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