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A note on the non-homogeneous initial boundary problem for an Ostrovsky-Hunter type 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 an Ostrovsky-Hunter type equation, which also includes the short pulse equation, or the Kozlov-Sazonov equation. We prove the well-posedness of the entropy solution for the non-homogeneous initial boundary value problem. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method.
References:
[1] |
C. Bardos, A. Y. Leroux and J. C.Nèdèlec,
First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations, 4 (1979), 1017-1034.
doi: 10.1080/03605307908820117. |
[2] |
R. Beals, M. Rabelo and K. Tenenblat,
Bäcklund transformations and inverse scattering solutions for some pseudospherical surface equations, Stud. Appl. Math., 81 (1989), 125-151.
doi: 10.1002/sapm1989812125. |
[3] |
J. C. Brunelli,, The short pulse hierarchy, J. Math. Phys., 46 (2005), 123507, 9pp.
doi: 10.1063/1.2146189. |
[4] |
I. E. Clarke,
Lectures on plane waves in reacting gases, Ann. Phyx Fr., 9 (1984), 211-216.
doi: 10.1051/anphys:0198400902021100. |
[5] |
G. M. Coclite and L. di Ruvo,
On the well-posedness of the exp-Rabelo equation, Ann. Mat. Pur. Appl., 195 (2016), 923-933.
doi: 10.1007/s10231-015-0497-8. |
[6] |
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. |
[7] |
G. M. Coclite and L. di Ruvo,
Dispersive and Diffusive limits for Ostrovsky-Hunter type equations, Nonlinear Differ. Equ. Appl., 22 (2015), 1733-1763.
doi: 10.1007/s00030-015-0342-1. |
[8] |
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. |
[9] |
G. M. Coclite and L. di Ruvo,
Wellposedness of bounded solutions of the non-homogeneous initial boundary value problem for the Ostrovsky-Hunter equation, J. Hyperbolic Differ. Equ., 12 (2015), 221-248.
doi: 10.1142/S021989161550006X. |
[10] |
G. M. Coclite and L. di Ruvo,
Wellposedness results for the short pulse equation, Z. Angew. Math. Phys., 66 (2015), 1529-1557.
doi: 10.1007/s00033-014-0478-6. |
[11] |
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. |
[12] |
G. M. Coclite and L. di Ruvo,
Well-posedness and dispersive/diffusive limit of a generalized Ostrovsky-Hunter equation, Milan J. Math., 86 (2018), 31-51.
doi: 10.1007/s00032-018-0278-0. |
[13] |
G. M. Coclite and L. di Ruvo,
Wellposedness of bounded solutions of the non-homogeneous initial boundary for the short pulse equation, Boll. Unione Mat. Ital., 8 (2015), 31-44.
doi: 10.1007/s40574-015-0023-3. |
[14] |
G. M. Coclite and L. di Ruvo,
A singular limit problem for conservation laws related to the Rosenau-Korteweg-de Vries equation, J. Math. Pures App., 107 (2017), 315-335.
doi: 10.1016/j.matpur.2016.07.002. |
[15] |
G. M. Coclite, L. di Ruvo and K. H. Karlsen,, Some wellposedness results for the Ostrovsky-Hunter equation, Hyperbolic Conservation laws and Related Analysis with Applications, 143–159, Springer Proc. Math. Stat., 49, Springer, Heidelberg, 2014.
doi: 10.1007/978-3-642-39007-4_7. |
[16] |
G. M. Coclite, L. di Ruvo and K. H. Karlsen,, The initial-boundary-value problem for an Ostrovsky-Hunter type equation, Non-linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis, 97–109, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2018. |
[17] |
G. M. Coclite, K. H. Karlsen and Y.-S. Kwon,
Initial-boundary value problems for conservation laws with source terms and the Degasperis-Procesi equation, J. Funct. Anal., 257 (2009), 3823-3857.
doi: 10.1016/j.jfa.2009.09.022. |
[18] |
G. M. Coclite, J. Ridder and H. Risebro,
A convergent finite difference scheme for the Ostrovsky-Hunter equation on a bounded domain, BIT Numer. Math., 57 (2017), 93-122.
doi: 10.1007/s10543-016-0625-x. |
[19] |
N. Costanzino, V. Manukian and C. K. R. T. Jones,
Solitary waves of the regularized short pulse and Ostrovsky equations, SIAM J. Math. Anal., 41 (2009), 2088-2106.
