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March  2017, 37(3): 1247-1282. doi: 10.3934/dcds.2017052

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

Giuseppe Maria Coclite 1,, and  Lorenzo Di Ruvo 2,

1. 

Dipartimento di Matematica, Università di Bari, via E. Orabona 4,70125 Bari, Italy
2. 

Dipartimento di Scienze e Metodi dell'Ingegneria, Università di Modena e Reggio Emilia, via G. Amendola 2,42122 Reggio Emilia, Italy

* Corresponding author: G. M. Coclite

Received  April 2016 Revised  May 2016 Published  December 2016

Fund Project: The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilitàe le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

We consider the high order Camassa-Holm equation, which is a non linear dispersive equation of the fifth order. We prove that as the diffusion and dispersion parameters tends to zero, the solutions converge to the entropy ones 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: Singular limit, compensated compactness, Camassa-Holm equation, entropy condition.
Mathematics Subject Classification: Primary:35G25, 35L65;Secondary:35L05.
Citation: 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
References:
[1]

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

[2]

R. M. Chen, Some nonlinear dispersive waves arising in compressible hyperelastic plates, Internat. J. Engrg. Sci., 44 (2006), 1188-1204.  doi: 10.1016/j.ijengsci.2006.08.003.  Google Scholar

[3]

G. M. Coclite and L. di Ruvo, A singural limit problem fro conservation laws related to the Rosenau equation, submitted. Google Scholar

[4]

G. M. Coclite and L. 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 Contin. Dynam. Systems, 36 (2016), 2981-2990.  doi: 10.3934/dcds.2016.36.2981.  Google Scholar

[5]

G. M. Coclite and L. di Ruvo, Convergence of the Ostrovsky equation to the Ostrovsky-Hunter one, J. Differential Equations, 256 (2014), 3245-3277.  doi: 10.1016/j.jde.2014.02.001.  Google Scholar

[6]

G. M. Coclite and L. di Ruvo, Dispersive and diffusive limits for Ostrovsky-Hunter type equations, Nonlinear Differ. Equ. Appl., 22 (2015), 1733-1763.  doi: 10.1007/s00030-015-0342-1.  Google Scholar

[7]

G. M. Coclite and L. di Ruvo, A singular limit problem for the Ibragimov-Shabat equation, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 661-673.  doi: 10.3934/dcdss.2016020.  Google Scholar

[8]

{G. M. Coclite and L. di Ruvo, A singular limit problem for the Rosenau-Korteweg-de Vries-regularized long wave and Rosenau-regularized long wave equations, Adv. Nonlinear Stud., 16 (2016), 421-437.  doi: 10.1515/ans-2015-5034.  Google Scholar

[9]

G. M. Coclite and L. di Ruvo, A singular limit problem for conservation laws related to the Kawahara-Korteweg-de Vries equation, Netw. Heterog. Media, 11 (2016), 281-300.  doi: 10.3934/nhm.2016.11.281.  Google Scholar

[10]

G. M. Coclite and L. di Ruvo, A singular limit problem for conservation laws related to the Kawahara equation, Bull. Sci. Math., 140 (2016), 303-338.  doi: 10.1016/j.bulsci.2015.12.003.  Google Scholar

[11]

G. M. Coclite and L. di Ruvo, Convergence of the Kuramoto-Sinelshchikov equation to the Burgers one, Acta Appl. Math., 145 (2016), 89-113.  doi: 10.1007/s10440-016-0049-2.  Google Scholar

[12]

G. M. Coclite and L. di Ruvo, Convergence of the solutions on the generalized Korteweg-de Vries equation, Math. Model. Anal., 21 (2016), 239-259.  doi: 10.3846/13926292.2016.1150358.  Google Scholar

[13]

G. M. Coclite and L. di Ruvo, Convergence results related to the modified Kawahara equation, Boll. Unione Mat. Ital. (9), 8 (2016), 265-286.  doi: 10.1007/s40574-015-0043-z.  Google Scholar

[14]

G. M. Coclite and L. di Ruvo, On the convergence of the modified Rosenau and the modified Benjamin-Bona-Mahony equations, Comput. Math. Appl., 145 (2016), 89-113.  doi: 10.1007/s10440-016-0049-2.  Google Scholar

[15]

G. M. Coclite and L. di Ruvo, Convergence of the regularized short pulse equation to the short pulse one, submitted. Google Scholar

[16]

G. M. Coclite and L. di Ruvo, A singular limit problem for the Kudryashov-Sinelshchikov equation, to appear on ZAMM Z. Angew. Math. Mech. Google Scholar

[17]

G. M. Coclite and L. di Ruvo, A singular limit problem for conservation laws related to the Rosenau-Korteweg-de Vries equation, to appear on J. Math. Pures Appl. Google Scholar

