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  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.

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. 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.

[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.

[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. 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.

[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.

[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.

[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. 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.

[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.

[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. 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.

[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. 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.

[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. 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.

[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.

[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. 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.

[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.

[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.

[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. 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.

[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.

[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. 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.

[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. 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.

[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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