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

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

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.
Citation: Giuseppe Maria Coclite, Lorenzo di Ruvo. A singular limit problem for conservation laws related to the Kawahara-Korteweg-de Vries equation. Networks & 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. Google Scholar

[2]

D. J. Benney, Long waves on liquid films,, J. Math. and Phys., 45 (1966), 150. doi: 10.1002/sapm1966451150. Google Scholar

[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. doi: 10.1017/S0956792504005625. Google Scholar

[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. Google Scholar

[5]

G. M. Coclite and L. di Ruvo, A singular limit problem for conservation laws related to the Rosenau-Korteweg-de Vries equation,, submitted., (). Google Scholar

[6]

G. M. Coclite and L. di Ruvo, Convergence of the generalized Kudryashov-Sinelshchikov equation to the Burgers one,, submitted., (). Google Scholar

[7]

G. M. Coclite and L. di Ruvo, A singular limit problem for conservation laws related to the Rosenau equation,, submitted., (). Google Scholar

[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., (). Google Scholar

[9]

G. M. Coclite and L. di Ruvo, Oleinik type estimate for the Ostrovsky-Hunter equation,, J. Math. Anal. Appl., 423 (2015), 162. doi: 10.1016/j.jmaa.2014.09.033. Google Scholar

[10]

G. M. Coclite and L. di Ruvo, Convergence of the Ostrovsky equation to the Ostrovsky-Hunter one,, J. Differential Equations, 256 (2014), 3245. doi: 10.1016/j.jde.2014.02.001. Google Scholar

[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. doi: 10.3934/nhm.2013.8.969. Google Scholar

[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, 49 (2014), 143. doi: 10.1007/978-3-642-39007-4_7. Google Scholar

[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. doi: 10.1080/03605300600781600. Google Scholar

[14]

A. Corli, C. Rohde and V. Schleper, Parabolic approximations of diffusive-dispersive equations,, J. Math. Anal. Appl., 414 (2014), 773. doi: 10.1016/j.jmaa.2014.01.049. Google Scholar

[15]

L. di Ruvo, Discontinuous Solutions for the Ostrovsky-Hunter Equation and Two Phase Flows,, Ph.D. thesis, (2013). Google Scholar

[16]

T. Kawahara, Oscillatory solitary waves in dispersive media,, J. Phys. Soc. Japan, 33 (1972), 260. doi: 10.1143/JPSJ.33.260. Google Scholar

[17]

T. Kakutani and H. Ono, Weak non-linear hydromagnetic waves in a cold collision free plasma,, J. Phys. Soc. Japan, 26 (1969), 1305. doi: 10.1143/JPSJ.26.1305. Google Scholar

[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. doi: 10.1016/j.mcm.2011.01.049. Google Scholar

[19]

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

[20]

E. Mahdavi, Exp-function method for finding some exact solutions of Rosenau Kawahara and Rosenau Korteweg-de Vries equations,, International Journal of Mathematical, 8 (2014), 993. Google Scholar

[21]

L. Molinet and Y. Wang, Dispersive limit from the Kawahara to the KdV equation,, J. Differential Equations, 255 (2013), 2196. doi: 10.1016/j.jde.2013.06.012. Google Scholar

[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. Google Scholar

[23]

F. Natali, A Note on the Stability for Kawahara-KdV Type Equations,, Appl. Math. Lett. 23 (2010), 23 (2010), 591. doi: 10.1016/j.aml.2010.01.017. Google Scholar

[24]

L. A. Ostrovsky, Nonlinear internal waves in a rotating ocean,, Okeanologia, 18 (1978), 181. Google Scholar

[25]

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

show all references

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. Google Scholar

[2]

D. J. Benney, Long waves on liquid films,, J. Math. and Phys., 45 (1966), 150. doi: 10.1002/sapm1966451150. Google Scholar

[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. doi: 10.1017/S0956792504005625. Google Scholar

[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. Google Scholar

[5]

G. M. Coclite and L. di Ruvo, A singular limit problem for conservation laws related to the Rosenau-Korteweg-de Vries equation,, submitted., (). Google Scholar

[6]

G. M. Coclite and L. di Ruvo, Convergence of the generalized Kudryashov-Sinelshchikov equation to the Burgers one,, submitted., (). Google Scholar

[7]

G. M. Coclite and L. di Ruvo, A singular limit problem for conservation laws related to the Rosenau equation,, submitted., (). Google Scholar

[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., (). Google Scholar

[9]

G. M. Coclite and L. di Ruvo, Oleinik type estimate for the Ostrovsky-Hunter equation,, J. Math. Anal. Appl., 423 (2015), 162. doi: 10.1016/j.jmaa.2014.09.033. Google Scholar

[10]

G. M. Coclite and L. di Ruvo, Convergence of the Ostrovsky equation to the Ostrovsky-Hunter one,, J. Differential Equations, 256 (2014), 3245. doi: 10.1016/j.jde.2014.02.001. Google Scholar

[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. doi: 10.3934/nhm.2013.8.969. Google Scholar

[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, 49 (2014), 143. doi: 10.1007/978-3-642-39007-4_7. Google Scholar

[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. doi: 10.1080/03605300600781600. Google Scholar

[14]

A. Corli, C. Rohde and V. Schleper, Parabolic approximations of diffusive-dispersive equations,, J. Math. Anal. Appl., 414 (2014), 773. doi: 10.1016/j.jmaa.2014.01.049. Google Scholar

[15]

L. di Ruvo, Discontinuous Solutions for the Ostrovsky-Hunter Equation and Two Phase Flows,, Ph.D. thesis, (2013). Google Scholar

[16]

T. Kawahara, Oscillatory solitary waves in dispersive media,, J. Phys. Soc. Japan, 33 (1972), 260. doi: 10.1143/JPSJ.33.260. Google Scholar

[17]

T. Kakutani and H. Ono, Weak non-linear hydromagnetic waves in a cold collision free plasma,, J. Phys. Soc. Japan, 26 (1969), 1305. doi: 10.1143/JPSJ.26.1305. Google Scholar

[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. doi: 10.1016/j.mcm.2011.01.049. Google Scholar

[19]

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

[20]

E. Mahdavi, Exp-function method for finding some exact solutions of Rosenau Kawahara and Rosenau Korteweg-de Vries equations,, International Journal of Mathematical, 8 (2014), 993. Google Scholar

[21]

L. Molinet and Y. Wang, Dispersive limit from the Kawahara to the KdV equation,, J. Differential Equations, 255 (2013), 2196. doi: 10.1016/j.jde.2013.06.012. Google Scholar

[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. Google Scholar

[23]

F. Natali, A Note on the Stability for Kawahara-KdV Type Equations,, Appl. Math. Lett. 23 (2010), 23 (2010), 591. doi: 10.1016/j.aml.2010.01.017. Google Scholar

[24]

L. A. Ostrovsky, Nonlinear internal waves in a rotating ocean,, Okeanologia, 18 (1978), 181. Google Scholar

[25]

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

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