On bilinear estimates for wave equations
Klainerman, Sergiù ; Foschi, Damiano
Journées équations aux dérivées partielles, (1999), p. 1-17 / Harvested from Numdam

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.

Publié le : 1999-01-01
@article{JEDP_1999____A20_0,
     author = {Klainerman, Sergi\`u and Foschi, Damiano},
     title = {On bilinear estimates for wave equations},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     year = {1999},
     pages = {1-17},
     zbl = {01810593},
     language = {en},
     url = {http://dml.mathdoc.fr/item/JEDP_1999____A20_0}
}
Klainerman, Sergiù; Foschi, Damiano. On bilinear estimates for wave equations. Journées équations aux dérivées partielles,  (1999), pp. 1-17. http://gdmltest.u-ga.fr/item/JEDP_1999____A20_0/

[1] Damiano Foschi and Sergiu Klainerman, Bilinear space-time estimates for homogeneous wave equations, preprint (1998).

[2] Jean Ginibre and Giorgio Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal. 133 (1995), no. 1, 50-68. | MR 97a:46047 | Zbl 0849.35064

[3] Lev Kapitanski, Weak and yet weaker solutions of semilinear wave equations, Comm. Partial Differential Equations 19 (1994), no. 9-10, 1629-1676. | MR 95j:35041 | Zbl 0831.35109

[4] Mark Keel and Terence Tao, Endpoint Strichartz estimates, American Journal of Mathematics (1998), to appear. | Zbl 0922.35028

[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. | MR 94h:35137 | Zbl 0803.35095

[6] Sergiu Klainerman and Matei Machedon, Finite energy solutions of the Yang-Mills equations in ℝ3+1, Annals of Mathematics 142 (1995), 39-119. | Zbl 0827.53056

[7] Sergiu Klainerman and Matei Machedon, Estimates for null forms and the spaces Hs, δ, Int. Math. Res. Not. (1996), no. 17, 853-865. | Zbl 0909.35095

[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. | MR 99c:35163 | Zbl 0884.35102

[9] Sergiu Klainerman and Daniel Tataru, On the optimal regularity for Yang-Mills equations in ℝ4+1, preprint (1998).

[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. | MR 96i:35087 | Zbl 0846.35085

[11] Hartmut Pecher, Nonlinear small data scattering for the wave and Klein-Gordon equation, Math. Z. 185 (1984), no. 2, 261-270. | MR 85h:35165 | Zbl 0538.35063

[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. | MR 58 #23577 | Zbl 0372.35001

[13] Terence Tao, Low regularity semi-linear wave equations, preprint (1998).

[14] Daniel Tataru, Local and global results for wave maps i, to appear, Comm. PDE (1998). | Zbl 0914.35083