A minicourse on global existence and blowup of classical solutions to multidimensional quasilinear wave equations
Journées équations aux dérivées partielles (2002), article no. 1, 33 p.

The aim of this mini-course is twofold: describe quickly the framework of quasilinear wave equation with small data; and give a detailed sketch of the proofs of the blowup theorems in this framework. The first chapter introduces the main tools and concepts, and presents the main results as solutions of natural conjectures. The second chapter gives a self-contained account of geometric blowup and of its applications to present problem.

@article{JEDP_2002____A1_0,
     author = {Alinhac, Serge},
     title = {A minicourse on global existence and blowup of classical solutions to multidimensional quasilinear wave equations},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {1},
     pages = {1--33},
     publisher = {Universit\'e de Nantes},
     year = {2002},
     doi = {10.5802/jedp.599},
     mrnumber = {1968197},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/jedp.599/}
}
TY  - JOUR
AU  - Alinhac, Serge
TI  - A minicourse on global existence and blowup of classical solutions to multidimensional quasilinear wave equations
JO  - Journées équations aux dérivées partielles
PY  - 2002
SP  - 1
EP  - 33
PB  - Université de Nantes
UR  - http://archive.numdam.org/articles/10.5802/jedp.599/
DO  - 10.5802/jedp.599
LA  - en
ID  - JEDP_2002____A1_0
ER  - 
%0 Journal Article
%A Alinhac, Serge
%T A minicourse on global existence and blowup of classical solutions to multidimensional quasilinear wave equations
%J Journées équations aux dérivées partielles
%D 2002
%P 1-33
%I Université de Nantes
%U http://archive.numdam.org/articles/10.5802/jedp.599/
%R 10.5802/jedp.599
%G en
%F JEDP_2002____A1_0
Alinhac, Serge. A minicourse on global existence and blowup of classical solutions to multidimensional quasilinear wave equations. Journées équations aux dérivées partielles (2002), article  no. 1, 33 p. doi : 10.5802/jedp.599. http://archive.numdam.org/articles/10.5802/jedp.599/

[1] Alinhac S., "Explosion géométrique pour des systèmes quasi-linéaires", Amer. J. Math. 117(4), 1995, 987-1017. | MR | Zbl

[2] Alinhac S., "Temps de vie précisé et explosion géométrique pour des systèmes hyperboliques quasilinéaires en dimension un d'espace", Ann. Scuola Norm. Sup. Pisa, Serie IV vol. XXII (3), 1995, 493-515. | Numdam | MR | Zbl

[3] Alinhac S., "Explosion des solutions d'une équation d'ondes quasi-linéaire en deux dimensions d'espace", Comm. PDE 21(5,6), 1996, 923-969. | MR | Zbl

[4] Alinhac S., "Blowup of small data solutions for a quasilinear wave equation in two space dimensions", Ann. Maths 149, 1999, 97-127. | MR | Zbl

[5] Alinhac S., "Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions II", Acta Mat. 182, 1999, 1-23. | MR | Zbl

[6] Alinhac S., "Rank two singular solutions for quasilinear wave equations", Int. Res. Math. Notices 18, 2000, 955-984. | MR | Zbl

[7] Alinhac S., "Remarks on the blowup rate of classical solutions to quasilinear multidimensional hyperbolic systems", J. Math. Pure Appl. 79, 2000, 839-854. I-30 | MR | Zbl

[8] Alinhac S., "Stability of geometric blowup", Arch. Rat. Mech. Analysis 150, 1999, 97-125. | MR | Zbl

[9] Alinhac S., "The null condition for quasilinear wave equations in two space dimensions I", Invent. Math. 145, (2001), 597-618. | MR | Zbl

[10] Alinhac S., "The null condition for quasilinear wave equations in two space dimensions II", Amer. J. Math. 123, (2000), 1-31. | MR | Zbl

[11] Alinhac S., "An Example of Blowup at Infinity for a Quasilinear Wave Equation", Preprint, Université Paris-Sud (Orsay), (2002).

[12] Alinhac S., " A remark on energy inequalities for perturbed wave equations", Preprint, Université Paris-Sud (Orsay), (2001). | MR

[13] Alinhac S., "Blowup for nonlinear hyperbolic equations", Progress in Nonlinear Differential Equations and their Applications, Birkhäuser, Boston, 1995. | MR | Zbl

[14] Alinhac S. and Gérard P., "Opérateurs pseudo-différentiels et théorème de Nash-Moser", InterEditions, Paris, 1991. | MR | Zbl

[15] Christodoulou D. and Klainerman S., " The global nonlinear stability of the Minkowski space", Princeton Math. Series 41, (1993). | MR | Zbl

[16] Hörmander L., "Lectures on Nonlinear hyperbolic differential equations", Math. et Appl. 26, (1997), Springer Verlag. | MR | Zbl

[17] Hoshiga A., "The initial value problems for quasilinear wave equations in two space dimensions with small data", Adv. Math. Sci. Appl. 5, (1995), 67-89. | MR | Zbl

[18] Klainerman S., "Uniform decay estimates and the Lorentz invariance of the classical wave equation", Comm. Pure Appl. Math. 38, (1985), 321-332. | MR | Zbl

[19] Klainerman S., "A Commuting Vectorfields Approach to Strichartz type Inequalities and Applications to Quasilinear Wave Equations", Int. Math. Res. Notices 5, (2001), 221-274. | MR | Zbl

[20] Klainerman S. and Sideris T., "On Almost Global Existence for Nonrelativistic Wave Equation in 3D", Comm. Pure Appl. Math. 49, (1996), 307-321. | MR | Zbl

[21] Kong De-Xing, "Cauchy Problem for Quasilinear Hyperbolic Systems", Memoirs Math. Soc. Japan 6, (2000). | MR | Zbl

[22] Ladhari R., "Petites solutions d'équations d'ondes quasi-linéaires en dimension deux d'espace", Thèse de Doctorat, Université Paris-Sud, (1999).

[23] Sideris T., "The null condition and global existence of nonlinear elastic waves", Invent. Math. 123, (1996) | MR | Zbl

Cited by Sources: