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

* Corresponding author

Both authors acknowledge the support by grant nos. 231668 and 250070 from the Research Council of Norway. The second author also acknowledges the support of the ANR project ANuI, and the partial support of grant no. 830018 from China

Received  July 2021 Revised  February 2022 Published  August 2022 Early access  April 2022

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 $\alpha$.

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
References:
[1]

L. AbdelouhabJ. L. BonaM. 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. BertiR. 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. DuchêneD. 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. EhrnströmM. 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. EhrnströmM. A. JohnsonO. 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. GermainN. 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. HunterM. 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. HunterM. IfrimD. 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. LinaresD. 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. MolinetD. 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. MolinetJ. 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. AbdelouhabJ. L. BonaM. 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. BertiR. 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. DuchêneD. 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. EhrnströmM. 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. EhrnströmM. A. JohnsonO. 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. GermainN. 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. HunterM. 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. HunterM. IfrimD. 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. LinaresD. 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. MolinetD. 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. MolinetJ. 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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