On bilinear estimates for wave equations

Sergiù Klainerman; Damiano Foschi

Journées équations aux dérivées partielles (1999)

  • page 1-17
  • ISSN: 0752-0360

Abstract

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I will start with a short review of the classical restriction theorem for the sphere and Strichartz estimates for the wave equation. I then plan to give a detailed presentation of their recent generalizations in the form of “Bilinear Estimates”. In addition to the L 2 theory, which is now quite well developed, I plan to discuss a more general point of view concerning the L p theory. By investigating simple examples I will derive necessary conditions for such estimates to be true. I also plan to discuss the relevance of these estimates to nonlinear wave equations.

How to cite

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Klainerman, Sergiù, and Foschi, Damiano. "On bilinear estimates for wave equations." Journées équations aux dérivées partielles (1999): 1-17. <http://eudml.org/doc/93378>.

@article{Klainerman1999,
abstract = {I will start with a short review of the classical restriction theorem for the sphere and Strichartz estimates for the wave equation. I then plan to give a detailed presentation of their recent generalizations in the form of “Bilinear Estimates”. In addition to the $L^2$ theory, which is now quite well developed, I plan to discuss a more general point of view concerning the $L^p$ theory. By investigating simple examples I will derive necessary conditions for such estimates to be true. I also plan to discuss the relevance of these estimates to nonlinear wave equations.},
author = {Klainerman, Sergiù, Foschi, Damiano},
journal = {Journées équations aux dérivées partielles},
keywords = {Strichartz estimates},
language = {eng},
pages = {1-17},
publisher = {Université de Nantes},
title = {On bilinear estimates for wave equations},
url = {http://eudml.org/doc/93378},
year = {1999},
}

TY - JOUR
AU - Klainerman, Sergiù
AU - Foschi, Damiano
TI - On bilinear estimates for wave equations
JO - Journées équations aux dérivées partielles
PY - 1999
PB - Université de Nantes
SP - 1
EP - 17
AB - I will start with a short review of the classical restriction theorem for the sphere and Strichartz estimates for the wave equation. I then plan to give a detailed presentation of their recent generalizations in the form of “Bilinear Estimates”. In addition to the $L^2$ theory, which is now quite well developed, I plan to discuss a more general point of view concerning the $L^p$ theory. By investigating simple examples I will derive necessary conditions for such estimates to be true. I also plan to discuss the relevance of these estimates to nonlinear wave equations.
LA - eng
KW - Strichartz estimates
UR - http://eudml.org/doc/93378
ER -

References

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  1. [1] Damiano Foschi and Sergiu Klainerman, Bilinear space-time estimates for homogeneous wave equations, preprint (1998). Zbl1011.35087
  2. [2] Jean Ginibre and Giorgio Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal. 133 (1995), no. 1, 50-68. Zbl0849.35064MR97a:46047
  3. [3] Lev Kapitanski, Weak and yet weaker solutions of semilinear wave equations, Comm. Partial Differential Equations 19 (1994), no. 9-10, 1629-1676. Zbl0831.35109MR95j:35041
  4. [4] Mark Keel and Terence Tao, Endpoint Strichartz estimates, American Journal of Mathematics (1998), to appear. Zbl0922.35028
  5. [5] Sergiu Klainerman and Matei Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math. 46 (1993), no. 9, 1221-1268. Zbl0803.35095MR94h:35137
  6. [6] Sergiu Klainerman and Matei Machedon, Finite energy solutions of the Yang-Mills equations in ℝ3+1, Annals of Mathematics 142 (1995), 39-119. Zbl0827.53056
  7. [7] Sergiu Klainerman and Matei Machedon, Estimates for null forms and the spaces Hs, δ, Int. Math. Res. Not. (1996), no. 17, 853-865. Zbl0909.35095
  8. [8] Sergiu Klainerman and Sigmund Selberg, Remark on the optimal regularity for equations of wave maps type, Comm. Partial Differential Equations 22 (1997), no. 5-6, 901-918. Zbl0884.35102MR99c:35163
  9. [9] Sergiu Klainerman and Daniel Tataru, On the optimal regularity for Yang-Mills equations in ℝ4+1, preprint (1998). 
  10. [10] Hans Lindblad and Christopher D. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal. 130 (1995), no. 2, 357-426. Zbl0846.35085MR96i:35087
  11. [11] Hartmut Pecher, Nonlinear small data scattering for the wave and Klein-Gordon equation, Math. Z. 185 (1984), no. 2, 261-270. Zbl0538.35063MR85h:35165
  12. [12] Robert S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), no. 3, 705-714. Zbl0372.35001MR58 #23577
  13. [13] Terence Tao, Low regularity semi-linear wave equations, preprint (1998). 
  14. [14] Daniel Tataru, Local and global results for wave maps i, to appear, Comm. PDE (1998). Zbl0914.35083

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