American Institute of Mathematical Sciences

Advanced
March  2021, 41(3): 1133-1155. doi: 10.3934/dcds.2020312

Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity

Jean-Claude Saut 1, and  Yuexun Wang 1,2,,

1. 

Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d'Orsay, 91405 Orsay, France
2. 

School of Mathematics and Statistics, Lanzhou University, 370000 Lanzhou, China

* Corresponding author

Received  April 2020 Revised  July 2020 Published  March 2021 Early access  August 2020

Fund Project: The work of both authors was partially supported by the ANR project ANuI (ANR-17-CE40-0035-02)

We prove global existence and modified scattering for the solutions of the Cauchy problem to the fractional Korteweg-de Vries equation with cubic nonlinearity for small, smooth and localized initial data.

Keywords: Fractional Korteweg-de Vries, cubic nonlinearity, global existence, modified scattering, small data.
Mathematics Subject Classification: Primary: 76B15, 76B03; Secondary: 35S30, 35A20.
Citation: Jean-Claude Saut, Yuexun Wang. Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1133-1155. doi: 10.3934/dcds.2020312
References:
[1]

J. Biello and J. K. Hunter, Nonlinear Hamiltonian waves with constant frequency and surface waves on vorticity discontinuities, Comm. Pure Appl. Math., 63 (2010), 303-336.  doi: 10.1002/cpa.20304.  Google Scholar

[2]

A. Bressan and K. T. Nguyen, Global existence of weak solutions for the Burgers-Hilbert equation, SIAM J. Math. Anal., 46 (2014), 2884-2904.  doi: 10.1137/140957536.  Google Scholar

[3]

A. Castro, D. Córdoba and F. Gancedo, Singularity formations for a surface wave model, Nonlinearity, 23 (2010), 2835-2847.  doi: 10.1088/0951-7715/23/11/006.  Google Scholar

[4]

D. Córdoba, J. Gómez-Serrano and A. D. Ionescu, Global solutions for the generalized SQG patch equation, Arch. Ration. Mech. Anal., 233 (2019), 1211-1251.  doi: 10.1007/s00205-019-01377-6.  Google Scholar

[5]

M. Ehrnström and Y. Wang, Enhanced existence time of solutions to the fractional Korteweg-de Vries equation, SIAM J. Math. Anal., 51 (2019), 3298-3323.  doi: 10.1137/19M1237867.  Google Scholar

[6]

P. Germain, N. Masmoudi and J. Shatah, Global solutions for 3D quadratic Schrödinger equations, Int. Math. Res. Not. IMRN., 2009 (2009), 414-432.  doi: 10.1093/imrn/rnn135.  Google Scholar

[7]

S. Gustafson, K. Nakanishi and T.-P. Tsai, Global dispersive solutions for the Gross-Pitaevskii equation in two and three dimensions, Ann. Henri Poincaré, 8 (2007), 1303-1331.  doi: 10.1007/s00023-007-0336-6.  Google Scholar

[8]

N. Hayashi and P. I. Naumkin, Large time behavior of solutions for the modified Korteweg-de Vries equation, Internat. Math. Res. Notices, 1999 (1999), 395-418.  doi: 10.1155/S1073792899000203.  Google Scholar

[9]

J. K. Hunter and M. Ifrim, Enhanced life span of smooth solutions of a Burgers-Hilbert equation, SIAM J. Math. Anal., 44 (2012), 2039-2052.  doi: 10.1137/110849791.  Google Scholar

[10]

J. K. Hunter, M. Ifrim, D. Tataru and T. K. Wong, Long time solutions for a Burgers-Hilbert equation via a modified energy method, Proc. Amer. Math. Soc., 143 (2015), 3407-3412.  doi: 10.1090/proc/12215.  Google Scholar

[11]

V. M. Hur and L. Tao, Wave breaking for the Whitham equation with fractional dispersion, Nonlinearity, 27 (2014), 2937-2949.  doi: 10.1088/0951-7715/27/12/2937.  Google Scholar

[12]

V. M. Hur, Wave breaking in the Whitham equation, Adv. Math., 317 (2017), 410-437.  doi: 10.1016/j.aim.2017.07.006.  Google Scholar

[13]

A. D. Ionescu and F. Pusateri, Global solutions for the gravity water waves system in 2d, Invent. Math., 199 (2015), 653-804.  doi: 10.1007/s00222-014-0521-4.  Google Scholar

[14]

A. D. Ionescu and F. Pusateri, Global analysis of a model for capillary water waves in two dimensions, Comm. Pure Appl. Math., 69 (2016), 2015-2071.  doi: 10.1002/cpa.21654.  Google Scholar

[15]

A. D. Ionescu and F. Pusateri, Nonlinear fractional Schrödinger equations in one dimension, J. Funct. Anal., 266 (2014), 139-176.  doi: 10.1016/j.jfa.2013.08.027.  Google Scholar

[16]

C. Klein, F. Linares, D. Pilod and J.-C. Saut, On Whitham and related equations, Stud. Appl. Math., 140 (2018), 133-177.  doi: 10.1111/sapm.12194.  Google Scholar

