• Previous Article
    Dynamical properties of almost repetitive Delone sets
  • DCDS Home
  • This Issue
  • Next Article
    Global weak solutions to the two-dimensional Navier-Stokes equations of compressible heat-conducting flows with symmetric data and forces
February  2014, 34(2): 557-566. doi: 10.3934/dcds.2014.34.557

Global existence of small-norm solutions in the reduced Ostrovsky equation

1. 

Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, United Kingdom

2. 

Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1

Received  November 2012 Revised  April 2013 Published  August 2013

We use a novel transformation of the reduced Ostrovsky equation to the integrable Tzitzéica equation and prove global existence of small-norm solutions in Sobolev space $H^3(\mathbb{R})$. This scenario is an alternative to finite-time wave breaking of large-norm solutions of the reduced Ostrovsky equation. We also discuss a sharp sufficient condition for the finite-time wave breaking.
Citation: Roger Grimshaw, Dmitry Pelinovsky. Global existence of small-norm solutions in the reduced Ostrovsky equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 557-566. doi: 10.3934/dcds.2014.34.557
References:
[1]

A. R. Aguirre, T. R. Araujo, J. F. Gomes and A. H. Zimerman, Type-II Bäcklund transformations via gauge transformations,, J. High Energy Phys., (2011). Google Scholar

[2]

J. P. Boyd, Microbreaking and polycnoidal waves in the Ostrovsky-Hunter equation,, Physics Letters A, 338 (2005), 36. doi: 10.1016/j.physleta.2005.02.017. Google Scholar

[3]

J. C. Brunelli and S. Sakovich, Hamiltonian structures for the Ostrovsky-Vakhnenko equation,, Comm. Nonlin. Sci. Numer. Simul., 18 (2013), 56. doi: 10.1016/j.cnsns.2012.06.018. Google Scholar

[4]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach,, Annales de l'Institute Fourier (Grenoble), 50 (2000), 321. doi: 10.5802/aif.1757. Google Scholar

[5]

A. Constantin, On the scattering problem for the Camassa-Holm equation,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 953. doi: 10.1098/rspa.2000.0701. Google Scholar

[6]

R. Iório and D. Pilod, Well-posedness for Hirota-Satsuma's equation,, Diff. Integr. Eqs., 21 (2008), 1177. Google Scholar

[7]

A. N. W. Hone and J. P. Wang, Prolongation algebras and Hamiltonian operators for peakon equations,, Inverse Problems, 19 (2003), 129. doi: 10.1088/0266-5611/19/1/307. Google Scholar

[8]

J. Hunter, Numerical solutions of some nonlinear dispersive wave equations,, in, 26 (1990), 301. Google Scholar

[9]

R. H. J. Grimshaw, K. Helfrich and E. R. Johnson, The reduced Ostrovsky equation: Integrability and breaking,, Stud. Appl. Math., 129 (2013), 414. doi: 10.1111/j.1467-9590.2012.00560.x. Google Scholar

[10]

G. Gui and Y. Liu, On the Cauchy problem for the Ostrovsky equation with positive dispersion,, Comm. Part. Diff. Eqs., 32 (2007), 1895. doi: 10.1080/03605300600987314. Google Scholar

[11]

R. Kraenkel, H. Leblond and M. A. Manna, An integrable evolution equation for surface waves in deep water,, (2011), (2011). Google Scholar

[12]

F. Linares and A. Milanés, Local and global well-posedness for the Ostrovsky equation,, J. Diff. Eqs., 222 (2006), 325. doi: 10.1016/j.jde.2005.07.023. Google Scholar

[13]

Y. Liu, D. Pelinovsky and A. Sakovich, Wave breaking in the short-pulse equation,, Dynamics of PDE, 6 (2009), 291. Google Scholar

[14]

Y. Liu, D. Pelinovsky and A. Sakovich, Wave breaking in the Ostrovsky-Hunter equation,, SIAM J. Math. Anal., 42 (2010), 1967. doi: 10.1137/09075799X. Google Scholar

[15]

