37]	F. Murat, L'injection du cône positif de $H^{-1}$ dans $W^{-1,q}$ est compacte pour tout $q<2$, J. Math. Pures Appl. (9), 60 (1981), 309-322. Google Scholar
[38]	G. Rodríguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., Ser. A: Theory Methods, 46 (2001), 309-327. doi: 10.1016/S0362-546X(01)00791-X. Google Scholar
[39]	M. E. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations, 7 (1982), 959-1000. doi: 10.1080/03605308208820242. Google Scholar
[40]	Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433. doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5. Google Scholar
[41]	Z. Xin and P. Zhang, On the uniqueness and large time behavior of the weak solutions to a shallow water equation, Comm. Partial Differential Equations, 27 (2002), 1815-1844. doi: 10.1081/PDE-120016129. Google Scholar

show all references

References:

[1]	A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z. Google Scholar
[2]	A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Analysis and Applications, 5 (2007), 1-27. doi: 10.1142/S0219530507000857. Google Scholar
[3]	R. Camassa and D. D. Holm, An integrable shallow water equation with peaked soliton, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661. Google Scholar
[4]	G. M. Coclite and L. di Ruvo, A singular limit problem for conservation laws related to the Rosenau equation,, submitted., (). Google Scholar
[5]	G. M. Coclite, F. Gargano and V. Sciacca, Analytic solutions and Singularity formation for the Peakon b-Family equations, Acta Appl. Math., 122 (2012), 419-434. doi: 10.1007/s10440-012-9753-8. Google Scholar
[6]	G. M. Coclite, H. Holden and K. H. Karlsen, Wellposedness of solutions of a parabolic-elliptic system, Discrete Cont. Dynam. Syst., 13 (2005), 659-682. doi: 10.3934/dcds.2005.13.659. Google Scholar
[7]	G. M. Coclite, H. Holden and K. H. Karlsen, Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. Math. Anal., 37 (2005), 1044-1069. doi: 10.1137/040616711. Google Scholar
[8]	G. M. Coclite, H. Holden and K. H. Karlsen, Well-posedness of higher-order Camassa-Holm equations, J. Differential Equations, 246 (2009), 929-963. doi: 10.1016/j.jde.2008.04.014. Google Scholar
[9]	G. M. Coclite, K. H. Karlsen and N. H. Risebro, A convergent finite difference scheme for the Camassa-Holm equation with general $H^1$ initial data, SIAM J. Numer. Anal., 46 (2008), 1554-1579. doi: 10.1137/060673242. Google Scholar
[10]	G. M. Coclite, K. H. Karlsen and N. H. Risebro, An explicit finite difference scheme for the Camassa-Holm equation, Adv. Differ. Equ., 13 (2008), 681-732. Google Scholar
[11]	G. M. Coclite and K. H. Karlsen, A singular limit problem for conservation laws related to the Camassa-Holm shallow water equation, Comm. Partial Differential Equations, 31 (2006), 1253-1272. doi: 10.1080/03605300600781600. Google Scholar
[12]	G. M. Coclite and K. H. Karlsen, A note on the Camassa-Holm equation, J. Differential Equations, 259 (2015), 2158-2166. doi: 10.1016/j.jde.2015.03.020. Google Scholar
[13]	A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362. doi: 10.5802/aif.1757. Google Scholar
[14]	A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5. Google Scholar
[15]	A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307. doi: 10.1093/imamat/hxs033. Google Scholar
[16]	A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 303-328. Google Scholar
[17]	A. Constantin and J. Escher, Global weak solutions for a shallow water equation, Indiana Univ. Math. J., 47 (1998), 1527-1545. doi: 10.1512/iumj.1998.47.1466. Google Scholar
[18]	A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431. doi: 10.1090/S0273-0979-07-01159-7. Google Scholar
[19]	A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12. Google Scholar
[20]	A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586. Google Scholar
[21]	A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2. Google Scholar
[22]	A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61. doi: 10.1007/s002200050801. Google Scholar
[23]	C. De Lellis, F. Otto and M. Westdickenberg, Minimal entropy conditions for Burgers equation, Quart. Appl. Math., 62 (2004), 687-700. Google Scholar
[24]	H.-H. Dai, Exact travelling-wave solutions of an integrable equation arising in hyperelastic rods, Wave Motion, 28 (1998), 367-381. doi: 10.1016/S0165-2125(98)00014-6. Google Scholar
[25]	H.-H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech., 127 (1998), 193-207. doi: 10.1007/BF01170373. Google Scholar
[26]	H. H. Dai and Y. Huo, Solitary shock waves and other travelling waves in a general compressible hyperelastic rod, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 456 (2000), 331-363. doi: 10.1098/rspa.2000.0520. Google Scholar
[27]	A. A. Himonas and G. Misiolek, The Cauchy problem for an integrable shallow-water equation, Differential Integral Equations, 14 (2001), 821-831. Google Scholar
[28]	H. Holden and X. Raynaud, Global conservative solutions of the generalized hyperelastic-rod wave equation, J. Differential Equations, 233 (2007), 448-484. doi: 10.1016/j.jde.2006.09.007. Google Scholar
[29]	H. Holden and X. Raynaud, Global conservative solutions of the Camassa-Holm equation-a Lagrangian point of view, Comm. Partial Differential Equations, 32 (2007), 1511-1549. doi: 10.1080/03605300601088674. Google Scholar
[30]	H. Holden and X. Raynaud, Dissipative solutions for the Camassa-Holm equation, Discrete Contin. Dyn. Syst., 24 (2009), 1047-1112. doi: 10.3934/dcds.2009.24.1047. Google Scholar
[31]	S. Hwang, Singular limit problem of the Camassa-Holm type equation, J. Differential Equations, 235 (2007), 74-84. doi: 10.1016/j.jde.2006.12.011. Google Scholar
[32]	S. Hwang and A. E. Tzavaras, Kinetic decomposition of approximate solutions to conservation laws: Application to relaxation and diffusion-dispersion approximations, Comm. Partial Differential Equations, 27 (2002), 1229-1254. doi: 10.1081/PDE-120004900. Google Scholar
[33]	D. Ionescu-Kruse, Variational derivation of the Camassa-Holm shallow water equation, J. Nonlinear Math. Phys., 14 (2007), 303-312. doi: 10.2991/jnmp.2007.14.3.1. Google Scholar
[34]	R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82. doi: 10.1017/S0022112001007224. Google Scholar
[35]	P. G. LeFloch and R. Natalini, Conservation laws with vanishing nonlinear diffusion and dispersion, Nonlinear Anal. Ser. A: Theory Methods, 36 (1999), 212-230. doi: 10.1016/S0362-546X(98)00012-1. Google Scholar
[36]	A. Y. Li and P. J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63. doi: 10.1006/jdeq.1999.3683. Google Scholar
[37]	F. Murat, L'injection du cône positif de $H^{-1}$ dans $W^{-1,q}$ est compacte pour tout $q<2$, J. Math. Pures Appl. (9), 60 (1981), 309-322. Google Scholar
[38]	G. Rodríguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., Ser. A: Theory Methods, 46 (2001), 309-327. doi: 10.1016/S0362-546X(01)00791-X. Google Scholar
[39]	M. E. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations, 7 (1982), 959-1000. doi: 10.1080/03605308208820242. Google Scholar
[40]	Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433. doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5. Google Scholar
[41]	Z. Xin and P. Zhang, On the uniqueness and large time behavior of the weak solutions to a shallow water equation, Comm. Partial Differential Equations, 27 (2002), 1815-1844. doi: 10.1081/PDE-120016129. Google Scholar

