American Institute of Mathematical Sciences

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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  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 - A, 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

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