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December  2019, 8(4): 709-735. doi: 10.3934/eect.2019035

Well-posedness and asymptotic behavior of the dissipative Ostrovsky equation

Hongwei Wang 1, and  Amin Esfahani 2,3,,

1. 

School of Mathematics and Statistics, Anyang Normal University, Anyang 455000, China
2. 

School of Mathematics and Computer Science, Damghan University, Damghan 36715-364, Iran
3. 

Institute for Research in Fundamental Sciences (IPM), Tehran 19395-5746, Iran

* Corresponding author: Amin Esfahani

Received  July 2018 Revised  January 2019 Published  December 2019 Early access  June 2019

Fund Project: The second author is partially supported by a grant from IPM (No. 96470043).

In this paper we study the global well-posedness and the large-time behavior of solutions to the initial-value problem for the dissipative Ostrovsky equation. We show that the associated solutions decay faster than the solutions of the dissipative KdV equation.

Keywords: Dissipative Ostrovsky equation, Bourgain spaces, Cauchy problem, asymptotic behavior.
Mathematics Subject Classification: Primary: 35Q53, 35B40; Secondary: 35G25, 35K55.
Citation: 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
References:
[1]

C. J. Amick, J. L. Bona and M. E. Schonbek, Decay of solutions of some nonlinear wave equations, J. Differential Equations, 81 (1989), 1-49.  doi: 10.1016/0022-0396(89)90176-9.  Google Scholar

[2]

I. Bejenaru and T. Tao, Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation, J. Funct. Anal., 233 (2006), 228-259.  doi: 10.1016/j.jfa.2005.08.004.  Google Scholar

[3]

O. Besov, V. Ilin and S. Nikolski, Integral Representation of Functions and Embedding Theorems. Vol. I., New York: J. Wiley, 1978.  Google Scholar

[4]

J. Bourgain, On the Cauchy problem for the Kadomtsev-Petviashvili equation, Geom. Funct. Anal., 3 (1993), 315-341.  doi: 10.1007/BF01896259.  Google Scholar

[5]

W. Chen, C. Miao and J. Li, On the well-posedness of the Cauchy problem for dissipative modified Korteweg-de Vries equations, Differential Integral Equations, 20 (2007), 1285-1301.   Google Scholar

[6]

A. Esfahani and S. Levandosky, Solitary waves of the rotation-generalized Benjamin-Ono equation, Discrete Contin. Dyn. Syst., 33 (2013), 663-700.  doi: 10.3934/dcds.2013.33.663.  Google Scholar

[7]

K. Fujiwara, V. Georgiev and T. Ozawa, Higher order fractional Leibniz rule, J. Fourier Anal. Appl., 24 (2018), 650-665.  doi: 10.1007/s00041-017-9541-y.  Google Scholar

[8]

V. N. Galkin and Y. A. Stepanyants, On the existence of stationary solitary waves in a rotating field, J. Appl. Math. Mech., 55 (1991), 939-943.  doi: 10.1016/0021-8928(91)90148-N.  Google Scholar

[9]

O. A. Gilman, R. Grimshaw and Y. A. Stepanyants, Approximate and numerical solutions of the stationary Ostrovsky equation, Stud. Appl. Math., 95 (1995), 115-126.  doi: 10.1002/sapm1995951115.  Google Scholar

[10]

J. Ginibre, Y. Tsutsumi and G. Velo, On the cauchy problem for the Zakharov system, J. Func. Anal., 151 (1997), 384-436.  doi: 10.1006/jfan.1997.3148.  Google Scholar

[11]

R. Grimshaw, Internal solitary waves, in: R. Grimshaw (Ed.), Environmental Stratified Flows, Kluwer, Boston, (2001), pp. 1–27. Google Scholar

[12]

Z. Guo and B. Wang, Global well-posedness and inviscid limit for the Korteweg-de Vries-Burgers equation, J. Differential Equations, 246 (2009), 3864-3901.  doi: 10.1016/j.jde.2009.03.006.  Google Scholar

[13]

Z. Huo and Y. Jia, Low-regularity solutions for the Ostrovsky equation, Proc. Edinb. Math. Soc., 49 (2006), 87-100.  doi: 10.1017/S0013091504000938.  Google Scholar

[14]

P. Isaza and J. Mejía, Local well-posedness and quantitative ill-posedness for the Ostrovsky equation, Nonlinear Anal., 70 (2009), 2306-2316.  doi: 10.1016/j.na.2008.03.010.  Google Scholar

[15]

G. Karch, Self-similar large time behavior of solutions to Korteweg-de Vries-Burgers equation, Nonlinear Anal., 35 (1999), 199-219.  doi: 10.1016/S0362-546X(97)00708-6.  Google Scholar

[16]

C. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603.  doi: 10.1090/S0894-0347-96-00200-7.  Google Scholar

[17]

S. Levandosky and Y. Liu, Stability of solitary waves of a generalized Ostrovsky equation, SIAM J. Math. Anal., 38 (2006), 985-1011.  doi: 10.1137/050638722.  Google Scholar

