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August
2022, 42(8): 3841-3860.
doi: 10.3934/dcds.2022035
Enhanced existence time of solutions to evolution equations of Whitham type
| 1. | Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway |
| 2. | School of Mathematics and Statistics, Lanzhou University, 730000 Lanzhou, China |
| 3. | Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d'Orsay, 91405 Orsay, France |
We show that Whitham type equations $u_t + u u_x -\mathcal{L} u_x = 0$, where $L$ is a general Fourier multiplier operator of order $\alpha \in [-1, 1]$, $\alpha\neq 0$, allow for small solutions to be extended beyond their ordinary existence time. The result is valid for a range of quadratic dispersive equations with inhomogenous symbols in the dispersive regime given by the parameter $\alpha$.
Mathematics Subject Classification:
Primary: 76B15, 76B03; Secondary: 35S30, 35A20.
Citation:
Mats Ehrnström, Yuexun Wang. Enhanced existence time of solutions to evolution equations of Whitham type. Discrete and Continuous Dynamical Systems,
2022, 42
(8)
: 3841-3860.
doi: 10.3934/dcds.2022035
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References:
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L. Abdelouhab, J. L. Bona, M. Felland and J.-C. Saut,
Nonlocal models for nonlinear, dispersive waves, Phys. D, 40 (1989), 360-392.
doi: 10.1016/0167-2789(89)90050-X. |
| [2] |
T. Alazard and J.-M. Delort, Sobolev Estimates for Two Dimensional Gravity Water Waves, Astérisque, 2015. |
| [3] |
M. N. Arnesen,
Existence of solitary-wave solutions to nonlocal equations, Discrete Contin. Dyn. Syst., 36 (2016), 3483-3510.
doi: 10.3934/dcds.2016.36.3483. |
| [4] |
M. Berti and J.-M. Delort, Almost Global Solutions of Capillary-Gravity Water Waves Equations on the Circle, Lecture Notes of the Unione Matematica Italiana, 24. Springer, Cham; Unione Matematica Italiana, [Bologna], 2018.
doi: 10.1007/978-3-319-99486-4. |
| [5] |
M. Berti, R. Feola and L. Franzoi,
Quadratic life span of periodic gravity-capillary water waves, Water Waves, 3 (2021), 85-115.
doi: 10.1007/s42286-020-00036-8. |
| [6] |
G. Bruell and R. Dhara,
Waves of maximal height for a class of nonlocal equations with homogeneous symbols, Indiana Univ. Math. J., 70 (2021), 711-742.
doi: 10.1512/iumj.2021.70.8368. |
| [7] |
J.-M. Delort and J. Szeftel, Long-time existence for small data nonlinear Klein-Gordon equations on tori and spheres, Int. Math. Res. Not., (2004), 1897–1966.
doi: 10.1155/S1073792804133321. |
| [8] |
V. Duchêne, D. Nilsson and E. Wahlén,
Solitary wave solutions to a class of modified Green-Naghdi systems, J. Math. Fluid Mech., 20 (2018), 1059-1091.
doi: 10.1007/s00021-017-0355-0. |
| [9] |
M. Ehrnström, M. D. Groves and E. Wahlén,
On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type, Nonlinearity, 25 (2012), 1-34.
doi: 10.1088/0951-7715/25/10/2903. |
| [10] |
M. Ehrnström, M. A. Johnson, O. I. H. Maehlen and F. Remonato,
On the bifurcation diagram of the capillary-gravity Whitham equation, Water Waves, 1 (2019), 275-313.
doi: 10.1007/s42286-019-00019-4. |
| [11] |
M. Ehrnström and E. Wahlén,
On Whitham's conjecture of a highest cusped wave for a nonlocal dispersive equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 1603-1637.
doi: 10.1016/j.anihpc.2019.02.006. |
| [12] |
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. |
| [13] |
R. Feola, B. Grébert and F. Iandoli, Long time solutions for quasi-linear hamiltonian perturbations of schrödinger and klein-gordon equations on tori, preprint, arXiv: 2009.07553. |
| [14] |
P. Germain, N. Masmoudi and J. Shatah,
Global solutions for the gravity water waves equation in dimension 3, Ann. of Math., 175 (2012), 691-754.
doi: 10.4007/annals.2012.175.2.6. |
| [15] |
F. Hildrum,
Solitary waves in dispersive evolution equations of Whitham type with nonlinearities of mild regularity, Nonlinearity, 33 (2020), 1594-1624.
doi: 10.1088/1361-6544/ab60d5. |
| [16] |
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. |
| [17] |
J. K. Hunter, M. Ifrim and D. Tataru,
Two dimensional water waves in holomorphic coordinates, Comm. Math. Phys., 346 (2016), 483-552.