[18]

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

[19]

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

[20]

A. Constantin, On the inverse spectral problem for the Camassa-Holm equation, J. Funct. Anal., 155 (1998), 352-363.  doi: 10.1006/jfan.1997.3231.  Google Scholar

[21]

A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804.  doi: 10.1007/s00014-003-0785-6.  Google Scholar

[22]

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

[23]

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

[24]

A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982.  doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D.  Google Scholar

[25]

A. Constantin and W. A. Strauss, Stability of a class of solitary waves in compressible elastic rods, Phys. Lett. A, 270 (2000), 140-148.  doi: 10.1016/S0375-9601(00)00255-3.  Google Scholar

[26]

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

[27]

L. di Ruvo, On the Rosenau-Kawahara type equation, submitted. Google Scholar

[28]

F. Gesztesy and H. Holden, Soliton Equations and Their Algebro-Geometric Solutions. Volume Ⅰ: (1 + 1) Dimensional Continuous Systems, Cambridge University Press, Cambridge, 2003 doi: 10.1017/CBO9780511546723.  Google Scholar

[29]

F. Gesztesy and H. Holden, Algebro-geometric solutions of the Camassa-Holm hiererachy, Rev. Mat. Iberoamericana, 19 (2003), 73-142.  doi: 10.4171/RMI/339.  Google Scholar

[30]

O. Glass and F. Sueur, Smoothness of the flow map for low-regularity solutions of the Camassa-Holm equations, Discrete Contin. Dyn. Syst., 33 (2013), 2791-2808.  doi: 10.3934/dcds.2013.33.2791.  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]

H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer, New York, 2011. doi: 10.1007/978-3-642-23911-3.  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]

S. Lai and Y. Wu, The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation, J. Differential Equations, 248 (2010), 2038-2063.  doi: 10.1016/j.jde.2010.01.008.  Google Scholar

[36]

P. G. LeFloch and R. Natalini, Conservation laws with vanishing nonlinear diffusion and dispersion, Nonlinear Anal. Ser. A: Theory Methods, 36 (1999), 213-230.  doi: 10.1016/S0362-546X(98)00012-1.  Google Scholar

[37]

P. Lax and C. D. Levermore, The zero dispersion limit for the Korteweg de Vries equation, Proc. Nat. Acad. Sci. U.S.A., 76 (1979), 3602-3606.  doi: 10.1073/pnas.76.8.3602.  Google Scholar

[38]

Y.G. Lu, Convergence of solutions to nonlinear dispersive equations without convexity conditions, Appl. Anal., 31 (1989), 239-246.  doi: 10.1080/00036818908839828.  Google Scholar

[39]

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

[40]

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

[41]

L. Tartar, Compensated compactness and applications to partial differential equations, In Nonlinear analysis and mechanics: Heriot-Watt Symposium, Pitman, Boston, Mass., 39 (1979), 136-212.   Google Scholar

show all references

References:
[1]

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

[2]

R. M. Chen, Some nonlinear dispersive waves arising in compressible hyperelastic plates, Internat. J. Engrg. Sci., 44 (2006), 1188-1204.  doi: 10.1016/j.ijengsci.2006.08.003.  Google Scholar

[3]

G. M. Coclite and L. di Ruvo, A singural limit problem fro conservation laws related to the Rosenau equation, submitted. Google Scholar

[4]

G. M. Coclite and L. 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 Contin. Dynam. Systems, 36 (2016), 2981-2990.  doi: 10.3934/dcds.2016.36.2981.  Google Scholar

[5]

G. M. Coclite and L. di Ruvo, Convergence of the Ostrovsky equation to the Ostrovsky-Hunter one, J. Differential Equations, 256 (2014), 3245-3277.  doi: 10.1016/j.jde.2014.02.001.  Google Scholar

[6]

G. M. Coclite and L. di Ruvo, Dispersive and diffusive limits for Ostrovsky-Hunter type equations, Nonlinear Differ. Equ. Appl., 22 (2015), 1733-1763.  doi: 10.1007/s00030-015-0342-1.  Google Scholar

[7]

G. M. Coclite and L. di Ruvo, A singular limit problem for the Ibragimov-Shabat equation, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 661-673.  doi: 10.3934/dcdss.2016020.  Google Scholar

[8]

{G. M. Coclite and L. di Ruvo, A singular limit problem for the Rosenau-Korteweg-de Vries-regularized long wave and Rosenau-regularized long wave equations, Adv. Nonlinear Stud., 16 (2016), 421-437.  doi: 10.1515/ans-2015-5034.  Google Scholar

[9]

G. M. Coclite and L. di Ruvo, A singular limit problem for conservation laws related to the Kawahara-Korteweg-de Vries equation, Netw. Heterog. Media, 11 (2016), 281-300.  doi: 10.3934/nhm.2016.11.281.  Google Scholar

