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Solution of the 33rd Palis-Pugh problem for gradient-like diffeomorphisms of a two-dimensional sphere
Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity
| 1. | Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d'Orsay, 91405 Orsay, France |
| 2. | School of Mathematics and Statistics, Lanzhou University, 370000 Lanzhou, China |
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.
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [12] |
V. M. Hur,
Wave breaking in the Whitham equation, Adv. Math., 317 (2017), 410-437.
doi: 10.1016/j.aim.2017.07.006. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [20] |
J.-C. Saut and Y. Wang, The wave breaking for Whitham-type equations revisited, preprint, arXiv: 2006.03803. |
| [21] |
R. Yang, Shock formation for the Burgers-Hilbert equation, preprint, arXiv: 2006.05568. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [12] |
V. M. Hur,
Wave breaking in the Whitham equation, Adv. Math., 317 (2017), 410-437.
doi: 10.1016/j.aim.2017.07.006. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [20] |
J.-C. Saut and Y. Wang, The wave breaking for Whitham-type equations revisited, preprint, arXiv: 2006.03803. |
| [21] |
R. Yang, Shock formation for the Burgers-Hilbert equation, preprint, arXiv: 2006.05568. |
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