A note on the non-homogeneous initial boundary problem for an Ostrovsky-Hunter type equation

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December 2020, 13(12): 3357-3389. doi: 10.3934/dcdss.2020236

A note on the non-homogeneous initial boundary problem for an Ostrovsky-Hunter type equation

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

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

^* Corresponding author: Giuseppe Maria Coclite

The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)

Received October 2018 Published December 2020 Early access January 2020

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.

Keywords: Existence, uniqueness, stability, entropy solutions, conservation laws, Ostrovsky-Hunter type equation, boundary value problems.

Mathematics Subject Classification: Primary: 35G15, 35L65, 35L05; Secondary: 35A05.

Citation: Giuseppe Maria Coclite, Lorenzo di Ruvo. A note on the non-homogeneous initial boundary problem for an Ostrovsky-Hunter type equation. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3357-3389. doi: 10.3934/dcdss.2020236

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/.
[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.
[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.
[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.
[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).

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/.
[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.
[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.
[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.
[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).

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