[17]

C. Klein and J.-C. Saut, A numerical approach to blow-up issues for dispersive perturbations of Burgers' equation, Phys. D, 295/296 (2015), 46-65.  doi: 10.1016/j.physd.2014.12.004.  Google Scholar

[18]

F. Linares, D. Pilod and J.-C. Saut, Dispersive perturbations of Burgers and hyperbolic equations Ⅰ: Local theory, SIAM J. Math. Anal., 46 (2014), 1505-1537.  doi: 10.1137/130912001.  Google Scholar

[19]

L. Molinet, D. Pilod and S. Vento, On well-posedness for some dispersive perturbations of Burgers' equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 1719-1756.  doi: 10.1016/j.anihpc.2017.12.004.  Google Scholar

[20]

J.-C. Saut and Y. Wang, The wave breaking for Whitham-type equations revisited, preprint, arXiv: 2006.03803. Google Scholar

[21]

R. Yang, Shock formation for the Burgers-Hilbert equation, preprint, arXiv: 2006.05568. Google Scholar

show all references

References:
[1]

J. Biello and J. K. Hunter, Nonlinear Hamiltonian waves with constant frequency and surface waves on vorticity discontinuities, Comm. Pure Appl. Math., 63 (2010), 303-336.  doi: 10.1002/cpa.20304.  Google Scholar

[2]

A. Bressan and K. T. Nguyen, Global existence of weak solutions for the Burgers-Hilbert equation, SIAM J. Math. Anal., 46 (2014), 2884-2904.  doi: 10.1137/140957536.  Google Scholar

[3]

A. Castro, D. Córdoba and F. Gancedo, Singularity formations for a surface wave model, Nonlinearity, 23 (2010), 2835-2847.  doi: 10.1088/0951-7715/23/11/006.  Google Scholar

[4]

D. Córdoba, J. Gómez-Serrano and A. D. Ionescu, Global solutions for the generalized SQG patch equation, Arch. Ration. Mech. Anal., 233 (2019), 1211-1251.  doi: 10.1007/s00205-019-01377-6.  Google Scholar

[5]

M. Ehrnström and Y. Wang, Enhanced existence time of solutions to the fractional Korteweg-de Vries equation, SIAM J. Math. Anal., 51 (2019), 3298-3323.  doi: 10.1137/19M1237867.  Google Scholar

[6]

P. Germain, N. Masmoudi and J. Shatah, Global solutions for 3D quadratic Schrödinger equations, Int. Math. Res. Not. IMRN., 2009 (2009), 414-432.  doi: 10.1093/imrn/rnn135.  Google Scholar

[7]

S. Gustafson, K. Nakanishi and T.-P. Tsai, Global dispersive solutions for the Gross-Pitaevskii equation in two and three dimensions, Ann. Henri Poincaré, 8 (2007), 1303-1331.  doi: 10.1007/s00023-007-0336-6.  Google Scholar

[8]

N. Hayashi and P. I. Naumkin, Large time behavior of solutions for the modified Korteweg-de Vries equation, Internat. Math. Res. Notices, 1999 (1999), 395-418.  doi: 10.1155/S1073792899000203.  Google Scholar

[9]

J. K. Hunter and M. Ifrim, Enhanced life span of smooth solutions of a Burgers-Hilbert equation, SIAM J. Math. Anal., 44 (2012), 2039-2052.  doi: 10.1137/110849791.  Google Scholar

[10]

J. K. Hunter, M. Ifrim, D. Tataru and T. K. Wong, Long time solutions for a Burgers-Hilbert equation via a modified energy method, Proc. Amer. Math. Soc., 143 (2015), 3407-3412.  doi: 10.1090/proc/12215.  Google Scholar

[11]

V. M. Hur and L. Tao, Wave breaking for the Whitham equation with fractional dispersion, Nonlinearity, 27 (2014), 2937-2949.  doi: 10.1088/0951-7715/27/12/2937.  Google Scholar

[12]

V. M. Hur, Wave breaking in the Whitham equation, Adv. Math., 317 (2017), 410-437.  doi: 10.1016/j.aim.2017.07.006.  Google Scholar

[13]

A. D. Ionescu and F. Pusateri, Global solutions for the gravity water waves system in 2d, Invent. Math., 199 (2015), 653-804.  doi: 10.1007/s00222-014-0521-4.  Google Scholar

[14]

A. D. Ionescu and F. Pusateri, Global analysis of a model for capillary water waves in two dimensions, Comm. Pure Appl. Math., 69 (2016), 2015-2071.  doi: 10.1002/cpa.21654.  Google Scholar

[15]

A. D. Ionescu and F. Pusateri, Nonlinear fractional Schrödinger equations in one dimension, J. Funct. Anal., 266 (2014), 139-176.  doi: 10.1016/j.jfa.2013.08.027.  Google Scholar

[16]

C. Klein, F. Linares, D. Pilod and J.-C. Saut, On Whitham and related equations, Stud. Appl. Math., 140 (2018), 133-177.  doi: 10.1111/sapm.12194.  Google Scholar

[17]

