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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 |
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. |
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), 135-145. |
[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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