Exemples d'instabilités pour des équations d'ondes non linéaires  [ Examples of instabilities for nonlinear wave equations ]
Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Talk no. 911, p. 63-75

The aim of the lecture is to provide a guide to a paper by Gilles Lebeau where it is shown that the Cauchy problem for the overcritical wave equation ( t 2 -Δ x )u+u p =0 is ill-posed in the sense of Hadamard in the energy space, for p7 in dimension 3. The proof relies on geometric optics constructions and analyses of instabilities in highly nonlinear regimes. We shall give the steps of the analysis, trying to place them in their more general context: construction of asymptotic solutions where eikonal equations and amplitude equations are linked, linear and nonlinear instability mechanisms by resonance and interactions.

Le but de l’exposé est de donner un guide de lecture pour un article de Gilles Lebeau où il est montré que le problème de Cauchy pour l’équation d’onde surcritique ( t 2 -Δ x )u+u p =0 est mal posé au sens de Hadamard dans l’espace d’énergie, pour p7 en dimension 3. La preuve repose sur des constructions d’optique géométrique et des analyses d’instabilité dans des régimes fortement non linéaires. On donnera les étapes de l’analyse en essayant de les situer dans leur contexte plus général : construction de solutions asymptotiques où équations eikonales et équations d’amplitudes sont liées, mécanismes d’instabilité linéaires et non linéaires par résonances et interactions.

Classification:  35L30
Keywords: nonlinear wave equations, geometric optics, instability
@incollection{SB_2002-2003__45__63_0,
     author = {M\'etivier, Guy},
     title = {Exemples d'instabilit\'es pour des \'equations d'ondes non lin\'eaires},
     booktitle = {S\'eminaire Bourbaki : volume 2002/2003, expos\'es 909-923},
     author = {Collectif},
     series = {Ast\'erisque},
     publisher = {Association des amis de Nicolas Bourbaki, Soci\'et\'e math\'ematique de France},
     address = {Paris},
     number = {294},
     year = {2004},
     note = {talk:911},
     pages = {63-75},
     zbl = {1215.35106},
     language = {fr},
     url = {http://www.numdam.org/item/SB_2002-2003__45__63_0}
}
Métivier, Guy. Exemples d'instabilités pour des équations d'ondes non linéaires, in Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Talk no. 911, pp. 63-75. http://www.numdam.org/item/SB_2002-2003__45__63_0/

[1] H. Bahouri & P. Gérard - “High frequency approximation of solutions to critical nonlinear wave equations”, Amer. J. Math. 121 (1999), p. 131-175. | MR 1705001 | Zbl 0919.35089

[2] C. Cheverry, O. Guès & G. Métivier - “Oscillations fortes sur un champ linéairement dégénéré” 36 (2003), p. 691-745. | Numdam | MR 2032985 | Zbl 1091.35039

[3] S. Friedlandler, W. Strauss & M. Vishik - “Nonlinear instability in an ideal fluid”, Ann. Inst. H. Poincaré. Anal. Non Linéaire 14 (1997), no. 2, p. 187-209. | Numdam | MR 1441392 | Zbl 0874.76026

[4] J. Ginibre, A. Soffer & G. Velo - “The global Cauchy problem for the critical nonlinear wave equation”, J. Funct. Anal. 110 (1992), p. 96-130. | MR 1190421 | Zbl 0813.35054

[5] E. Grenier - “On the nonlinear instability of Euler and Prandtl equations”, Comm. Pure Appl. Math. 53 (2000), no. 9, p. 1067-1091. | MR 1761409 | Zbl 1048.35081

[6] M. Grillakis - “Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity”, Ann. of Math. 132 (1990), p. 485-509. | MR 1078267 | Zbl 0736.35067

[7] J. Hunter, A. Majda & R. Rosales - “Resonantly interacting weakly nonlinear hyperbolic waves II : several space variables”, Stud. Appl. Math. 75 (1986), p. 187-226. | MR 867874 | Zbl 0657.35084

[8] J.-L. Joly, G. Métivier & J. Rauch - “Recent results in non-linear geometric optics”, in Hyperbolic problems : theory, numerics, applications, Vol. II, (Zürich 1998), Internat. Ser. Numer. Math., vol. 130, Birkhäuser Verlag, Basel, 1999, p. 723-736. | MR 1717244 | Zbl 0938.35034

[9] -, “Transparent Nonlinear Geometric Optics and Maxwell-Bloch Equations”, J. Differential Equations 166 (2000), p. 175-250. | MR 1779260 | Zbl 1170.78311

[10] G. Lebeau - “Nonlinear optic and supercritical wave equation”, preprint et “Optique non linéaire et ondes surcritiques”, Séminaire EDP École Polytechnique, 1999. | Numdam | MR 1813167 | Zbl 1069.35512

[11] H. Lindblad & C. Sogge - “On existence and scattering with minimal regularity for semilinear wave equations”, J. Funct. Anal. 130 (1995), p. 357-426. | MR 1335386 | Zbl 0846.35085

[12] V. Maslov & G. Omel'Yanov - Geometric Asymptotics for Nonlinear PDE, Trans. Mathematical Monographs, vol. 202, AMS, 2000. | Zbl 0984.35001

[13] J. Rauch - “The u 5 Klein-Gordon equation. II. Anomalous singularities for semilinear wave equations”, in Nonlinear partial differential equations and their applications, Vol. I, Collège de France Seminar, Paris, 1978-1979, Res. notes in math., Pitman (Advanced Publishing Program) Publ., 1981, p. 335-364. | MR 631403 | Zbl 0473.35055

[14] D. Serre - “Oscillations non linéaires des systèmes hyperboliques : méthodes et résultats qualitatifs”, Ann. Inst. H. Poincaré. Anal. Non Linéaire 8 (1991), p. 351-417. | Numdam | MR 1127931 | Zbl 0810.35060

[15] J. Shatah & M. Struwe - “Well posedness in the energy estimate for semi-linear wave equations with critical growth”, I.M.R.N. (1994), p. 303-309. | MR 1283026 | Zbl 0830.35086

[16] M. Struwe - “Globally regular solutions to the u 5 Klein-Gordon equation”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15 (1988), p. 495-513. | Numdam | MR 1015805 | Zbl 0728.35072

[17] G. B. Whitham - Linear and Nonlinear Waves, Pure and Applied Mathematics, John Wiley and Sons, 1974. | MR 483954 | Zbl 0940.76002