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 theory, which is now quite well developed, I plan to discuss a more general point of view concerning the 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.
@incollection{JEDP_1999____A20_0,
author = {Sergi\`u Klainerman and Damiano Foschi},
title = {On bilinear estimates for wave equations},
booktitle = {},
series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
eid = {20},
pages = {1--17},
year = {1999},
publisher = {Universit\'e de Nantes},
doi = {10.5802/jedp.564},
zbl = {01810593},
language = {en},
url = {https://proceedings.centre-mersenne.org/articles/10.5802/jedp.564/}
}
TY - JOUR AU - Sergiù Klainerman AU - Damiano Foschi TI - On bilinear estimates for wave equations JO - Journées équations aux dérivées partielles PY - 1999 SP - 1 EP - 17 PB - Université de Nantes UR - https://proceedings.centre-mersenne.org/articles/10.5802/jedp.564/ DO - 10.5802/jedp.564 LA - en ID - JEDP_1999____A20_0 ER -
%0 Journal Article %A Sergiù Klainerman %A Damiano Foschi %T On bilinear estimates for wave equations %J Journées équations aux dérivées partielles %D 1999 %P 1-17 %I Université de Nantes %U https://proceedings.centre-mersenne.org/articles/10.5802/jedp.564/ %R 10.5802/jedp.564 %G en %F JEDP_1999____A20_0
Sergiù Klainerman; Damiano Foschi. On bilinear estimates for wave equations. Journées équations aux dérivées partielles (1999), article no. 20, 17 p.. doi: 10.5802/jedp.564
[1] and , Bilinear space-time estimates for homogeneous wave equations, preprint (1998).
[2] and , Generalized Strichartz inequalities for the wave equation, J. Funct. Anal. 133 (1995), no. 1, 50-68. | MR | Zbl
[3] , Weak and yet weaker solutions of semilinear wave equations, Comm. Partial Differential Equations 19 (1994), no. 9-10, 1629-1676. | MR | Zbl
[4] and , Endpoint Strichartz estimates, American Journal of Mathematics (1998), to appear. | Zbl
[5] and , Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math. 46 (1993), no. 9, 1221-1268. | MR | Zbl
[6] and , Finite energy solutions of the Yang-Mills equations in ℝ3+1, Annals of Mathematics 142 (1995), 39-119. | Zbl
[7] and , Estimates for null forms and the spaces Hs, δ, Int. Math. Res. Not. (1996), no. 17, 853-865. | Zbl
[8] and , Remark on the optimal regularity for equations of wave maps type, Comm. Partial Differential Equations 22 (1997), no. 5-6, 901-918. | MR | Zbl
[9] and , On the optimal regularity for Yang-Mills equations in ℝ4+1, preprint (1998).
[10] and , On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal. 130 (1995), no. 2, 357-426. | MR | Zbl
[11] , Nonlinear small data scattering for the wave and Klein-Gordon equation, Math. Z. 185 (1984), no. 2, 261-270. | MR | EuDML | Zbl
[12] , Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), no. 3, 705-714. | MR | Zbl
[13] , Low regularity semi-linear wave equations, preprint (1998).
[14] , Local and global results for wave maps i, to appear, Comm. PDE (1998). | Zbl
Cité par Sources :