[10]

G. M. Coclite and L. di Ruvo, A singular limit problem for conservation laws related to the Kawahara equation, Bull. Sci. Math., 140 (2016), 303-338.  doi: 10.1016/j.bulsci.2015.12.003.  Google Scholar

[11]

G. M. Coclite and L. di Ruvo, Convergence of the Kuramoto-Sinelshchikov equation to the Burgers one, Acta Appl. Math., 145 (2016), 89-113.  doi: 10.1007/s10440-016-0049-2.  Google Scholar

[12]

G. M. Coclite and L. di Ruvo, Convergence of the solutions on the generalized Korteweg-de Vries equation, Math. Model. Anal., 21 (2016), 239-259.  doi: 10.3846/13926292.2016.1150358.  Google Scholar

[13]

G. M. Coclite and L. di Ruvo, Convergence results related to the modified Kawahara equation, Boll. Unione Mat. Ital. (9), 8 (2016), 265-286.  doi: 10.1007/s40574-015-0043-z.  Google Scholar

[14]

G. M. Coclite and L. di Ruvo, On the convergence of the modified Rosenau and the modified Benjamin-Bona-Mahony equations, Comput. Math. Appl., 145 (2016), 89-113.  doi: 10.1007/s10440-016-0049-2.  Google Scholar

[15]

G. M. Coclite and L. di Ruvo, Convergence of the regularized short pulse equation to the short pulse one, submitted. Google Scholar

[16]

G. M. Coclite and L. di Ruvo, A singular limit problem for the Kudryashov-Sinelshchikov equation, to appear on ZAMM Z. Angew. Math. Mech. Google Scholar

[17]

G. M. Coclite and L. di Ruvo, A singular limit problem for conservation laws related to the Rosenau-Korteweg-de Vries equation, to appear on J. Math. Pures Appl. Google Scholar

[18]

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

[19]

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

[20]

A. Constantin, On the inverse spectral problem for the Camassa-Holm equation, J. Funct. Anal., 155 (1998), 352-363.  doi: 10.1006/jfan.1997.3231.  Google Scholar

[21]

A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804.  doi: 10.1007/s00014-003-0785-6.  Google Scholar

[22]

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

[23]

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

[24]

A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982.  doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D.  Google Scholar

[25]

A. Constantin and W. A. Strauss, Stability of a class of solitary waves in compressible elastic rods, Phys. Lett. A, 270 (2000), 140-148.  doi: 10.1016/S0375-9601(00)00255-3.  Google Scholar

[26]

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

[27]

L. di Ruvo, On the Rosenau-Kawahara type equation, submitted. Google Scholar

[28]

F. Gesztesy and H. Holden, Soliton Equations and Their Algebro-Geometric Solutions. Volume Ⅰ: (1 + 1) Dimensional Continuous Systems, Cambridge University Press, Cambridge, 2003 doi: 10.1017/CBO9780511546723.  Google Scholar

[29]

F. Gesztesy and H. Holden, Algebro-geometric solutions of the Camassa-Holm hiererachy, Rev. Mat. Iberoamericana, 19 (2003), 73-142.  doi: 10.4171/RMI/339.  Google Scholar

[30]

O. Glass and F. Sueur, Smoothness of the flow map for low-regularity solutions of the Camassa-Holm equations, Discrete Contin. Dyn. Syst., 33 (2013), 2791-2808.  doi: 10.3934/dcds.2013.33.2791.  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]

H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer, New York, 2011. doi: 10.1007/978-3-642-23911-3.  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]

S. Lai and Y. Wu, The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation, J. Differential Equations, 248 (2010), 2038-2063.  doi: 10.1016/j.jde.2010.01.008.  Google Scholar

[36]

P. G. LeFloch and R. Natalini, Conservation laws with vanishing nonlinear diffusion and dispersion, Nonlinear Anal. Ser. A: Theory Methods, 36 (1999), 213-230.  doi: 10.1016/S0362-546X(98)00012-1.  Google Scholar

[37]

P. Lax and C. D. Levermore, The zero dispersion limit for the Korteweg de Vries equation, Proc. Nat. Acad. Sci. U.S.A., 76 (1979), 3602-3606.  doi: 10.1073/pnas.76.8.3602.  Google Scholar

[38]

Y.G. Lu, Convergence of solutions to nonlinear dispersive equations without convexity conditions, Appl. Anal., 31 (1989), 239-246.  doi: 10.1080/00036818908839828.  Google Scholar

[39]

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

[40]

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

[41]

L. Tartar, Compensated compactness and applications to partial differential equations, In Nonlinear analysis and mechanics: Heriot-Watt Symposium, Pitman, Boston, Mass., 39 (1979), 136-212.   Google Scholar

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