doi: 10.1137/080734327. |
[20] |
L. di Ruvo,, Discontinuous solutions for the Ostrovsky–Hunter equation and two phase flows, Phd Thesis, University of Bari, 2013. http:www.dm.uniba.it/home/dottorato/dottorato/tesi/. Google Scholar |
[21] |
J. Hunter,, Numerical solutions of some nonlinear dispersive wave equations, in Computational Solution of Nonlinear Systems of Equations, Lectures in Applied Mathematics, Vol. 26 American Mathematical Society, Providence, RI, (1990), 301–316. |
[22] |
J. Hunter and K. P. Tan,, Weakly dispersive short waves, Proceedings of the IVth international Congress on Waves and Stability in Continuous Media, Sicily, 1987. Google Scholar |
[23] |
S. A. Kozlov and S. V. Sazonov,
Nonlinear propagation of optical pulses of a few oscillations duration in dielectric media, J. Exp. Theor. Phys., 84 (1997), 221-228.
doi: 10.1134/1.558109. |
[24] |
S. N. Kruzkov,
First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81(123) (1970), 228-255.
|
[25] |
C. Lattanzio and P. Marcati,
Global well-posedness and relaxation limits of a model for radiating gas, J. Differential Equations, 190 (2013), 439-465.
doi: 10.1016/S0022-0396(02)00158-4. |
[26] |
Y. Liu, D. Pelinovsky and A. Sakovich,
Wave breaking in the short-pulse equation, Dynamics of PDE, 6 (2009), 291-310.
doi: 10.4310/DPDE.2009.v6.n4.a1. |
[27] |
A. J. Morrison, E. J. Parkes and V. O. Vakhnenko,
The N loop soliton solutions of the Vakhnenko equation, Nonlinearity, 12 (1999), 1427-1437.
doi: 10.1088/0951-7715/12/5/314. |
[28] |
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.
|
[29] |
S. P. Nikitenkova, Yu. A. Stepanyants and L. M. Chikhladze,
Solutions of the modified Ostrovskii equation with cubic non-linearity, J. Appl. Maths Mechs, 64 (2000), 267-274.
doi: 10.1016/S0021-8928(00)00048-4. |
[30] |
L. A. Ostrovsky, Nonlinear internal waves in a rotating ocean, Okeanologia, 18 (1978), 181-191. Google Scholar |
[31] |
E. J. Parkes,
Explicit solutions of the reduced Ostrovsky equation, Chaos Solitons Fractals, 31 (2007), 602-610.
doi: 10.1016/j.chaos.2005.10.028. |
[32] |
E. J. Parkes and V. O. Vakhnenko,
The calculation of multi-soliton solutions of the Vakhnenko equation by the inverse scattering method, Chaos Solitons Fractals, 13 (2002), 1819-18126.
doi: 10.1016/S0960-0779(01)00200-4. |
[33] |
D. Pelinovsky and G. Schneider,
Rigorous justification of the short-pulse equation, Nonlinear Differ. Equ. Appl., 20 (2013), 1277-1294.
doi: 10.1007/s00030-012-0208-8. |
[34] |
M. Rabelo,
On equations which describe pseudospherical surfaces, Stud. Appl. Math, 81 (1989), 221-248.
doi: 10.1002/sapm1989813221. |
[35] |
A. Sakovich and S. Sakovich, The short pulse equation is integrable, J. Phys. Soc. Jpn., 74 (2005), 239-241. Google Scholar |
[36] |
A. Sakovich and S. Sakovich,, On the transformations of the Rabelo equations, SIGMA, 3 (2007), Paper 086, 8 pp.
doi: 10.3842/SIGMA.2007.086. |
[37] |
T. Schäfer and C. E. Wayne,
Propagation of ultra-short optical pulses in cubic nonlinear media, Physica D, 196 (2004), 90-105.
doi: 10.1016/j.physd.2004.04.007. |
[38] |
M. E. Schonbek,
Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations, 7 (1982), 959-1000.
doi: 10.1080/03605308208820242. |
[39] |
D. Serre,
$L^1$-stability of constants in a model for radiating gases, Commun. Math. Sci., 1 (2003), 197-205.
doi: 10.4310/CMS.2003.v1.n1.a12. |
[40] |
L. Tartar,, Compensated compactness and applications to partial differential equations, In Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV, pages 136–212. Res. Notes in Math., 39, Pitman, Boston, Mass.-London, 1979. |
[41] |
N. L. Tsitsasa, T. P. Horikisb, Y. Shen, P. G. Kevrekidisc, N. Whitakerc and D. J. Frantzeskakisd,
Short pulse equations and localized structures in frequency band gaps of nonlinear metamaterials, Physics Letters A, 374 (2010), 1384-1388.
doi: 10.1016/j.physleta.2010.01.004. |
[42] |
V. A. Vakhnenko,
Solitons in a nonlinear model medium, J. Phys. A: Math. Gen., 25 (1992), 4181-4187.
