On bilinear estimates for wave equations
Journées équations aux dérivées partielles (1999), article no. 20, 17 p.
complète On bilinear restriction type estimates and applications to nonlinear wave equations

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.

@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},
     eid = {20},
     pages = {1--17},
     publisher = {Universit\'e de Nantes},
     year = {1999},
     zbl = {01810593},
     language = {en},
     url = {http://www.numdam.org/item/JEDP_1999____A20_0/}
}
TY  - JOUR
AU  - Klainerman, Sergiù
AU  - Foschi, Damiano
TI  - On bilinear estimates for wave equations
JO  - Journées équations aux dérivées partielles
PY  - 1999
SP  - 1
EP  - 17
PB  - Université de Nantes
UR  - http://www.numdam.org/item/JEDP_1999____A20_0/
LA  - en
ID  - JEDP_1999____A20_0
ER  - 
%0 Journal Article
%A Klainerman, Sergiù
%A Foschi, Damiano
%T On bilinear estimates for wave equations
%J Journées équations aux dérivées partielles
%D 1999
%P 1-17
%I Université de Nantes
%U http://www.numdam.org/item/JEDP_1999____A20_0/
%G en
%F JEDP_1999____A20_0
Klainerman, Sergiù; Foschi, Damiano. On bilinear estimates for wave equations. Journées équations aux dérivées partielles (1999), article  no. 20, 17 p. http://www.numdam.org/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 | Zbl

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

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

[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 | Zbl

[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

[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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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