doi: 10.1007/s00220-016-2708-6. |
| [18] |
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. |
| [19] |
V. M. Hur,
Wave breaking in the Whitham equation, Adv. Math., 317 (2017), 410-437.
doi: 10.1016/j.aim.2017.07.006. |
| [20] |
M. Ifrim and D. Tataru,
The lifespan of small data solutions in two dimensional capillary water waves, Arch. Ration. Mech. Anal., 225 (2017), 1279-1346.
doi: 10.1007/s00205-017-1126-z. |
| [21] |
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. |
| [22] |
A. D. Ionescu and F. Pusateri, Global Regularity for 2D Water Waves with Surface Tension, Mem. Amer. Math. Soc. 256, 2018
doi: 10.1090/memo/1227. |
| [23] |
A. D. Ionescu and F. Pusateri,
Long-time existence for multi-dimensional periodic water waves, Geom. Funct. Anal., 29 (2019), 811-870.
doi: 10.1007/s00039-019-00490-8. |
| [24] |
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. |
| [25] |
D. Lannes, The Water Waves Problem, Mathematical Surveys and Monographs, 188. American Mathematical Society, Providence, RI, 2013.
doi: 10.1090/surv/188. |
| [26] |
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. |
| [27] |
O. I. H. Maehlen,
Solitary waves for weakly dispersive equations with inhomogeneous nonlinearities, Discrete Contin. Dyn. Syst., 40 (2020), 4113-4130.
doi: 10.3934/dcds.2020174. |
| [28] |
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. |
| [29] |
L. Molinet, J. C. Saut and N. Tzvetkov,
Ill-posedness issues for the Benjamin-Ono and related equations, SIAM J. Math. Anal., 33 (2001), 982-988.
doi: 10.1137/S0036141001385307. |
| [30] |
D. Nilsson, Extended lifespan of the fractional BBM equation, Asymptotic Analysis, (2021), 1–21. |
| [31] |
J.-C. Saut,
Sur quelques généralisations de l'équation de Korteweg-de Vries, J. Math. Pures Appl., 58 (1979), 21-61.
|
| [32] |
J.-C. Saut and Y. Wang, The wave breaking for Whitham-type equations revisited, arXiv: 2006.03803, to appear in SIAM. J. Math. Anal. |
| [33] |
J. Shatah,
Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math., 38 (1985), 685-696.
doi: 10.1002/cpa.3160380516. |
| [34] |
A. Stefanov and J. D. Wright,
Small amplitude traveling waves in the full-dispersion Whitham equation, J. Dynam. Differential Equations, 32 (2020), 85-99.
doi: 10.1007/s10884-018-9713-8. |
| [35] |
S. Wu,
Almost global wellposedness of the 2-D full water wave problem, Invent. Math., 177 (2009), 45-135.
doi: 10.1007/s00222-009-0176-8. |
show all references
References:
| [1] |
L. Abdelouhab, J. L. Bona, M. Felland and J.-C. Saut,
Nonlocal models for nonlinear, dispersive waves, Phys. D, 40 (1989), 360-392.
doi: 10.1016/0167-2789(89)90050-X. |
| [2] |
T. Alazard and J.-M. Delort, Sobolev Estimates for Two Dimensional Gravity Water Waves, Astérisque, 2015. |
| [3] |
M. N. Arnesen,
Existence of solitary-wave solutions to nonlocal equations, Discrete Contin. Dyn. Syst., 36 (2016), 3483-3510.
doi: 10.3934/dcds.2016.36.3483. |
| [4] |
M. Berti and J.-M. Delort, Almost Global Solutions of Capillary-Gravity Water Waves Equations on the Circle, Lecture Notes of the Unione Matematica Italiana, 24. Springer, Cham; Unione Matematica Italiana, [Bologna], 2018.
doi: 10.1007/978-3-319-99486-4. |
| [5] |
M. Berti, R. Feola and L. Franzoi,
Quadratic life span of periodic gravity-capillary water waves, Water Waves, 3 (2021), 85-115.
doi: 10.1007/s42286-020-00036-8. |
| [6] |
G. Bruell and R. Dhara,
Waves of maximal height for a class of nonlocal equations with homogeneous symbols, Indiana Univ. Math. J., 70 (2021), 711-742.
doi: 10.1512/iumj.2021.70.8368. |
| [7] |
J.-M. Delort and J. Szeftel, Long-time existence for small data nonlinear Klein-Gordon equations on tori and spheres, Int. Math. Res. Not., (2004), 1897–1966.
