American Institute of Mathematical Sciences

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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

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 - A, 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.5802/aif.1757. Google Scholar

[14]

A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523. 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. 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. Google Scholar

[17]

A. Constantin and J. Escher, Global weak solutions for a shallow water equation,, Indiana Univ. Math. J., 47 (1998), 1527. 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. 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. 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. 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. doi: 10.1007/s00205-008-0128-2. Google Scholar

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A. Constantin and L. Molinet, Global weak solutions for a shallow water equation,, Comm. Math. Phys., 211 (2000), 45. 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. Google Scholar

[24]

H.-H. Dai, Exact travelling-wave solutions of an integrable equation arising in hyperelastic rods,, Wave Motion, 28 (1998), 367. 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. 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. 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. Google Scholar

[28]

H. Holden and X. Raynaud, Global conservative solutions of the generalized hyperelastic-rod wave equation,, J. Differential Equations, 233 (2007), 448. 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. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar

[38]

G. Rodríguez-Blanco, On the Cauchy problem for the Camassa-Holm equation,, Nonlinear Anal., 46 (2001), 309. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.5802/aif.1757. Google Scholar

[14]

A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523. 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. 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. Google Scholar

[17]

A. Constantin and J. Escher, Global weak solutions for a shallow water equation,, Indiana Univ. Math. J., 47 (1998), 1527. 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. 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. 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. 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. 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. 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. Google Scholar

[24]

H.-H. Dai, Exact travelling-wave solutions of an integrable equation arising in hyperelastic rods,, Wave Motion, 28 (1998), 367. 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. 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. 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. Google Scholar

[28]

H. Holden and X. Raynaud, Global conservative solutions of the generalized hyperelastic-rod wave equation,, J. Differential Equations, 233 (2007), 448. 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. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar

[38]

G. Rodríguez-Blanco, On the Cauchy problem for the Camassa-Holm equation,, Nonlinear Anal., 46 (2001), 309. 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. 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. 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. doi: 10.1081/PDE-120016129. Google Scholar

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