A singular limit problem for conservation laws related to the Kawahara-Korteweg-de Vries equation

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June 2016, 11(2): 281-300. doi: 10.3934/nhm.2016.11.281

A singular limit problem for conservation laws related to the Kawahara-Korteweg-de Vries equation

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

1.	Department of Mathematics, University of Bari, Via E. Orabona 4, I--70125 Bari
2.	Department of Science and Methods for Engineering, University of Modena and Reggio Emilia, via G. Amendola 2, 42122 Reggio Emilia, Italy

Received April 2015 Revised October 2015 Published March 2016

Abstract
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We consider the Kawahara-Korteweg-de Vries equation, which contains nonlinear dispersive effects. We prove that as the dispersion parameter tends to zero, the solutions of the dispersive equation converge to discontinuous weak solutions of the Burgers equation. 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, Kawahara-Korteweg-de Vries equation, entropy condition., compensated compactness.

Mathematics Subject Classification: Primary: 35G25, 35L65; Secondary: 35L0.

Citation: Giuseppe Maria Coclite, Lorenzo di Ruvo. A singular limit problem for conservation laws related to the Kawahara-Korteweg-de Vries equation. Networks and Heterogeneous Media, 2016, 11 (2) : 281-300. doi: 10.3934/nhm.2016.11.281

References:

[1]	A. H. Badali, M. S. Hashemi and M. Ghahremani, Lie symmetry analysis for Kawahara-KdV equations, Computational Methods for Differential Equations, 1 (2013), 135-145.
[2]	D. J. Benney, Long waves on liquid films, J. Math. and Phys., 45 (1966), 150-155. doi: 10.1002/sapm1966451150.
[3]	J. Boyd, Ostrovsky and Hunter's generic wave equation for weakly dispersive waves: matched asymptotic and pseudospectral study of the paraboloidal travelling waves (corner and near-corner waves), Euro. Jnl. of Appl. Math., 16 (2005), 65-81. doi: 10.1017/S0956792504005625.
[4]	G. M. Coclite and L. di Ruvo, A singular limit problem for conservation laws related to the Kawahara equation,, Bull. Sci. Math., (). doi: 10.1016/j.bulsci.2015.12.003.
[5]	G. M. Coclite and L. di Ruvo, A singular limit problem for conservation laws related to the Rosenau-Korteweg-de Vries equation,, submitted., ().
[6]	G. M. Coclite and L. di Ruvo, Convergence of the generalized Kudryashov-Sinelshchikov equation to the Burgers one,, submitted., ().
[7]	G. M. Coclite and L. di Ruvo, A singular limit problem for conservation laws related to the Rosenau equation,, submitted., ().
[8]	G. M. Coclite and L. di Ruvo, A singular limit problem for conservation laws related to the Kudryashov-Sinelshchikov equation,, ZAMM Z. Angew. Math. Mech., ().
[9]	G. M. Coclite and L. di Ruvo, Oleinik type estimate for the Ostrovsky-Hunter equation, J. Math. Anal. Appl., 423 (2015), 162-190. doi: 10.1016/j.jmaa.2014.09.033.
[10]	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.
[11]	G. M. Coclite, L. di Ruvo, J. Ernest and S. Mishra, Convergence of vanishing capillarity approximations for scalar conservation laws with discontinuous fluxes, Netw. Heterog. Media, 8 (2013), 969-984. doi: 10.3934/nhm.2013.8.969.
[12]	G. M. Coclite, L. di Ruvo and K. H. Karlsen, Some wellposedness results for the Ostrovsky-Hunter Equation, in Hyperbolic conservation laws and related analysis with applications, Springer Proc. Math. Stat., Springer, Heidelberg, 49 (2014), 143-159. doi: 10.1007/978-3-642-39007-4_7.
[13]	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.
[14]	A. Corli, C. Rohde and V. Schleper, Parabolic approximations of diffusive-dispersive equations, J. Math. Anal. Appl., 414 (2014), 773-798. doi: 10.1016/j.jmaa.2014.01.049.
[15]	L. di Ruvo, Discontinuous Solutions for the Ostrovsky-Hunter Equation and Two Phase Flows, Ph.D. thesis, University of Bari, 2013.
[16]	T. Kawahara, Oscillatory solitary waves in dispersive media, J. Phys. Soc. Japan, 33 (1972), 260-264. doi: 10.1143/JPSJ.33.260.
[17]	T. Kakutani and H. Ono, Weak non-linear hydromagnetic waves in a cold collision free plasma, J. Phys. Soc. Japan, 26 (1969), 1305-1318. doi: 10.1143/JPSJ.26.1305.
[18]	C. M. Khalique and K. R. Adem, Exact solution of the $(2+1)-$dimensional Zakharov-Kuznetsov modified Equal width equation using Lie group analysis, Computer modelling, 54 (2011), 184-189. doi: 10.1016/j.mcm.2011.01.049.
[19]	P. G. LeFloch and R. Natalini, Conservation laws with vanishing nonlinear diffusion and dispersion, Nonlinear Anal. Ser. A: Theory Methods, 36 (1992), 212-230. doi: 10.1016/S0362-546X(98)00012-1.
[20]	E. Mahdavi, Exp-function method for finding some exact solutions of Rosenau Kawahara and Rosenau Korteweg-de Vries equations, International Journal of Mathematical, Computational, Physical and Quantum Engineering, 8 (2014), 993-999.
[21]	L. Molinet and Y. Wang, Dispersive limit from the Kawahara to the KdV equation, J. Differential Equations, 255 (2013), 2196-2219. doi: 10.1016/j.jde.2013.06.012.
[22]	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.
[23]	F. Natali, A Note on the Stability for Kawahara-KdV Type Equations, Appl. Math. Lett. 23 (2010), 591-596. doi: 10.1016/j.aml.2010.01.017.
[24]	L. A. Ostrovsky, Nonlinear internal waves in a rotating ocean, Okeanologia, 18 (1978), 181-191.
[25]	M. E. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations, 7 (1982), 959-1000. doi: 10.1080/03605308208820242.

