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June  2019, 24(6): 2865-2899. doi: 10.3934/dcdsb.2018290

A note on the convergence of the solution of the Novikov equation

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

1. 

Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, via E. Orabona 4, 70125 Bari, Italy
2. 

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

* Corresponding author: Giuseppe Maria Coclite

Received  July 2017 Revised  June 2018 Published  June 2019 Early access  October 2018

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

We consider the Novikov and Camass-Holm equations, which contain 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: Singular limit, compensated compactness, Camassa-Holm equation, Novikov 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 Novikov equation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2865-2899. doi: 10.3934/dcdsb.2018290
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]

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.

[3]

G. M. Coclite and L. di Ruvo, A singural limit problem fro conservation laws related to the Rosenau equation, Jour. Abstr. differ. Equ. Appl., 8 (2017), 24-47. 

[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, A note on convergence of the solutions of the Benjamin-Bona-Mahony type equations, Nonlinear Anal. Real World Appl., 40 (2018), 64-81.  doi: 10.1016/j.nonrwa.2017.07.014.

[6]

G. M. Coclite and L. di Ruvo, On the convergence of the modified Rosenau and the modified Benjamin-Bona-Mahony equations, Comput. Math. Appl., 74 (2017), 899-919.  doi: 10.1016/j.camwa.2016.02.016.

[7]

G. M. Coclite and L. di Ruvo, Convergence of the regularized short pulse equation to the short pulse one, Math. Nachr., 291 (2018), 774-792.  doi: 10.1002/mana.201600301.

[8]

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.

[9]

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.

[10]

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.

[11]

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. 

[12]

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.

[13]

A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61.  doi: 10.1007/s002200050801.

[14]

P. L. da Silva and I. L. Freire, An equation unifying both Camassa-Holm and Noviokv equation, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications. 10th AIMS Conference. Suppl., (2015), 304–311. doi: 10.3934/proc.2015.0304.

[15]

H. H. Dai and Y. Dai, Solitary shock waves and other travelling waves in a general compressible hyperelastic rod, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 456 (1994), 331-363.  doi: 10.1098/rspa.2000.0520.

[16]

C. De Lellis and F. Otto, Minimal entropy conditions for Burgers equation, Quart. Appl. Math., 62 (2004), 687-700.  doi: 10.1090/qam/2104269.

[17]

A. A. Himonas and C. Holliman, The Cauchy problem for the Novikov equation, Nonlinearity, 25 (2012), 449-479.  doi: 10.1088/0951-7715/25/2/449.

[18]

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

[19]

A. N. W. Hone, H. Lundmark and J. Szmigielski, Explicit multipeakon solutions of Novikovs cubically nonlinear integrable Camassa-Holm type equation, Dyn. Partial Differ. Equ., 6 (2009), 253-289.  doi: 10.4310/DPDE.2009.v6.n3.a3.

[20]

A. N. W. Hone and J. P. Wang, Integrable peakon equations with cubic nonlinearity, J. Phys. A: Math. Theor., 41 (2008), 372002-372012.  doi: 10.1088/1751-8113/41/37/372002.

[21]

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.

[22]

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.

[23]

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.

[24]

P. G. LeFloch and R. Natalini, Conservation laws with vanishing nonlinear diffusion and dispersion, Nonlinear Anal., 36 (1992), 212-230.  doi: 10.1016/S0362-546X(98)00012-1.

[25]

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.

[26]

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. 

[27]

L. Ni and Y. Zhou, Well-posedness and persistence properties for the Novikov equation, J. Diff. Equ., 250 (2011), 3002-3021.  doi: 10.1016/j.jde.2011.01.030.

[28]

V. S. Novikov, Generalizations of the Camassa-Holm equation, J. Phys. A: Math. Theor., 42 (2009), 342002, 14pp. doi: 10.1088/1751-8113/42/34/342002.

[29]

G. Rodriguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46 (2001), 309-327.  doi: 10.1016/S0362-546X(01)00791-X.

[30]

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

[31]

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.

