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

Well-posedness and asymptotic behavior of the dissipative Ostrovsky equation

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.

Citation: Hongwei Wang, Amin Esfahani. Well-posedness and asymptotic behavior of the dissipative Ostrovsky equation. Evolution Equations and Control Theory, 2019, 8 (4) : 709-735. doi: 10.3934/eect.2019035
References:
[1]

C. J. AmickJ. 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.

[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.

[3]

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

[4]

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

[5]

W. ChenC. 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. 

[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.

[7]

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

[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.

[9]

O. A. GilmanR. 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.

[10]

J. GinibreY. 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.

[11]

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

[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.

[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.

[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.

[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.

[16]

C. KenigG. 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.

[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.

[18]

Y. LiJ. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[25]

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

[26]

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

[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.

[28]

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

[29]

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

[30]

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

show all references

References:
[1]

C. J. AmickJ. 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.

[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.

[3]

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

[4]

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

[5]

W. ChenC. 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. 

[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.

[7]

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

[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.

[9]

O. A. GilmanR. 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.

[10]

J. GinibreY. 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.

[11]

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

[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.

[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.

[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.

[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.

[16]

C. KenigG. 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.

[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.

[18]

Y. LiJ. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[25]

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

[26]

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

[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.

[28]

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

[29]

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

[30]

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

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