doi: 10.1088/0305-4470/25/15/025. |
[43] |
G. P. Yasnikov and V. S. Belousov,, Effective thermodynamic gas functions with hard panicles, J. Eng. Phys., 34 (1978), 1085–1089(in Russian). Google Scholar |
show all references
References:
[1] |
C. Bardos, A. Y. Leroux and J. C.Nèdèlec,
First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations, 4 (1979), 1017-1034.
doi: 10.1080/03605307908820117. |
[2] |
R. Beals, M. Rabelo and K. Tenenblat,
Bäcklund transformations and inverse scattering solutions for some pseudospherical surface equations, Stud. Appl. Math., 81 (1989), 125-151.
doi: 10.1002/sapm1989812125. |
[3] |
J. C. Brunelli,, The short pulse hierarchy, J. Math. Phys., 46 (2005), 123507, 9pp.
doi: 10.1063/1.2146189. |
[4] |
I. E. Clarke,
Lectures on plane waves in reacting gases, Ann. Phyx Fr., 9 (1984), 211-216.
doi: 10.1051/anphys:0198400902021100. |
[5] |
G. M. Coclite and L. di Ruvo,
On the well-posedness of the exp-Rabelo equation, Ann. Mat. Pur. Appl., 195 (2016), 923-933.
doi: 10.1007/s10231-015-0497-8. |
[6] |
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. |
[7] |
G. M. Coclite and L. di Ruvo,
Dispersive and Diffusive limits for Ostrovsky-Hunter type equations, Nonlinear Differ. Equ. Appl., 22 (2015), 1733-1763.
doi: 10.1007/s00030-015-0342-1. |
[8] |
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. |
[9] |
G. M. Coclite and L. di Ruvo,
Wellposedness of bounded solutions of the non-homogeneous initial boundary value problem for the Ostrovsky-Hunter equation, J. Hyperbolic Differ. Equ., 12 (2015), 221-248.
doi: 10.1142/S021989161550006X. |
[10] |
G. M. Coclite and L. di Ruvo,
Wellposedness results for the short pulse equation, Z. Angew. Math. Phys., 66 (2015), 1529-1557.
doi: 10.1007/s00033-014-0478-6. |
[11] |
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. |
[12] |
G. M. Coclite and L. di Ruvo,
Well-posedness and dispersive/diffusive limit of a generalized Ostrovsky-Hunter equation, Milan J. Math., 86 (2018), 31-51.
doi: 10.1007/s00032-018-0278-0. |
[13] |
G. M. Coclite and L. di Ruvo,
Wellposedness of bounded solutions of the non-homogeneous initial boundary for the short pulse equation, Boll. Unione Mat. Ital., 8 (2015), 31-44.
doi: 10.1007/s40574-015-0023-3. |
[14] |
G. M. Coclite and L. di Ruvo,
A singular limit problem for conservation laws related to the Rosenau-Korteweg-de Vries equation, J. Math. Pures App., 107 (2017), 315-335.
doi: 10.1016/j.matpur.2016.07.002. |
[15] |
G. M. Coclite, L. di Ruvo and K. H. Karlsen,, Some wellposedness results for the Ostrovsky-Hunter equation, Hyperbolic Conservation laws and Related Analysis with Applications, 143–159, Springer Proc. Math. Stat., 49, Springer, Heidelberg, 2014.
doi: 10.1007/978-3-642-39007-4_7. |
[16] |
G. M. Coclite, L. di Ruvo and K. H. Karlsen,, The initial-boundary-value problem for an Ostrovsky-Hunter type equation, Non-linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis, 97–109, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2018. |
[17] |
G. M. Coclite, K. H. Karlsen and Y.-S. Kwon,
Initial-boundary value problems for conservation laws with source terms and the Degasperis-Procesi equation, J. Funct. Anal., 257 (2009), 3823-3857.
doi: 10.1016/j.jfa.2009.09.022. |
[18] |
G. M. Coclite, J. Ridder and H. Risebro,
A convergent finite difference scheme for the Ostrovsky-Hunter equation on a bounded domain, BIT Numer. Math., 57 (2017), 93-122.
doi: 10.1007/s10543-016-0625-x. |
[19] |
N. Costanzino, V. Manukian and C. K. R. T. Jones,
Solitary waves of the regularized short pulse and Ostrovsky equations, SIAM J. Math. Anal., 41 (2009), 2088-2106.
