A minicourse on global existence and blowup of classical solutions to multidimensional quasilinear wave equations
Alinhac, Serge
Journées équations aux dérivées partielles, (2002), p. 1-33 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

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},
     year = {2002},
     pages = {1-33},
     doi = {10.5802/jedp.599},
     mrnumber = {1968197},
     language = {en},
     url = {http://dml.mathdoc.fr/item/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), pp. 1-33. doi : 10.5802/jedp.599. http://gdmltest.u-ga.fr/item/JEDP_2002____A1_0/

[1] Alinhac S., "Explosion géométrique pour des systèmes quasi-linéaires", Amer. J. Math. 117(4), 1995, 987-1017. | MR 1342838 | Zbl 0840.35060

[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 1360547 | Zbl 0840.35059

[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 1391528 | Zbl 0858.35082

[4] Alinhac S., "Blowup of small data solutions for a quasilinear wave equation in two space dimensions", Ann. Maths 149, 1999, 97-127. | MR 1680539 | Zbl 01278975

[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 1687180 | Zbl 0973.35135

[6] Alinhac S., "Rank two singular solutions for quasilinear wave equations", Int. Res. Math. Notices 18, 2000, 955-984. | MR 1792284 | Zbl 0971.35053

[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 1782105 | Zbl 0979.35092

[8] Alinhac S., "Stability of geometric blowup", Arch. Rat. Mech. Analysis 150, 1999, 97-125. | MR 1736700 | Zbl 0962.35119

[9] Alinhac S., "The null condition for quasilinear wave equations in two space dimensions I", Invent. Math. 145, (2001), 597-618. | MR 1856402 | Zbl 01682128

[10] Alinhac S., "The null condition for quasilinear wave equations in two space dimensions II", Amer. J. Math. 123, (2000), 1-31. | MR 1867312 | Zbl 1112.35342

[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 2076683

[13] Alinhac S., "Blowup for nonlinear hyperbolic equations", Progress in Nonlinear Differential Equations and their Applications, Birkhäuser, Boston, 1995. | MR 1339762 | Zbl 0820.35001

[14] Alinhac S. and Gérard P., "Opérateurs pseudo-différentiels et théorème de Nash-Moser", InterEditions, Paris, 1991. | MR 1172111 | Zbl 0791.47044

[15] Christodoulou D. and Klainerman S., " The global nonlinear stability of the Minkowski space", Princeton Math. Series 41, (1993). | MR 1316662 | Zbl 0827.53055

[16] Hörmander L., "Lectures on Nonlinear hyperbolic differential equations", Math. et Appl. 26, (1997), Springer Verlag. | MR 1466700 | Zbl 0881.35001

[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 1325960 | Zbl 0829.35080

[18] Klainerman S., "Uniform decay estimates and the Lorentz invariance of the classical wave equation", Comm. Pure Appl. Math. 38, (1985), 321-332. | MR 784477 | Zbl 0635.35059

[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 1820023 | Zbl 0993.35022

[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 1374174 | Zbl 0867.35064

[21] Kong De-Xing, "Cauchy Problem for Quasilinear Hyperbolic Systems", Memoirs Math. Soc. Japan 6, (2000). | MR 1797837 | Zbl 0959.35003

[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 1374204 | Zbl 0844.73016