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

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  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.

Citation: Giuseppe Maria Coclite, Lorenzo di Ruvo. A note on the non-homogeneous initial boundary problem for an Ostrovsky-Hunter type equation. Discrete & Continuous Dynamical Systems - S, 2020, 13 (12) : 3357-3389. doi: 10.3934/dcdss.2020236
References:
[1]

C. BardosA. 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.  Google Scholar

[2]

R. BealsM. 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.  Google Scholar

[3]

J. C. Brunelli,, The short pulse hierarchy, J. Math. Phys., 46 (2005), 123507, 9pp. doi: 10.1063/1.2146189.  Google Scholar

[4]

I. E. Clarke, Lectures on plane waves in reacting gases, Ann. Phyx Fr., 9 (1984), 211-216.  doi: 10.1051/anphys:0198400902021100.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

G. M. CocliteK. 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.  Google Scholar

[18]

G. M. CocliteJ. 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.  Google Scholar

[19]

N. CostanzinoV. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

S. N. Kruzkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81(123) (1970), 228-255.   Google Scholar

[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.  Google Scholar

[26]

Y. LiuD. 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.  Google Scholar

[27]

A. J. MorrisonE. 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.  Google Scholar

[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.   Google Scholar

[29]

S. P. NikitenkovaYu. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

M. Rabelo, On equations which describe pseudospherical surfaces, Stud. Appl. Math, 81 (1989), 221-248.  doi: 10.1002/sapm1989813221.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

M. E. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations, 7 (1982), 959-1000.  doi: 10.1080/03605308208820242.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

N. L. TsitsasaT. P. HorikisbY. ShenP. G. KevrekidiscN. 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.  Google Scholar

[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.  Google Scholar

[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. BardosA. 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.  Google Scholar

[2]

R. BealsM. 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.  Google Scholar

[3]

J. C. Brunelli,, The short pulse hierarchy, J. Math. Phys., 46 (2005), 123507, 9pp. doi: 10.1063/1.2146189.  Google Scholar

[4]

I. E. Clarke, Lectures on plane waves in reacting gases, Ann. Phyx Fr., 9 (1984), 211-216.  doi: 10.1051/anphys:0198400902021100.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

G. M. CocliteK. 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.  Google Scholar

[18]

G. M. CocliteJ. 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.  Google Scholar

[19]

N. CostanzinoV. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

S. N. Kruzkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81(123) (1970), 228-255.   Google Scholar

[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.  Google Scholar

[26]

Y. LiuD. 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.  Google Scholar

[27]

A. J. MorrisonE. 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.  Google Scholar

[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.   Google Scholar

[29]

S. P. NikitenkovaYu. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

M. Rabelo, On equations which describe pseudospherical surfaces, Stud. Appl. Math, 81 (1989), 221-248.  doi: 10.1002/sapm1989813221.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

M. E. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations, 7 (1982), 959-1000.  doi: 10.1080/03605308208820242.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

N. L. TsitsasaT. P. HorikisbY. ShenP. G. KevrekidiscN. 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.  Google Scholar

[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.  Google Scholar

[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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