C. Klein and J.-C. Saut, A numerical approach to blow-up issues for dispersive perturbations of Burgers' equation, Phys. D, 295/296 (2015), 46-65.  doi: 10.1016/j.physd.2014.12.004.  Google Scholar

[18]

F. Linares, D. Pilod and J.-C. Saut, Dispersive perturbations of Burgers and hyperbolic equations Ⅰ: Local theory, SIAM J. Math. Anal., 46 (2014), 1505-1537.  doi: 10.1137/130912001.  Google Scholar

[19]

L. Molinet, D. Pilod and S. Vento, On well-posedness for some dispersive perturbations of Burgers' equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 1719-1756.  doi: 10.1016/j.anihpc.2017.12.004.  Google Scholar

[20]

J.-C. Saut and Y. Wang, The wave breaking for Whitham-type equations revisited, preprint, arXiv: 2006.03803. Google Scholar

[21]

R. Yang, Shock formation for the Burgers-Hilbert equation, preprint, arXiv: 2006.05568. Google Scholar

[1]

Guolian Wang, Boling Guo. Stochastic Korteweg-de Vries equation driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5255-5272. doi: 10.3934/dcds.2015.35.5255
[2]

Eduardo Cerpa. Control of a Korteweg-de Vries equation: A tutorial. Mathematical Control & Related Fields, 2014, 4 (1) : 45-99. doi: 10.3934/mcrf.2014.4.45
[3]

M. Agrotis, S. Lafortune, P.G. Kevrekidis. On a discrete version of the Korteweg-De Vries equation. Conference Publications, 2005, 2005 (Special) : 22-29. doi: 10.3934/proc.2005.2005.22
[4]

Shou-Fu Tian. Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval. Communications on Pure & Applied Analysis, 2018, 17 (3) : 923-957. doi: 10.3934/cpaa.2018046
[5]

Anne de Bouard, Eric Gautier. Exit problems related to the persistence of solitons for the Korteweg-de Vries equation with small noise. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 857-871. doi: 10.3934/dcds.2010.26.857
[6]

Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Global stabilization of a coupled system of two generalized Korteweg-de Vries type equations posed on a finite domain. Mathematical Control & Related Fields, 2011, 1 (3) : 353-389. doi: 10.3934/mcrf.2011.1.353
[7]

Muhammad Usman, Bing-Yu Zhang. Forced oscillations of the Korteweg-de Vries equation on a bounded domain and their stability. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1509-1523. doi: 10.3934/dcds.2010.26.1509
[8]

Eduardo Cerpa, Emmanuelle Crépeau. Rapid exponential stabilization for a linear Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 655-668. doi: 10.3934/dcdsb.2009.11.655
[9]

Pierre Garnier. Damping to prevent the blow-up of the korteweg-de vries equation. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1455-1470. doi: 10.3934/cpaa.2017069
[10]

Ahmat Mahamat Taboye, Mohamed Laabissi. Exponential stabilization of a linear Korteweg-de Vries equation with input saturation. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021052
[11]

Hironobu Sasaki. Small data scattering for the Klein-Gordon equation with cubic convolution nonlinearity. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 973-981. doi: 10.3934/dcds.2006.15.973
[12]

Ivonne Rivas, Muhammad Usman, Bing-Yu Zhang. Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain. Mathematical Control & Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61
[13]

Netra Khanal, Ramjee Sharma, Jiahong Wu, Juan-Ming Yuan. A dual-Petrov-Galerkin method for extended fifth-order Korteweg-de Vries type equations. Conference Publications, 2009, 2009 (Special) : 442-450. doi: 10.3934/proc.2009.2009.442
[14]

Eduardo Cerpa, Emmanuelle Crépeau, Julie Valein. Boundary controllability of the Korteweg-de Vries equation on a tree-shaped network. Evolution Equations & Control Theory, 2020, 9 (3) : 673-692. doi: 10.3934/eect.2020028
[15]

Ludovick Gagnon. Qualitative description of the particle trajectories for the N-solitons solution of the Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1489-1507. doi: 10.3934/dcds.2017061
[16]

Arnaud Debussche, Jacques Printems. Convergence of a semi-discrete scheme for the stochastic Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 761-781. doi: 10.3934/dcdsb.2006.6.761
[17]

Brian Pigott. Polynomial-in-time upper bounds for the orbital instability of subcritical generalized Korteweg-de Vries equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 389-418. doi: 10.3934/cpaa.2014.13.389
[18]

Roberto A. Capistrano-Filho, Shuming Sun, Bing-Yu Zhang. General boundary value problems of the Korteweg-de Vries equation on a bounded domain. Mathematical Control & Related Fields, 2018, 8 (3&4) : 583-605. doi: 10.3934/mcrf.2018024
[19]

Qifan Li. Local well-posedness for the periodic Korteweg-de Vries equation in analytic Gevrey classes. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1097-1109. doi: 10.3934/cpaa.2012.11.1097
[20]

Olivier Goubet. Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations. Discrete & Continuous Dynamical Systems, 2000, 6 (3) : 625-644. doi: 10.3934/dcds.2000.6.625

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (250)
  • HTML views (251)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences