A minicourse on global existence and blowup of classical solutions to multidimensional quasilinear wave equations

Serge Alinhac

Journées équations aux dérivées partielles (2002)

  • page 1-33
  • ISSN: 0752-0360

Abstract

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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.

How to cite

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Alinhac, Serge. "A minicourse on global existence and blowup of classical solutions to multidimensional quasilinear wave equations." Journées équations aux dérivées partielles (2002): 1-33. <http://eudml.org/doc/93428>.

@article{Alinhac2002,
abstract = {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.},
author = {Alinhac, Serge},
journal = {Journées équations aux dérivées partielles},
language = {eng},
pages = {1-33},
publisher = {Université de Nantes},
title = {A minicourse on global existence and blowup of classical solutions to multidimensional quasilinear wave equations},
url = {http://eudml.org/doc/93428},
year = {2002},
}

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
PB - Université de Nantes
SP - 1
EP - 33
AB - 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.
LA - eng
UR - http://eudml.org/doc/93428
ER -

References

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  1. [1] Alinhac S., "Explosion géométrique pour des systèmes quasi-linéaires", Amer. J. Math. 117(4), 1995, 987-1017. Zbl0840.35060MR1342838
  2. [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. Zbl0840.35059MR1360547
  3. [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. Zbl0858.35082MR1391528
  4. [4] Alinhac S., "Blowup of small data solutions for a quasilinear wave equation in two space dimensions", Ann. Maths 149, 1999, 97-127. Zbl1080.35043MR1680539
  5. [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. Zbl0973.35135MR1687180
  6. [6] Alinhac S., "Rank two singular solutions for quasilinear wave equations", Int. Res. Math. Notices 18, 2000, 955-984. Zbl0971.35053MR1792284
  7. [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 Zbl0979.35092MR1782105
  8. [8] Alinhac S., "Stability of geometric blowup", Arch. Rat. Mech. Analysis 150, 1999, 97-125. Zbl0962.35119MR1736700
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  11. [11] Alinhac S., "An Example of Blowup at Infinity for a Quasilinear Wave Equation", Preprint, Université Paris-Sud (Orsay), (2002). Zbl1292.35059
  12. [12] Alinhac S., " A remark on energy inequalities for perturbed wave equations", Preprint, Université Paris-Sud (Orsay), (2001). MR2076683
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  20. [20] Klainerman S. and Sideris T., "On Almost Global Existence for Nonrelativistic Wave Equation in 3D", Comm. Pure Appl. Math. 49, (1996), 307-321. Zbl0867.35064MR1374174
  21. [21] Kong De-xing, "Cauchy Problem for Quasilinear Hyperbolic Systems", Memoirs Math. Soc. Japan 6, (2000). Zbl0959.35003MR1797837
  22. [22] Ladhari R., "Petites solutions d'équations d'ondes quasi-linéaires en dimension deux d'espace", Thèse de Doctorat, Université Paris-Sud, (1999). 
  23. [23] Sideris T., "The null condition and global existence of nonlinear elastic waves", Invent. Math. 123, (1996) Zbl0844.73016MR1374204

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