[32]

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.

[33]

X. Wu and Z. Yin, Global weak solutions for the Novikov equation, J. Phys. A: Math. Theor, 44 (2011), 055202, 17pp. doi: 10.1088/1751-8113/44/5/055202.

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]

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.

[3]

G. M. Coclite and L. di Ruvo, A singural limit problem fro conservation laws related to the Rosenau equation, Jour. Abstr. differ. Equ. Appl., 8 (2017), 24-47. 

[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, A note on convergence of the solutions of the Benjamin-Bona-Mahony type equations, Nonlinear Anal. Real World Appl., 40 (2018), 64-81.  doi: 10.1016/j.nonrwa.2017.07.014.

[6]

G. M. Coclite and L. di Ruvo, On the convergence of the modified Rosenau and the modified Benjamin-Bona-Mahony equations, Comput. Math. Appl., 74 (2017), 899-919.  doi: 10.1016/j.camwa.2016.02.016.

[7]

G. M. Coclite and L. di Ruvo, Convergence of the regularized short pulse equation to the short pulse one, Math. Nachr., 291 (2018), 774-792.  doi: 10.1002/mana.201600301.

[8]

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.

[9]

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.

[10]

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.

[11]

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. 

[12]

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.

[13]

A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61.  doi: 10.1007/s002200050801.

[14]

P. L. da Silva and I. L. Freire, An equation unifying both Camassa-Holm and Noviokv equation, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications. 10th AIMS Conference. Suppl., (2015), 304–311. doi: 10.3934/proc.2015.0304.

[15]

H. H. Dai and Y. Dai, Solitary shock waves and other travelling waves in a general compressible hyperelastic rod, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 456 (1994), 331-363.  doi: 10.1098/rspa.2000.0520.

[16]

C. De Lellis and F. Otto, Minimal entropy conditions for Burgers equation, Quart. Appl. Math., 62 (2004), 687-700.  doi: 10.1090/qam/2104269.

[17]

A. A. Himonas and C. Holliman, The Cauchy problem for the Novikov equation, Nonlinearity, 25 (2012), 449-479.  doi: 10.1088/0951-7715/25/2/449.

[18]

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

[19]

A. N. W. Hone, H. Lundmark and J. Szmigielski, Explicit multipeakon solutions of Novikovs cubically nonlinear integrable Camassa-Holm type equation, Dyn. Partial Differ. Equ., 6 (2009), 253-289.  doi: 10.4310/DPDE.2009.v6.n3.a3.

[20]

A. N. W. Hone and J. P. Wang, Integrable peakon equations with cubic nonlinearity, J. Phys. A: Math. Theor., 41 (2008), 372002-372012.  doi: 10.1088/1751-8113/41/37/372002.

[21]

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.

[22]

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.

[23]

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.

[24]

P. G. LeFloch and R. Natalini, Conservation laws with vanishing nonlinear diffusion and dispersion, Nonlinear Anal., 36 (1992), 212-230.  doi: 10.1016/S0362-546X(98)00012-1.

[25]

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.

[26]

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. 

[27]

L. Ni and Y. Zhou, Well-posedness and persistence properties for the Novikov equation, J. Diff. Equ., 250 (2011), 3002-3021.  doi: 10.1016/j.jde.2011.01.030.

[28]

V. S. Novikov, Generalizations of the Camassa-Holm equation, J. Phys. A: Math. Theor., 42 (2009), 342002, 14pp. doi: 10.1088/1751-8113/42/34/342002.

[29]

G. Rodriguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46 (2001), 309-327.  doi: 10.1016/S0362-546X(01)00791-X.

[30]

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

[31]

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.

[32]

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.

[33]

X. Wu and Z. Yin, Global weak solutions for the Novikov equation, J. Phys. A: Math. Theor, 44 (2011), 055202, 17pp. doi: 10.1088/1751-8113/44/5/055202.

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