Space-Time Resonances and the Null Condition for Wave Equations

Fabio Pusateri

Bollettino dell'Unione Matematica Italiana (2013)

  • Volume: 6, Issue: 3, page 513-529
  • ISSN: 0392-4041

Abstract

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In this note we describe a recent result obtained by the author and Shatah [26], concerning global existence and scattering for small solutions of nonlinear wave equations. Based on the analysis of space-time resonances, we formulate a very natural non-resonance condition for quadratic nonlinearities that guarantees the existence of global solutions with linear asymptotic behavior. This non-resonance condition turns out to be a generalization of the null condition given by Klainerman in his seminal work [21].

How to cite

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Pusateri, Fabio. "Space-Time Resonances and the Null Condition for Wave Equations." Bollettino dell'Unione Matematica Italiana 6.3 (2013): 513-529. <http://eudml.org/doc/294036>.

@article{Pusateri2013,
abstract = {In this note we describe a recent result obtained by the author and Shatah [26], concerning global existence and scattering for small solutions of nonlinear wave equations. Based on the analysis of space-time resonances, we formulate a very natural non-resonance condition for quadratic nonlinearities that guarantees the existence of global solutions with linear asymptotic behavior. This non-resonance condition turns out to be a generalization of the null condition given by Klainerman in his seminal work [21].},
author = {Pusateri, Fabio},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {513-529},
publisher = {Unione Matematica Italiana},
title = {Space-Time Resonances and the Null Condition for Wave Equations},
url = {http://eudml.org/doc/294036},
volume = {6},
year = {2013},
}

TY - JOUR
AU - Pusateri, Fabio
TI - Space-Time Resonances and the Null Condition for Wave Equations
JO - Bollettino dell'Unione Matematica Italiana
DA - 2013/10//
PB - Unione Matematica Italiana
VL - 6
IS - 3
SP - 513
EP - 529
AB - In this note we describe a recent result obtained by the author and Shatah [26], concerning global existence and scattering for small solutions of nonlinear wave equations. Based on the analysis of space-time resonances, we formulate a very natural non-resonance condition for quadratic nonlinearities that guarantees the existence of global solutions with linear asymptotic behavior. This non-resonance condition turns out to be a generalization of the null condition given by Klainerman in his seminal work [21].
LA - eng
UR - http://eudml.org/doc/294036
ER -