M. A. Manna and A. Neveu, Short-wave dynamics in the Euler equations,, Inverse Problems, 17 (2001), 855. doi: 10.1088/0266-5611/17/4/317. Google Scholar

[16]

A. J. Morrison, E. J. Parkes and V. O. Vakhnenko, The $N$ loop soliton solutions of the Vakhnenko equation,, Nonlinearity, 12 (1999), 1427. doi: 10.1088/0951-7715/12/5/314. Google Scholar

[17]

L. A. Ostrovsky, Nonlinear internal waves in a rotating ocean,, Okeanologia, 18 (1978), 181. Google Scholar

[18]

D. Pelinovsky and A. Sakovich, Global well-posedness of the short-pulse and sine-Gordon equations in energy space,, Comm. Part. Diff. Eqs., 35 (2010), 613. doi: 10.1080/03605300903509104. Google Scholar

[19]

J. Satsuma and D. J. Kaup, A Bäcklund transformation for a higher-order Korteweg-de Vries equation,, J. Phys. Soc. Japan, 43 (1977), 692. doi: 10.1143/JPSJ.43.692. Google Scholar

[20]

A. Stefanov, Y. Shen and P. G. Kevrekidis, Well-posedness and small data scattering for the generalized Ostrovsky equation,, J. Diff. Eqs., 249 (2010), 2600. doi: 10.1016/j.jde.2010.05.015. Google Scholar

[21]

K. Tsugawa, Well-posedness and weak rotation limit for the Ostrovsky equation,, J. Diff. Eqs., 247 (2009), 3163. doi: 10.1016/j.jde.2009.09.009. Google Scholar

[22]

G. Tzitzeica, Sur une nouvelle classe des surfaces,, C. R. Acad. Sci. Paris, 150 (1910), 955. Google Scholar

[23]

V. A. Vakhnenko, Solitons in a nonlinear model medium,, J. Phys. A, 25 (1992), 4181. doi: 10.1088/0305-4470/25/15/025. Google Scholar

[24]

V. O. Vakhnenko and E. J. Parkes, The calculation of multi-soliton solutions of the Vakhnenko equation by the inverse scattering method,, Chaos, 13 (2002), 1819. doi: 10.1016/S0960-0779(01)00200-4. Google Scholar

[25]

V. O. Vakhnenko, E. J. Parkes and A. J. Morrison, A Bäcklund transformation and the inverse scattering transform method for the generalised Vakhnenko equation,, Chaos, 17 (2003), 683. doi: 10.1016/S0960-0779(02)00483-6. Google Scholar

[26]

V. Varlamov and Y. Liu, Cauchy problem for the Ostrovsky equation,, Discr. Cont. Dyn. Syst., 10 (2004), 731. doi: 10.3934/dcds.2004.10.731. Google Scholar

show all references

References:
[1]

A. R. Aguirre, T. R. Araujo, J. F. Gomes and A. H. Zimerman, Type-II Bäcklund transformations via gauge transformations,, J. High Energy Phys., (2011). Google Scholar

[2]

J. P. Boyd, Microbreaking and polycnoidal waves in the Ostrovsky-Hunter equation,, Physics Letters A, 338 (2005), 36. doi: 10.1016/j.physleta.2005.02.017. Google Scholar

[3]

J. C. Brunelli and S. Sakovich, Hamiltonian structures for the Ostrovsky-Vakhnenko equation,, Comm. Nonlin. Sci. Numer. Simul., 18 (2013), 56. doi: 10.1016/j.cnsns.2012.06.018. Google Scholar

[4]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach,, Annales de l'Institute Fourier (Grenoble), 50 (2000), 321. doi: 10.5802/aif.1757. Google Scholar

[5]

A. Constantin, On the scattering problem for the Camassa-Holm equation,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 953. doi: 10.1098/rspa.2000.0701. Google Scholar

[6]

R. Iório and D. Pilod, Well-posedness for Hirota-Satsuma's equation,, Diff. Integr. Eqs., 21 (2008), 1177. Google Scholar

[7]

A. N. W. Hone and J. P. Wang, Prolongation algebras and Hamiltonian operators for peakon equations,, Inverse Problems, 19 (2003), 129. doi: 10.1088/0266-5611/19/1/307. Google Scholar

