Cubic Quasilinear wave equation and bilinear estimates
Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2000-2001), Exposé no. 17, 15 p.
Bahouri, Hajer 1 ; Chemin, Jean-Yves 2

1 Université de Tunis, Département de Mathématiques, 1060 Tunis, Tunisia
2 Université Pierre-et-Marie-Curie, Analyse numérique 4, place Jussieu, 75230 Paris Cedex 05, France
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     title = {Cubic {Quasilinear} wave equation and bilinear estimates},
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Bahouri, Hajer; Chemin, Jean-Yves. Cubic Quasilinear wave equation and bilinear estimates. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2000-2001), Exposé no. 17, 15 p. http://www.numdam.org/item/SEDP_2000-2001____A17_0/

[1] H. Bahouri and J.-Y. Chemin, Équations d’ondes quasilinéaires et inégalités de Strichartz, American Journal of Mathematics, 121, 1999, pages 1337–1377. | Zbl

[2] H. Bahouri and J.-Y. Chemin, Équations d’ondes quasilinéaires et effet dispersif, International Mathematical Research News, 21, 1999, pages 1141–1178. | Zbl

[3] H. Bahouri and J.-Y. Chemin, Microlocal analysis, bilinear estimates and cubic quasilinear wave equation, in preparation.

[4] J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Annales de l’École Normale Supérieure, 14, 1981, pages 209–246. | EuDML | Numdam | Zbl

[5] J.-M. Bony, personnal communication.

[6] J.-M. Bony and J.-Y. Chemin, Espaces fonctionnels associés au calcul de Weyl-Hörmander, Bulletin de la Société Mathématique de France, 122, 1994, pages 77–118. | EuDML | Numdam | MR | Zbl

[7] J.-M. Bony and N. Lerner, Quantification asymptotique et microlocalisation d’ordre supérieur, Annales de l’École Normale Supérieure, 22, 1989, pages 377–433. | EuDML | Numdam | Zbl

[8] J.-Y. Chemin and C.-J. Xu, Inclusions de Sobolev en calcul de Weyl-Hörmander et systèmes sous-elliptiques, Annales de l’École Normale Supérieure, 30, 1997, pages 719–751. | EuDML | Numdam | Zbl

[9] J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, Journal of Functional Analysis, 133, 1995, page 50–68. | MR | Zbl

[10] L. Hörmander, The analysis of linear partial differential equations, Springer Verlag, 1983.

[11] L. Kapitanski, Some generalization of the Strichartz-Brenner inequality, Leningrad Mathematical Journal, 1, 1990, pages 693–721. | Zbl

[12] S. Klainerman, The null condition and global existence to non linear wave equations, Communications in Pure and Applied Mathematics, 38, 1985, pages 631–641. | Zbl

[13] S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence theorem, Communications in Pure and Applied Mathematics, 46, 1993, pages 1221–1268. | MR | Zbl

[14] S. Klainerman and M. Machedon, Smoothing estimates for null forms and applications, Duke Mathematical Journal, 81, 1995, pages 99–133. | MR | Zbl

[15] S. Klainerman and M. Machedon, On the regularity properties of a model problem relates to wave maps, Duke Mathematical Journal, 87, 1997, pages 553–589. | MR | Zbl

[16] S. Klainerman and M. Machedon, Estimates for null forms and the spaces H s,δ , International Mathematical Research News, 15, 1998, pages 756–774.

[17] S. Klainerman and D. Tataru, On the optimal local regularity for the Yang-Mills equations in R 4+1 , Journal of the American Mathematical Society, 12, 1999, pages 93–116. | MR | Zbl

[18] H. Lindblad, A sharp couternexample to local existence of low regularity solutions to non linear wave equations, Duke Mathematical Journal, 72, 1993, pages 503–539. | MR | Zbl

[19] G. Ponce and T. Sideris, Local regularity of non linear wave equations in three space dimensions, Communications in Partial Differential Equations, 18, 1993, pages 169–177. | MR | Zbl

[20] H. Smith, A parametrix construction for wave equation with C 1,1 coefficients, Annales de l’Institut Fourier, 48, 1998, pages 797–835. | Numdam | Zbl

[21] D. Tataru, Local and global results for wave maps I, Communications in Partial Differential Equations, 23, 1998, pages 1781–1793. | MR | Zbl

[22] D. Tataru, Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients III, preprint | MR | Zbl