References

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  1. ALINHAC, S., Semilinear hyperbolic systems with blowup at infinity. Indiana Univ. Math. J., 55 (3) (2006),1209-1232. Zbl1122.35068MR2244605DOI10.1512/iumj.2006.55.2671
  2. CHOQUET-BRUHAT, Y. - CHRISTODOULOU, D., Existence of global solutions of the classical equations of gauge theories, C. R. Acad. Sci. Paris Sér. I Math., 293, no. 3 (1981), 195-199. Zbl0478.58027MR635980
  3. CHRISTODOULOU, D., Global solutions of nonlinear hyperbolic equations for small initial data. Comm. Pure Appl. Math., 39 (2) (1986), 267-282. Zbl0612.35090MR820070DOI10.1002/cpa.3160390205
  4. COIFMAN, R. - MEYER, Y., Au delà des opérateurs pseudo-différentiels. Astérisque, 57 (1978). Zbl0483.35082MR518170
  5. GERMAIN, P., Global existence for coupled Klein-Gordon equations with different speeds. ArXiv:1005.5238v1, 2010. Zbl1255.35162MR2915571DOI10.1007/s10440-011-9662-2
  6. GERMAIN, P. - MASMOUDI, N., Global existence for the Euler-Maxwell system, ArXiv:1107.1595v1, 2011. MR3239096DOI10.24033/asens.2219
  7. GERMAIN, P. - MASMOUDI, N. - SHATAH, J., Global solutions for 3D quadratic Schrödinger equations. Int. Math. Res. Not. IMRN, (3) (2009), 414-432. Zbl1156.35087MR2482120DOI10.1093/imrn/rnn135
  8. GERMAIN, P. - MASMOUDI, N. - SHATAH, J., Global solutions for the gravity surface water waves equation in dimension 3. Ann. of Math., 175, no. 2 (2012), 691-754. Zbl1241.35003MR2993751DOI10.4007/annals.2012.175.2.6
  9. GERMAIN, P. - MASMOUDI, N. - SHATAH, J., Global solutions for a class of 2d quadratic Schrödinger equations. J. Math. Pures Appl., 97, no. 5 (2012), 505-543. Zbl1244.35134MR2914945DOI10.1016/j.matpur.2011.09.008
  10. GERMAIN, P. - MASMOUDI, N. - SHATAH, J., Global existence for capillary waves. To appear in Comm. Pure Appl. Math., 2012. Zbl1244.35134MR3318019DOI10.1002/cpa.21535
  11. JOHN, F., Blow-up of solutions of nonlinear wave equations in three space dimensions. Manuscripta Math., 28 (1-3) (1979), 235-268. Zbl0406.35042MR535704DOI10.1007/BF01647974
  12. JOHN, F., Blow-up for quasilinear wave equations in three space dimensions. Comm. on Pure Appl. Math., 34 (1) (1981), 29-51. Zbl0453.35060MR600571DOI10.1002/cpa.3160340103
  13. JOHN, F. - KLAINERMAN, S., Almost global existence to nonlinear wave equations in three space dimensions. Comm. Pure Appl. Math., 37 (4) (1984), 443-455. Zbl0599.35104MR745325DOI10.1002/cpa.3160370403
  14. KATAYAMA, S., Asymptotic pointwise behavior for systems of semilinear wave equations in three space dimensions. arXiv:1101.3657, J. Hyperbolic Differ. Equ.9 (2) (2012), 263-323. Zbl1254.35142MR2928109DOI10.1142/S0219891612500099
  15. KATAYAMA, S. - YOKOYAMA, K., Global small amplitude solutions to systems of non-linear wave equations with multiple speeds. Osaka J. Math., 43, no. 2 (2006), 283-326. Zbl1195.35225MR2262337
  16. KATO, J. - PUSATERI, F., A new proof of long range scattering for critical nonlinear Schrödinger equations. Diff. Int. Equations, 24 (9-10) (2011), 923-940. MR2850346
  17. KEEL, M. - SMITH, H. - SOGGE, C., Almost global existence for some semilinear wave equations. Dedicated to the memory of Thomas H. Wolff. J. Anal. Math., 87 (2002), 265-279. MR1945285DOI10.1007/BF02868477
  18. KEEL, M. - SMITH, H. - SOGGE, C., Almost global existence for quasilinear wave equations in three space dimensions. J. Amer. Math. Soc., 17 (1) (2004), 109-153. Zbl1055.35075MR2015331DOI10.1090/S0894-0347-03-00443-0
  19. KLAINERMAN, S., Uniform decay estimates and the Lorentz invariance of the classical wave equation. Comm. Pure Appl. Math., 38 (3) (1985), 321-332. Zbl0635.35059MR784477DOI10.1002/cpa.3160380305
  20. KLAINERMAN, S., Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions. Comm. Pure Appl. Math., 38 (5) (1985), 631-641. Zbl0597.35100MR803252DOI10.1002/cpa.3160380512
  21. KLAINERMAN, S., The null condition and global existence to nonlinear wave equations. Nonlinear systems of partial differential equations in applied mathematics, Part 1 (Santa Fe, N.M., 1984). Lectures in Appl. Math., 23 (1986), 293-326. MR837683
  22. KLAINERMAN, S. - SIDERIS, T., On almost global existence for nonrelativistic wave equations in 3D. Comm. Pure Appl. Math., 49 (3) (1996), 307-321. Zbl0867.35064MR1374174DOI10.1002/(SICI)1097-0312(199603)49:3<307::AID-CPA4>3.0.CO;2-H
  23. LINDBLAD, H., Global solutions of nonlinear wave equations. Comm. Pure Appl. Math.45 (9) (1992), 1063-1096,. Zbl0840.35065MR1177476DOI10.1002/cpa.3160450902
  24. LINDBLAD, H. - RODNIANSKI, I., The weak null condition for Einstein's equations. C. R. Acad. Sci. Paris. Ser. I, 336 (11) (2003), 901-906. Zbl1045.35101MR1994592DOI10.1016/S1631-073X(03)00231-0
  25. LINDBLAD, H. - RODNIANSKI, I., Global existence for the Einstein vacuum equations in wave coordinates. Comm. Math. Phys., 256(1) (2005), 43-110. Zbl1081.83003MR2134337DOI10.1007/s00220-004-1281-6
  26. PUSATERI, F. - SHATAH, J., Space-time resonances and the null condition for (first order) systems of wave equations. arXiv:1109.5662 (2011), Comm. Pure and Appl. Math., 66 (2) (2013), 1495-1540. Zbl1284.35261MR3084697DOI10.1002/cpa.21461
  27. SIDERIS, T., The null condition and global existence of nonlinear elastic waves. Invent. Math.123, no. 2 (1996), 323-342. Zbl0844.73016MR1374204DOI10.1007/s002220050030
  28. SIDERIS, T., Nonresonance and global existence of prestressed nonlinear elastic waves. Ann. of Math. (2), 151, no. 2 (2000), 849-874. Zbl0957.35126MR1765712DOI10.2307/121050
  29. SIDERIS, T. - TU, S., Global existence for systems of nonlinear wave equations in 3D with multiple speeds. SIAM J. Math. Anal.33, no. 2 (2001), 477-488. Zbl1002.35091MR1857981DOI10.1137/S0036141000378966
  30. SHATAH, J., Normal forms and quadratic nonlinear Klein-Gordon equations. Comm. Pure Appl. Math., 38 (5) (1985), 685-696. Zbl0597.35101MR803256DOI10.1002/cpa.3160380516
  31. SHATAH, J. - STRUWE, M., Geometric wave equations, volume 2 of Courant Lecture Notes in Mathematics. New York University, Courant Institute of Mathematical Sciences, New York, 1998. Zbl0993.35001MR1674843
  32. SIMON, J., A wave operator for a nonlinear Klein-Gordon equation. Lett. Math. Phys.7, no. 5 (1983), 387-398. MR719852DOI10.1007/BF00398760

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