Well-posedness and asymptotic behavior of the dissipative Ostrovsky equation

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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 and 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.
[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. 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.
[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. 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.
[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. 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.
[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.
[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. 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.
[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. 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.
[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. 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.
[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. 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.
[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. 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.
[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. 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.
[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.
[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. 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.
[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. 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.
[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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