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

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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

Abstract
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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:

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[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

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

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