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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 and 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.

[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.

[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.

[4]

G. M. Coclite and L. di Ruvo, A singular limit problem for conservation laws related to the Rosenau equation, submitted.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.

[15]

A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307. doi: 10.1093/imamat/hxs033.

[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.

[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.

[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.

[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.

[20]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586.

[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.

[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.

[23]

C. De Lellis, F. Otto and M. Westdickenberg, Minimal entropy conditions for Burgers equation, Quart. Appl. Math., 62 (2004), 687-700.

[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.

[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.

[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.

[27]

A. A. Himonas and G. Misiolek, The Cauchy problem for an integrable shallow-water equation, Differential Integral Equations, 14 (2001), 821-831.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[39]

M. E. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations, 7 (1982), 959-1000. doi: 10.1080/03605308208820242.

[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.

[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.

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.

[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.

[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.

[4]

G. M. Coclite and L. di Ruvo, A singular limit problem for conservation laws related to the Rosenau equation, submitted.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.

[15]

A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307. doi: 10.1093/imamat/hxs033.

[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.

[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.

[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.

[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.

[20]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586.

[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.

[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.

[23]

C. De Lellis, F. Otto and M. Westdickenberg, Minimal entropy conditions for Burgers equation, Quart. Appl. Math., 62 (2004), 687-700.

[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.

[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.

[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.

[27]

A. A. Himonas and G. Misiolek, The Cauchy problem for an integrable shallow-water equation, Differential Integral Equations, 14 (2001), 821-831.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[39]

M. E. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations, 7 (1982), 959-1000. doi: 10.1080/03605308208820242.

[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.

[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.

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