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A penalty decomposition method for nuclear norm minimization with l1 norm fidelity term
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 |
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.
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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