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Functional envelopes relative to the point-open topology on a subset
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 |
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.
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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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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