[18]

Y. Li, J. Huang and W. Yan, The Cauchy problem for the Ostrovsky equation with negative dispersion at the critical regularity, J. Differential Equations, 259 (2015), 1379-1408.  doi: 10.1016/j.jde.2015.03.007.  Google Scholar

[19]

F. Linares and A. Milanes, Local and global well-posedness for the Ostrovsky equation, J. Differential Equations, 222 (2006), 325-340.  doi: 10.1016/j.jde.2005.07.023.  Google Scholar

[20]

B. Melinand, Long wave approximation for water waves under a Coriolis forcing and the Ostrovsky equation, Proc. Roy. Soc. Edinburgh Sect. A, 148 (2018), 1201-1237.  doi: 10.1017/S0308210518000136.  Google Scholar

[21]

H. Mitsudera and R. Grimshaw, Effects of friction on a localized structure in a baroclinic current, J. Physical Oceanography, 23 (1993), 2265-2292.  doi: 10.1175/1520-0485(1993)023<2265:EOFOAL>2.0.CO;2.  Google Scholar

[22]

L. Molinet and F. Ribaud, The Cauchy problem for dissipative Korteweg-de Vries equations in Sobolev spaces of negative order, Indiana Univ. Math. J., 50 (2001), 1745-1776.  doi: 10.1512/iumj.2001.50.2135.  Google Scholar

[23]

L. Molinet and F. Ribaud, On the low regularity of the Korteweg-de Vries-Burgers equation, Inter. Math. Research Notices, 37 (2002), 1979-2005.  doi: 10.1155/S1073792802112104.  Google Scholar

[24]

L. Molinet and F. Ribaud, The global Cauchy problem in Bourgain's-type spaces for a dispersive dissipative semilinear equation, SIAM J. Math. Anal., 33 (2002), 1269-1296.  doi: 10.1137/S0036141000374634.  Google Scholar

[25]

L. A. Ostrovsky, Nonlinear internal waves in a rotating ocean, Oceanologia, 18 (1978), 181-191.   Google Scholar

[26]

E. Ott and R. N. Sudan, Damping of solitary waves, Physics of Fluids, 13 (1970), 1432-1434.  doi: 10.1063/1.1693097.  Google Scholar

[27]

T. Tao, Multilinear weighted convolution of $L^2$ functions, and applications to non-linear dispersive equations, Amer. J. Math., 123 (2001), 839-908.  doi: 10.1353/ajm.2001.0035.  Google Scholar

[28]

S. Vento, Asymptotic behavior of solutions to dissipative Korteweg-de Vries equations, Asymptot. Anal., 68 (2010), 155-186.   Google Scholar

[29]

S. Vento, Global well-posedness for dissipative Korteweg-de Vries equations, Funkcial. Ekvac., 54 (2011), 119-138.  doi: 10.1619/fesi.54.119.  Google Scholar

[30]

S. Vento, Well-posedness and ill-posedness results for dissipative Benjamin-Ono equations, Osaka J. Math., 48 (2011), 933-958.   Google Scholar

show all references

References:
[1]

C. J. Amick, J. L. Bona and M. E. Schonbek, Decay of solutions of some nonlinear wave equations, J. Differential Equations, 81 (1989), 1-49.  doi: 10.1016/0022-0396(89)90176-9.  Google Scholar

[2]

I. Bejenaru and T. Tao, Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation, J. Funct. Anal., 233 (2006), 228-259.  doi: 10.1016/j.jfa.2005.08.004.  Google Scholar

[3]

O. Besov, V. Ilin and S. Nikolski, Integral Representation of Functions and Embedding Theorems. Vol. I., New York: J. Wiley, 1978.  Google Scholar

[4]

J. Bourgain, On the Cauchy problem for the Kadomtsev-Petviashvili equation, Geom. Funct. Anal., 3 (1993), 315-341.  doi: 10.1007/BF01896259.  Google Scholar

[5]

W. Chen, C. Miao and J. Li, On the well-posedness of the Cauchy problem for dissipative modified Korteweg-de Vries equations, Differential Integral Equations, 20 (2007), 1285-1301.   Google Scholar

[6]

A. Esfahani and S. Levandosky, Solitary waves of the rotation-generalized Benjamin-Ono equation, Discrete Contin. Dyn. Syst., 33 (2013), 663-700.  doi: 10.3934/dcds.2013.33.663.  Google Scholar

[7]

K. Fujiwara, V. Georgiev and T. Ozawa, Higher order fractional Leibniz rule, J. Fourier Anal. Appl., 24 (2018), 650-665.  doi: 10.1007/s00041-017-9541-y.  Google Scholar

[8]

V. N. Galkin and Y. A. Stepanyants, On the existence of stationary solitary waves in a rotating field, J. Appl. Math. Mech., 55 (1991), 939-943.  doi: 10.1016/0021-8928(91)90148-N.  Google Scholar

[9]