doi: 10.1155/S1073792804133321. |
| [8] |
V. Duchêne, D. Nilsson and E. Wahlén,
Solitary wave solutions to a class of modified Green-Naghdi systems, J. Math. Fluid Mech., 20 (2018), 1059-1091.
doi: 10.1007/s00021-017-0355-0. |
| [9] |
M. Ehrnström, M. D. Groves and E. Wahlén,
On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type, Nonlinearity, 25 (2012), 1-34.
doi: 10.1088/0951-7715/25/10/2903. |
| [10] |
M. Ehrnström, M. A. Johnson, O. I. H. Maehlen and F. Remonato,
On the bifurcation diagram of the capillary-gravity Whitham equation, Water Waves, 1 (2019), 275-313.
doi: 10.1007/s42286-019-00019-4. |
| [11] |
M. Ehrnström and E. Wahlén,
On Whitham's conjecture of a highest cusped wave for a nonlocal dispersive equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 1603-1637.
doi: 10.1016/j.anihpc.2019.02.006. |
| [12] |
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. |
| [13] |
R. Feola, B. Grébert and F. Iandoli, Long time solutions for quasi-linear hamiltonian perturbations of schrödinger and klein-gordon equations on tori, preprint, arXiv: 2009.07553. |
| [14] |
P. Germain, N. Masmoudi and J. Shatah,
Global solutions for the gravity water waves equation in dimension 3, Ann. of Math., 175 (2012), 691-754.
doi: 10.4007/annals.2012.175.2.6. |
| [15] |
F. Hildrum,
Solitary waves in dispersive evolution equations of Whitham type with nonlinearities of mild regularity, Nonlinearity, 33 (2020), 1594-1624.
doi: 10.1088/1361-6544/ab60d5. |
| [16] |
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. |
| [17] |
J. K. Hunter, M. Ifrim and D. Tataru,
Two dimensional water waves in holomorphic coordinates, Comm. Math. Phys., 346 (2016), 483-552.
doi: 10.1007/s00220-016-2708-6. |
| [18] |
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. |
| [19] |
V. M. Hur,
Wave breaking in the Whitham equation, Adv. Math., 317 (2017), 410-437.
doi: 10.1016/j.aim.2017.07.006. |
| [20] |
M. Ifrim and D. Tataru,
The lifespan of small data solutions in two dimensional capillary water waves, Arch. Ration. Mech. Anal., 225 (2017), 1279-1346.
doi: 10.1007/s00205-017-1126-z. |
| [21] |
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. |
| [22] |
A. D. Ionescu and F. Pusateri, Global Regularity for 2D Water Waves with Surface Tension, Mem. Amer. Math. Soc. 256, 2018
doi: 10.1090/memo/1227. |
| [23] |
A. D. Ionescu and F. Pusateri,
Long-time existence for multi-dimensional periodic water waves, Geom. Funct. Anal., 29 (2019), 811-870.
doi: 10.1007/s00039-019-00490-8. |
| [24] |
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. |
| [25] |
D. Lannes, The Water Waves Problem, Mathematical Surveys and Monographs, 188. American Mathematical Society, Providence, RI, 2013.
doi: 10.1090/surv/188. |
| [26] |
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. |
| [27] |
O. I. H. Maehlen,
Solitary waves for weakly dispersive equations with inhomogeneous nonlinearities, Discrete Contin. Dyn. Syst., 40 (2020), 4113-4130.
doi: 10.3934/dcds.2020174. |
| [28] |
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. |
| [29] |
L. Molinet, J. C. Saut and N. Tzvetkov,
Ill-posedness issues for the Benjamin-Ono and related equations, SIAM J. Math. Anal., 33 (2001), 982-988.
doi: 10.1137/S0036141001385307. |
| [30] |
D. Nilsson, Extended lifespan of the fractional BBM equation, Asymptotic Analysis, (2021), 1–21. |
| [31] |
J.-C. Saut,
Sur quelques généralisations de l'équation de Korteweg-de Vries, J. Math. Pures Appl., 58 (1979), 21-61.
|
| [32] |
J.-C. Saut and Y. Wang, The wave breaking for Whitham-type equations revisited, arXiv: 2006.03803, to appear in SIAM. J. Math. Anal. |
| [33] |
J. Shatah,
Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math., 38 (1985), 685-696.
doi: 10.1002/cpa.3160380516. |
| [34] |
A. Stefanov and J. D. Wright,
Small amplitude traveling waves in the full-dispersion Whitham equation, J. Dynam. Differential Equations, 32 (2020), 85-99.
doi: 10.1007/s10884-018-9713-8. |
| [35] |
S. Wu,
Almost global wellposedness of the 2-D full water wave problem, Invent. Math., 177 (2009), 45-135.
doi: 10.1007/s00222-009-0176-8. |
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