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References:

[1]	A. H. Badali, M. S. Hashemi and M. Ghahremani, Lie symmetry analysis for Kawahara-KdV equations, Computational Methods for Differential Equations, 1 (2013), 135-145.
[2]	D. J. Benney, Long waves on liquid films, J. Math. and Phys., 45 (1966), 150-155. doi: 10.1002/sapm1966451150.
[3]	J. Boyd, Ostrovsky and Hunter's generic wave equation for weakly dispersive waves: matched asymptotic and pseudospectral study of the paraboloidal travelling waves (corner and near-corner waves), Euro. Jnl. of Appl. Math., 16 (2005), 65-81. doi: 10.1017/S0956792504005625.
[4]	G. M. Coclite and L. di Ruvo, A singular limit problem for conservation laws related to the Kawahara equation,, Bull. Sci. Math., (). doi: 10.1016/j.bulsci.2015.12.003.
[5]	G. M. Coclite and L. di Ruvo, A singular limit problem for conservation laws related to the Rosenau-Korteweg-de Vries equation,, submitted., ().
[6]	G. M. Coclite and L. di Ruvo, Convergence of the generalized Kudryashov-Sinelshchikov equation to the Burgers one,, submitted., ().
[7]	G. M. Coclite and L. di Ruvo, A singular limit problem for conservation laws related to the Rosenau equation,, submitted., ().
[8]	G. M. Coclite and L. di Ruvo, A singular limit problem for conservation laws related to the Kudryashov-Sinelshchikov equation,, ZAMM Z. Angew. Math. Mech., ().
[9]	G. M. Coclite and L. di Ruvo, Oleinik type estimate for the Ostrovsky-Hunter equation, J. Math. Anal. Appl., 423 (2015), 162-190. doi: 10.1016/j.jmaa.2014.09.033.
[10]	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.
[11]	G. M. Coclite, L. di Ruvo, J. Ernest and S. Mishra, Convergence of vanishing capillarity approximations for scalar conservation laws with discontinuous fluxes, Netw. Heterog. Media, 8 (2013), 969-984. doi: 10.3934/nhm.2013.8.969.
[12]	G. M. Coclite, L. di Ruvo and K. H. Karlsen, Some wellposedness results for the Ostrovsky-Hunter Equation, in Hyperbolic conservation laws and related analysis with applications, Springer Proc. Math. Stat., Springer, Heidelberg, 49 (2014), 143-159. doi: 10.1007/978-3-642-39007-4_7.
[13]	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.
[14]	A. Corli, C. Rohde and V. Schleper, Parabolic approximations of diffusive-dispersive equations, J. Math. Anal. Appl., 414 (2014), 773-798. doi: 10.1016/j.jmaa.2014.01.049.
[15]	L. di Ruvo, Discontinuous Solutions for the Ostrovsky-Hunter Equation and Two Phase Flows, Ph.D. thesis, University of Bari, 2013.
[16]	T. Kawahara, Oscillatory solitary waves in dispersive media, J. Phys. Soc. Japan, 33 (1972), 260-264. doi: 10.1143/JPSJ.33.260.
[17]	T. Kakutani and H. Ono, Weak non-linear hydromagnetic waves in a cold collision free plasma, J. Phys. Soc. Japan, 26 (1969), 1305-1318. doi: 10.1143/JPSJ.26.1305.
[18]	C. M. Khalique and K. R. Adem, Exact solution of the $(2+1)-$dimensional Zakharov-Kuznetsov modified Equal width equation using Lie group analysis, Computer modelling, 54 (2011), 184-189. doi: 10.1016/j.mcm.2011.01.049.
[19]	P. G. LeFloch and R. Natalini, Conservation laws with vanishing nonlinear diffusion and dispersion, Nonlinear Anal. Ser. A: Theory Methods, 36 (1992), 212-230. doi: 10.1016/S0362-546X(98)00012-1.
[20]	E. Mahdavi, Exp-function method for finding some exact solutions of Rosenau Kawahara and Rosenau Korteweg-de Vries equations, International Journal of Mathematical, Computational, Physical and Quantum Engineering, 8 (2014), 993-999.
[21]	L. Molinet and Y. Wang, Dispersive limit from the Kawahara to the KdV equation, J. Differential Equations, 255 (2013), 2196-2219. doi: 10.1016/j.jde.2013.06.012.
[22]	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.
[23]	F. Natali, A Note on the Stability for Kawahara-KdV Type Equations, Appl. Math. Lett. 23 (2010), 591-596. doi: 10.1016/j.aml.2010.01.017.
[24]	L. A. Ostrovsky, Nonlinear internal waves in a rotating ocean, Okeanologia, 18 (1978), 181-191.
[25]	M. E. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations, 7 (1982), 959-1000. doi: 10.1080/03605308208820242.

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