[1]	H. A. Erbay, S. Erbay, A. Erkip. The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6101-6116. doi: 10.3934/dcds.2016066
[2]	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, 2017, 37 (3) : 1247-1282. doi: 10.3934/dcds.2017052
[3]	Yongsheng Mi, Boling Guo, Chunlai Mu. Persistence properties for the generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1623-1630. doi: 10.3934/dcdsb.2019243
[4]	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
[5]	Yongsheng Mi, Boling Guo, Chunlai Mu. On an $N$-Component Camassa-Holm equation with peakons. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1575-1601. doi: 10.3934/dcds.2017065
[6]	Helge Holden, Xavier Raynaud. Dissipative solutions for the Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2009, 24 (4) : 1047-1112. doi: 10.3934/dcds.2009.24.1047
[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]	Defu Chen, Yongsheng Li, Wei Yan. On the Cauchy problem for a generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 871-889. doi: 10.3934/dcds.2015.35.871
[9]	Milena Stanislavova, Atanas Stefanov. Attractors for the viscous Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2007, 18 (1) : 159-186. doi: 10.3934/dcds.2007.18.159
[10]	Aiyong Chen, Xinhui Lu. Orbital stability of elliptic periodic peakons for the modified Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1703-1735. doi: 10.3934/dcds.2020090
[11]	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
[12]	Li Yang, Zeng Rong, Shouming Zhou, Chunlai Mu. Uniqueness of conservative solutions to the generalized Camassa-Holm equation via characteristics. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 5205-5220. doi: 10.3934/dcds.2018230
[13]	Shouming Zhou, Chunlai Mu. Global conservative and dissipative solutions of the generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1713-1739. doi: 10.3934/dcds.2013.33.1713
[14]	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
[15]	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
[16]	Feng Wang, Fengquan Li, Zhijun Qiao. On the Cauchy problem for a higher-order μ-Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 4163-4187. doi: 10.3934/dcds.2018181
[17]	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
[18]	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
[19]	Stephen Anco, Daniel Kraus. Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4449-4465. doi: 10.3934/dcds.2018194
[20]	David F. Parker. Higher-order shallow water equations and the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 629-641. doi: 10.3934/dcdsb.2007.7.629

2020 Impact Factor: 1.392

American Institute of Mathematical Sciences

A note on the convergence of the solutions of the Camassa-Holm equation to the entropy ones of a scalar conservation law

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

A note on the convergence of the solutions of the Camassa-Holm equation to the entropy ones of a scalar conservation law

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors