Improved global well-posedness for the quartic Korteweg-de Vries equation

Correia, Simão

doi:10.1090/proc/16911

Improved global well-posedness for the quartic Korteweg-de Vries equation
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by Simão Correia;

Proc. Amer. Math. Soc. 152 (2024), 5117-5136

DOI: https://doi.org/10.1090/proc/16911

Published electronically: September 26, 2024

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Abstract:

We prove that the quartic Korteweg-de Vries equation is globally well-posed for real-valued initial data in $H^s(\mathbb {R})$, $s>-1/24$.

References

Jean Bourgain, A remark on normal forms and the “$I$-method” for periodic NLS, J. Anal. Math. 94 (2004), 125–157. MR 2124457, DOI 10.1007/BF02789044
J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Global well-posedness for KdV in Sobolev spaces of negative index, Electron. J. Differential Equations (2001), No. 26, 7. MR 1824796
J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal. 33 (2001), no. 3, 649–669. MR 1871414, DOI 10.1137/S0036141001384387
J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, A refined global well-posedness result for Schrödinger equations with derivative, SIAM J. Math. Anal. 34 (2002), no. 1, 64–86. MR 1950826, DOI 10.1137/S0036141001394541
J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\Bbb R$ and $\Bbb T$, J. Amer. Math. Soc. 16 (2003), no. 3, 705–749. MR 1969209, DOI 10.1090/S0894-0347-03-00421-1
J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Resonant decompositions and the $I$-method for the cubic nonlinear Schrödinger equation on $\Bbb R^2$, Discrete Contin. Dyn. Syst. 21 (2008), no. 3, 665–686. MR 2399431, DOI 10.3934/dcds.2008.21.665
Adán J. Corcho and Mahendra Panthee, Global well-posedness for a coupled modified KdV system, Bull. Braz. Math. Soc. (N.S.) 43 (2012), no. 1, 27–57. MR 2909922, DOI 10.1007/s00574-012-0004-4
Simão Correia, Filipe Oliveira, and Jorge Drumond Silva, Sharp local well-posedness and nonlinear smoothing for dispersive equations through frequency-restricted estimates, SIAM J. Math. Anal. 56 (2024), no. 4, 5604–5633. MR 4782510, DOI 10.1137/23M156923X
Luiz Gustavo Farah, Global rough solutions to the critical generalized KdV equation, J. Differential Equations 249 (2010), no. 8, 1968–1985. MR 2679011, DOI 10.1016/j.jde.2010.05.010
Axel Grünrock, A bilinear Airy-estimate with application to gKdV-3, Differential Integral Equations 18 (2005), no. 12, 1333–1339. MR 2174975
Axel Grünrock, Mahendra Panthee, and Jorge Drumond Silva, A remark on global well-posedness below $L^2$ for the GKDV-3 equation, Differential Integral Equations 20 (2007), no. 11, 1229–1236. MR 2372424
Zihua Guo, Soonsik Kwon, and Tadahiro Oh, Poincaré-Dulac normal form reduction for unconditional well-posedness of the periodic cubic NLS, Comm. Math. Phys. 322 (2013), no. 1, 19–48. MR 3073156, DOI 10.1007/s00220-013-1755-5
Nobu Kishimoto, Resonant decomposition and the $I$-method for the two-dimensional Zakharov system, Discrete Contin. Dyn. Syst. 33 (2013), no. 9, 4095–4122. MR 3038054, DOI 10.3934/dcds.2013.33.4095
Soonsik Kwon, Tadahiro Oh, and Haewon Yoon, Normal form approach to unconditional well-posedness of nonlinear dispersive PDEs on the real line, Ann. Fac. Sci. Toulouse Math. (6) 29 (2020), no. 3, 649–720 (English, with English and French summaries). MR 4196700, DOI 10.5802/afst.1643
Terence Tao, Scattering for the quartic generalised Korteweg-de Vries equation, J. Differential Equations 232 (2007), no. 2, 623–651. MR 2286393, DOI 10.1016/j.jde.2006.07.019
Nikolaos Tzirakis, The Cauchy problem for the semilinear quintic Schrödinger equation in one dimension, Differential Integral Equations 18 (2005), no. 8, 947–960. MR 2150447