[8]

J. Hunter, Numerical solutions of some nonlinear dispersive wave equations,, in, 26 (1990), 301. Google Scholar

[9]

R. H. J. Grimshaw, K. Helfrich and E. R. Johnson, The reduced Ostrovsky equation: Integrability and breaking,, Stud. Appl. Math., 129 (2013), 414. doi: 10.1111/j.1467-9590.2012.00560.x. Google Scholar

[10]

G. Gui and Y. Liu, On the Cauchy problem for the Ostrovsky equation with positive dispersion,, Comm. Part. Diff. Eqs., 32 (2007), 1895. doi: 10.1080/03605300600987314. Google Scholar

[11]

R. Kraenkel, H. Leblond and M. A. Manna, An integrable evolution equation for surface waves in deep water,, (2011), (2011). Google Scholar

[12]

F. Linares and A. Milanés, Local and global well-posedness for the Ostrovsky equation,, J. Diff. Eqs., 222 (2006), 325. doi: 10.1016/j.jde.2005.07.023. Google Scholar

[13]

Y. Liu, D. Pelinovsky and A. Sakovich, Wave breaking in the short-pulse equation,, Dynamics of PDE, 6 (2009), 291. Google Scholar

[14]

Y. Liu, D. Pelinovsky and A. Sakovich, Wave breaking in the Ostrovsky-Hunter equation,, SIAM J. Math. Anal., 42 (2010), 1967. doi: 10.1137/09075799X. Google Scholar

[15]

M. A. Manna and A. Neveu, Short-wave dynamics in the Euler equations,, Inverse Problems, 17 (2001), 855. doi: 10.1088/0266-5611/17/4/317. Google Scholar

[16]

A. J. Morrison, E. J. Parkes and V. O. Vakhnenko, The $N$ loop soliton solutions of the Vakhnenko equation,, Nonlinearity, 12 (1999), 1427. doi: 10.1088/0951-7715/12/5/314. Google Scholar

[17]

L. A. Ostrovsky, Nonlinear internal waves in a rotating ocean,, Okeanologia, 18 (1978), 181. Google Scholar

[18]

D. Pelinovsky and A. Sakovich, Global well-posedness of the short-pulse and sine-Gordon equations in energy space,, Comm. Part. Diff. Eqs., 35 (2010), 613. doi: 10.1080/03605300903509104. Google Scholar

[19]

J. Satsuma and D. J. Kaup, A Bäcklund transformation for a higher-order Korteweg-de Vries equation,, J. Phys. Soc. Japan, 43 (1977), 692. doi: 10.1143/JPSJ.43.692. Google Scholar

[20]

A. Stefanov, Y. Shen and P. G. Kevrekidis, Well-posedness and small data scattering for the generalized Ostrovsky equation,, J. Diff. Eqs., 249 (2010), 2600. doi: 10.1016/j.jde.2010.05.015. Google Scholar

[21]

K. Tsugawa, Well-posedness and weak rotation limit for the Ostrovsky equation,, J. Diff. Eqs., 247 (2009), 3163. doi: 10.1016/j.jde.2009.09.009. Google Scholar

[22]

G. Tzitzeica, Sur une nouvelle classe des surfaces,, C. R. Acad. Sci. Paris, 150 (1910), 955. Google Scholar

[23]

V. A. Vakhnenko, Solitons in a nonlinear model medium,, J. Phys. A, 25 (1992), 4181. doi: 10.1088/0305-4470/25/15/025. Google Scholar

[24]

V. O. Vakhnenko and E. J. Parkes, The calculation of multi-soliton solutions of the Vakhnenko equation by the inverse scattering method,, Chaos, 13 (2002), 1819. doi: 10.1016/S0960-0779(01)00200-4. Google Scholar

[25]

V. O. Vakhnenko, E. J. Parkes and A. J. Morrison, A Bäcklund transformation and the inverse scattering transform method for the generalised Vakhnenko equation,, Chaos, 17 (2003), 683. doi: 10.1016/S0960-0779(02)00483-6. Google Scholar