O. A. Gilman, R. Grimshaw and Y. A. Stepanyants, Approximate and numerical solutions of the stationary Ostrovsky equation, Stud. Appl. Math., 95 (1995), 115-126.  doi: 10.1002/sapm1995951115.  Google Scholar

[10]

J. Ginibre, Y. Tsutsumi and G. Velo, On the cauchy problem for the Zakharov system, J. Func. Anal., 151 (1997), 384-436.  doi: 10.1006/jfan.1997.3148.  Google Scholar

[11]

R. Grimshaw, Internal solitary waves, in: R. Grimshaw (Ed.), Environmental Stratified Flows, Kluwer, Boston, (2001), pp. 1–27. Google Scholar

[12]

Z. Guo and B. Wang, Global well-posedness and inviscid limit for the Korteweg-de Vries-Burgers equation, J. Differential Equations, 246 (2009), 3864-3901.  doi: 10.1016/j.jde.2009.03.006.  Google Scholar

[13]

Z. Huo and Y. Jia, Low-regularity solutions for the Ostrovsky equation, Proc. Edinb. Math. Soc., 49 (2006), 87-100.  doi: 10.1017/S0013091504000938.  Google Scholar

[14]

P. Isaza and J. Mejía, Local well-posedness and quantitative ill-posedness for the Ostrovsky equation, Nonlinear Anal., 70 (2009), 2306-2316.  doi: 10.1016/j.na.2008.03.010.  Google Scholar

[15]

G. Karch, Self-similar large time behavior of solutions to Korteweg-de Vries-Burgers equation, Nonlinear Anal., 35 (1999), 199-219.  doi: 10.1016/S0362-546X(97)00708-6.  Google Scholar

[16]

C. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603.  doi: 10.1090/S0894-0347-96-00200-7.  Google Scholar

[17]

S. Levandosky and Y. Liu, Stability of solitary waves of a generalized Ostrovsky equation, SIAM J. Math. Anal., 38 (2006), 985-1011.  doi: 10.1137/050638722.  Google Scholar

[18]

Y. Li, J. Huang and W. Yan, The Cauchy problem for the Ostrovsky equation with negative dispersion at the critical regularity, J. Differential Equations, 259 (2015), 1379-1408.  doi: 10.1016/j.jde.2015.03.007.  Google Scholar

[19]

F. Linares and A. Milanes, Local and global well-posedness for the Ostrovsky equation, J. Differential Equations, 222 (2006), 325-340.  doi: 10.1016/j.jde.2005.07.023.  Google Scholar

[20]

B. Melinand, Long wave approximation for water waves under a Coriolis forcing and the Ostrovsky equation, Proc. Roy. Soc. Edinburgh Sect. A, 148 (2018), 1201-1237.  doi: 10.1017/S0308210518000136.  Google Scholar

[21]

H. Mitsudera and R. Grimshaw, Effects of friction on a localized structure in a baroclinic current, J. Physical Oceanography, 23 (1993), 2265-2292.  doi: 10.1175/1520-0485(1993)023<2265:EOFOAL>2.0.CO;2.  Google Scholar

[22]

L. Molinet and F. Ribaud, The Cauchy problem for dissipative Korteweg-de Vries equations in Sobolev spaces of negative order, Indiana Univ. Math. J., 50 (2001), 1745-1776.  doi: 10.1512/iumj.2001.50.2135.  Google Scholar

[23]

L. Molinet and F. Ribaud, On the low regularity of the Korteweg-de Vries-Burgers equation, Inter. Math. Research Notices, 37 (2002), 1979-2005.  doi: 10.1155/S1073792802112104.  Google Scholar

[24]

L. Molinet and F. Ribaud, The global Cauchy problem in Bourgain's-type spaces for a dispersive dissipative semilinear equation, SIAM J. Math. Anal., 33 (2002), 1269-1296.  doi: 10.1137/S0036141000374634.  Google Scholar

[25]

L. A. Ostrovsky, Nonlinear internal waves in a rotating ocean, Oceanologia, 18 (1978), 181-191.   Google Scholar

[26]

E. Ott and R. N. Sudan, Damping of solitary waves, Physics of Fluids, 13 (1970), 1432-1434.  doi: 10.1063/1.1693097.  Google Scholar

[27]

T. Tao, Multilinear weighted convolution of $L^2$ functions, and applications to non-linear dispersive equations, Amer. J. Math., 123 (2001), 839-908.  doi: 10.1353/ajm.2001.0035.  Google Scholar

[28]

S. Vento, Asymptotic behavior of solutions to dissipative Korteweg-de Vries equations, Asymptot. Anal., 68 (2010), 155-186.   Google Scholar

[29]

S. Vento, Global well-posedness for dissipative Korteweg-de Vries equations, Funkcial. Ekvac., 54 (2011), 119-138.  doi: 10.1619/fesi.54.119.  Google Scholar

[30]

S. Vento, Well-posedness and ill-posedness results for dissipative Benjamin-Ono equations, Osaka J. Math., 48 (2011), 933-958.   Google Scholar

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