doi: 10.1137/080734327. |
[20] |
L. di Ruvo,, Discontinuous solutions for the Ostrovsky–Hunter equation and two phase flows, Phd Thesis, University of Bari, 2013. http:www.dm.uniba.it/home/dottorato/dottorato/tesi/. Google Scholar |
[21] |
J. Hunter,, Numerical solutions of some nonlinear dispersive wave equations, in Computational Solution of Nonlinear Systems of Equations, Lectures in Applied Mathematics, Vol. 26 American Mathematical Society, Providence, RI, (1990), 301–316. |
[22] |
J. Hunter and K. P. Tan,, Weakly dispersive short waves, Proceedings of the IVth international Congress on Waves and Stability in Continuous Media, Sicily, 1987. Google Scholar |
[23] |
S. A. Kozlov and S. V. Sazonov,
Nonlinear propagation of optical pulses of a few oscillations duration in dielectric media, J. Exp. Theor. Phys., 84 (1997), 221-228.
doi: 10.1134/1.558109. |
[24] |
S. N. Kruzkov,
First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81(123) (1970), 228-255.
|
[25] |
C. Lattanzio and P. Marcati,
Global well-posedness and relaxation limits of a model for radiating gas, J. Differential Equations, 190 (2013), 439-465.
doi: 10.1016/S0022-0396(02)00158-4. |
[26] |
Y. Liu, D. Pelinovsky and A. Sakovich,
Wave breaking in the short-pulse equation, Dynamics of PDE, 6 (2009), 291-310.
doi: 10.4310/DPDE.2009.v6.n4.a1. |
[27] |
A. J. Morrison, E. J. Parkes and V. O. Vakhnenko,
The N loop soliton solutions of the Vakhnenko equation, Nonlinearity, 12 (1999), 1427-1437.
doi: 10.1088/0951-7715/12/5/314. |
[28] |
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.
|
[29] |
S. P. Nikitenkova, Yu. A. Stepanyants and L. M. Chikhladze,
Solutions of the modified Ostrovskii equation with cubic non-linearity, J. Appl. Maths Mechs, 64 (2000), 267-274.
doi: 10.1016/S0021-8928(00)00048-4. |
[30] |
L. A. Ostrovsky, Nonlinear internal waves in a rotating ocean, Okeanologia, 18 (1978), 181-191. Google Scholar |
[31] |
E. J. Parkes,
Explicit solutions of the reduced Ostrovsky equation, Chaos Solitons Fractals, 31 (2007), 602-610.
doi: 10.1016/j.chaos.2005.10.028. |
[32] |
E. J. Parkes and V. O. Vakhnenko,
The calculation of multi-soliton solutions of the Vakhnenko equation by the inverse scattering method, Chaos Solitons Fractals, 13 (2002), 1819-18126.
doi: 10.1016/S0960-0779(01)00200-4. |
[33] |
D. Pelinovsky and G. Schneider,
Rigorous justification of the short-pulse equation, Nonlinear Differ. Equ. Appl., 20 (2013), 1277-1294.
doi: 10.1007/s00030-012-0208-8. |
[34] |
M. Rabelo,
On equations which describe pseudospherical surfaces, Stud. Appl. Math, 81 (1989), 221-248.
doi: 10.1002/sapm1989813221. |
[35] |
A. Sakovich and S. Sakovich, The short pulse equation is integrable, J. Phys. Soc. Jpn., 74 (2005), 239-241. Google Scholar |
[36] |
A. Sakovich and S. Sakovich,, On the transformations of the Rabelo equations, SIGMA, 3 (2007), Paper 086, 8 pp.
doi: 10.3842/SIGMA.2007.086. |
[37] |
T. Schäfer and C. E. Wayne,
Propagation of ultra-short optical pulses in cubic nonlinear media, Physica D, 196 (2004), 90-105.
doi: 10.1016/j.physd.2004.04.007. |
[38] |
M. E. Schonbek,
Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations, 7 (1982), 959-1000.
doi: 10.1080/03605308208820242. |
[39] |
D. Serre,
$L^1$-stability of constants in a model for radiating gases, Commun. Math. Sci., 1 (2003), 197-205.
doi: 10.4310/CMS.2003.v1.n1.a12. |
[40] |
L. Tartar,, Compensated compactness and applications to partial differential equations, In Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV, pages 136–212. Res. Notes in Math., 39, Pitman, Boston, Mass.-London, 1979. |
[41] |
N. L. Tsitsasa, T. P. Horikisb, Y. Shen, P. G. Kevrekidisc, N. Whitakerc and D. J. Frantzeskakisd,
Short pulse equations and localized structures in frequency band gaps of nonlinear metamaterials, Physics Letters A, 374 (2010), 1384-1388.
doi: 10.1016/j.physleta.2010.01.004. |
[42] |
V. A. Vakhnenko,
Solitons in a nonlinear model medium, J. Phys. A: Math. Gen., 25 (1992), 4181-4187.
doi: 10.1088/0305-4470/25/15/025. |
[43] |
G. P. Yasnikov and V. S. Belousov,, Effective thermodynamic gas functions with hard panicles, J. Eng. Phys., 34 (1978), 1085–1089(in Russian). Google Scholar |
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