[26]

V. Varlamov and Y. Liu, Cauchy problem for the Ostrovsky equation,, Discr. Cont. Dyn. Syst., 10 (2004), 731. doi: 10.3934/dcds.2004.10.731. Google Scholar

[1]

Long Wei, Zhijun Qiao, Yang Wang, Shouming Zhou. Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1733-1748. doi: 10.3934/dcds.2017072

[2]

V. Varlamov, Yue Liu. Cauchy problem for the Ostrovsky equation. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 731-753. doi: 10.3934/dcds.2004.10.731

[3]

Perikles G. Papadopoulos, Nikolaos M. Stavrakakis. Global existence for a wave equation on $R^n$. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 139-149. doi: 10.3934/dcdss.2008.1.139

[4]

Ying Zhang. Wave breaking and global existence for the periodic rotation-Camassa-Holm system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2243-2257. doi: 10.3934/dcds.2017097

[5]

Xue Yang, Xinglong Wu. Wave breaking and persistent decay of solution to a shallow water wave equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2149-2165. doi: 10.3934/dcdss.2016089

[6]

Günther Hörmann. Wave breaking of periodic solutions to the Fornberg-Whitham equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1605-1613. doi: 10.3934/dcds.2018066

[7]

Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361

[8]

Boyan Jonov, Thomas C. Sideris. Global and almost global existence of small solutions to a dissipative wave equation in 3D with nearly null nonlinear terms. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1407-1442. doi: 10.3934/cpaa.2015.14.1407

[9]

Sonomi Kakizaki, Akiko Fukuda, Yusaku Yamamoto, Masashi Iwasaki, Emiko Ishiwata, Yoshimasa Nakamura. Conserved quantities of the integrable discrete hungry systems. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 889-899. doi: 10.3934/dcdss.2015.8.889

[10]

Belkacem Said-Houari, Flávio A. Falcão Nascimento. Global existence and nonexistence for the viscoelastic wave equation with nonlinear boundary damping-source interaction. Communications on Pure & Applied Analysis, 2013, 12 (1) : 375-403. doi: 10.3934/cpaa.2013.12.375

[11]

Nicholas J. Kass, Mohammad A. Rammaha. Local and global existence of solutions to a strongly damped wave equation of the $ p $-Laplacian type. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1449-1478. doi: 10.3934/cpaa.2018070

[12]

Steve Levandosky, Yue Liu. Stability and weak rotation limit of solitary waves of the Ostrovsky equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 793-806. doi: 10.3934/dcdsb.2007.7.793

[13]

Hongwei Wang, Amin Esfahani. Well-posedness and asymptotic behavior of the dissipative Ostrovsky equation. Evolution Equations & Control Theory, 2019, 8 (4) : 709-735. doi: 10.3934/eect.2019035

[14]

Jian-Guo Liu, Jinhuan Wang. Global existence for a thin film equation with subcritical mass. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1461-1492. doi: 10.3934/dcdsb.2017070

[15]

Yong Chen, Hongjun Gao. Global existence for the stochastic Degasperis-Procesi equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5171-5184. doi: 10.3934/dcds.2015.35.5171

[16]

Barbara Kaltenbacher, Irena Lasiecka. Global existence and exponential decay rates for the Westervelt equation. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 503-523. doi: 10.3934/dcdss.2009.2.503

[17]

Jun Zhou. Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1175-1185. doi: 10.3934/dcdss.2017064

[18]

Perla El Kettani, Danielle Hilhorst, Kai Lee. A stochastic mass conserved reaction-diffusion equation with nonlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5615-5648. doi: 10.3934/dcds.2018246

[19]

Keisuke Matsuya, Tetsuji Tokihiro. Existence and non-existence of global solutions for a discrete semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 209-220. doi: 10.3934/dcds.2011.31.209

[20]

Alfonso Castro, Benjamin Preskill. Existence of solutions for a semilinear wave equation with non-monotone nonlinearity. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 649-658. doi: 10.3934/dcds.2010.28.649

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (16)

Other articles
by authors

[Back to Top]