# American Institute of Mathematical Sciences

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

 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.

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 and Continuous Dynamical Systems, 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. [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. [3] G. M. Coclite and L. di Ruvo, A singural limit problem fro conservation laws related to the Rosenau equation, submitted. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [15] G. M. Coclite and L. di Ruvo, Convergence of the regularized short pulse equation to the short pulse one, submitted. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [27] L. di Ruvo, On the Rosenau-Kawahara type equation, submitted. [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. [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. [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. [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] H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer, New York, 2011. doi: 10.1007/978-3-642-23911-3. [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] 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. [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. [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. [38] Y.G. Lu, Convergence of solutions to nonlinear dispersive equations without convexity conditions, Appl. Anal., 31 (1989), 239-246.  doi: 10.1080/00036818908839828. [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. [40] M. E. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations, 7 (1982), 959-1000.  doi: 10.1080/03605308208820242. [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.

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. [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. [3] G. M. Coclite and L. di Ruvo, A singural limit problem fro conservation laws related to the Rosenau equation, submitted. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [15] G. M. Coclite and L. di Ruvo, Convergence of the regularized short pulse equation to the short pulse one, submitted. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [27] L. di Ruvo, On the Rosenau-Kawahara type equation, submitted. [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. [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. [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. [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] H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer, New York, 2011. doi: 10.1007/978-3-642-23911-3. [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] 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. [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. [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. [38] Y.G. Lu, Convergence of solutions to nonlinear dispersive equations without convexity conditions, Appl. Anal., 31 (1989), 239-246.  doi: 10.1080/00036818908839828. [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. [40] M. E. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations, 7 (1982), 959-1000.  doi: 10.1080/03605308208820242. [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.
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