Perte de régularité pour les équations d'ondes sur-critiques

Lebeau, Gilles

doi:10.24033/bsmf.2482

Lebeau, Gilles

Bulletin de la Société Mathématique de France, Tome 133 (2005) no. 1, pp. 145-157.

Résumé
Abstract

On prouve que le problème de Cauchy local pour l’équation d’onde sur-critique dans $ℝ^{d}$ , $□ u + u^{p} = 0$ , $p$ impair, avec $d \geq 3$ et $p > (d + 2) / (d - 2)$ , est mal posé dans $H^{σ}$ pour tout $σ \in] 1, σ_{crit} [$ , où $σ_{crit} = d / 2 - 2 / (p - 1)$ est l’exposant critique.

We prove that the local Cauchy problem for the supercritical wave equation in $ℝ^{d}$ , $□ u + u^{p} = 0$ , with $d \geq 3$ , $p > 3$ and $p > (d + 2) / (d - 2)$ , is ill-posed in $H^{σ}$ for every $σ \in] 1, σ_{c} [$ , where $σ_{c} = d / 2 - 2 / (p - 1)$ is the critical exponent.

MR Zbl | 2 citations dans Numdam

DOI : 10.24033/bsmf.2482

Classification : 35L05, 35L15
Mot clés : analyse microlocale, équations d'ondes non-linéaires
Keywords: microlocal analysis, nonlinear wave equations

@article{BSMF_2005__133_1_145_0,
     author = {Lebeau, Gilles},
     title = {Perte de r\'egularit\'e pour les \'equations d'ondes sur-critiques},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {145--157},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {133},
     number = {1},
     year = {2005},
     doi = {10.24033/bsmf.2482},
     mrnumber = {2145023},
     zbl = {1071.35020},
     language = {fr},
     url = {http://www.numdam.org/articles/10.24033/bsmf.2482/}
}

TY  - JOUR
AU  - Lebeau, Gilles
TI  - Perte de régularité pour les équations d'ondes sur-critiques
JO  - Bulletin de la Société Mathématique de France
PY  - 2005
SP  - 145
EP  - 157
VL  - 133
IS  - 1
PB  - Société mathématique de France
UR  - http://www.numdam.org/articles/10.24033/bsmf.2482/
DO  - 10.24033/bsmf.2482
LA  - fr
ID  - BSMF_2005__133_1_145_0
ER  -

%0 Journal Article
%A Lebeau, Gilles
%T Perte de régularité pour les équations d'ondes sur-critiques
%J Bulletin de la Société Mathématique de France
%D 2005
%P 145-157
%V 133
%N 1
%I Société mathématique de France
%U http://www.numdam.org/articles/10.24033/bsmf.2482/
%R 10.24033/bsmf.2482
%G fr
%F BSMF_2005__133_1_145_0

Lebeau, Gilles. Perte de régularité pour les équations d'ondes sur-critiques. Bulletin de la Société Mathématique de France, Tome 133 (2005) no. 1, pp. 145-157. doi : 10.24033/bsmf.2482. http://www.numdam.org/articles/10.24033/bsmf.2482/

Bibliographie
Cité par

[1] P. Brenner & W. Von Wahl - « Global classical solutions of non-linear wave equations », Math. Z. 176 (1981), p. 87-121. | MR | Zbl

[2] J. Ginibre & G. Velo - « The global Cauchy problem for the non linear Klein-Gordon equation », Math.Z. 189 (1985), p. 487-505. | MR | Zbl

[3] H. Hochstadt - « On the Determination of a Hill's Equation from its Spectrum », Arch. Rat. Mech. Anal. 19 (1965), p. 353-362. | MR | Zbl

[4] K. Jörgens - « Das Anfangswertproblem in Grossen fur eine Klasse nichtlinearer Wellingleichungen », Math. Z. 77 (1961), p. 295-308. | MR | Zbl

[5] G. Lebeau - « Non linear optic and supercritical wave equation », Bull. Soc. Math. Liège 70 (2001), p. 267-306. | MR | Zbl

[6] J.-L. Lions - Quelques méthodes de résolution des problèmes aux limites non-linéaires, Dunod Gauthier-Villars, Paris, 1969. | MR | Zbl

[7] I. Segal - « The Global Cauchy Problem for a Relativistic Scalar Field with Power Interaction », Bull. Soc. Math. France 91 (1963), p. 129-135. | Numdam | MR | Zbl

[8] J. Shatah & M. Struwe - « Well-Posedness in the Energy Space for Semilinear Wave Equation with Critical Growth », I.M.R.N. 7 (1994), p. 303-309. | MR | Zbl

[9] W. Strauss - « Nonlinear invariant wave equations », Invariant wave equations (Proc. “Ettore Majorana” Internat. School of Math. Phys., Erice, 1977), Lecture Notes in Phys., vol. 73, Springer, 1978, p. 197-249. | MR

Córdoba, Diego; Martínez-Zoroa, Luis; Ożański, Wojciech S. Instantaneous gap loss of Sobolev regularity for the 2D incompressible Euler equations, Duke Mathematical Journal, Volume 173 (2024) no. 10, pp. 1931-1971 | DOI:10.1215/00127094-2023-0052 | Zbl:1547.35518
Mainini, Edoardo; Percivale, Danilo On the weighted inertia-energy approach to forced wave equations, Journal of Differential Equations, Volume 385 (2024), pp. 121-154 | DOI:10.1016/j.jde.2023.12.005 | Zbl:1532.49020
de Roubin, Pierre; Okamoto, Mamoru Norm inflation for the viscous nonlinear wave equation, NoDEA. Nonlinear Differential Equations and Applications, Volume 31 (2024) no. 4, p. 38 (Id/No 52) | DOI:10.1007/s00030-024-00944-5 | Zbl:1541.35373
Tzvetkov, Nikolay Nonlinear PDE in the presence of singular randomness, European Mathematical Society Magazine, Volume 129 (2023), pp. 5-12 | DOI:10.4171/mag/164 | Zbl:1521.35003
Bhimani, Divyang G.; Haque, Saikatul Strong ill-posedness for fractional Hartree and cubic NLS equations, Journal of Functional Analysis, Volume 285 (2023) no. 11, p. 47 (Id/No 110157) | DOI:10.1016/j.jfa.2023.110157 | Zbl:1527.35365
Merle, Frank; Raphaël, Pierre; Rodnianski, Igor; Szeftel, Jeremie On blow up for the energy super critical defocusing nonlinear Schrödinger equations, Inventiones Mathematicae, Volume 227 (2022) no. 1, pp. 247-413 | DOI:10.1007/s00222-021-01067-9 | Zbl:1487.35353
Bhimani, Divyang G.; Haque, Saikatul Norm inflation with infinite loss of regularity at general initial data for nonlinear wave equations in Wiener amalgam and Fourier amalgam spaces, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 223 (2022), p. 14 (Id/No 113076) | DOI:10.1016/j.na.2022.113076 | Zbl:1503.35111
Xia, Bo Generic IllPosedness for Wave Equation of Power Type on Three-Dimensional Torus, International Mathematics Research Notices, Volume 2021 (2021) no. 20, p. 15533 | DOI:10.1093/imrn/rnaa068
Bhimani, Divyang G.; Haque, Saikatul Norm Inflation for Benjamin–Bona–Mahony Equation in Fourier Amalgam and Wiener Amalgam Spaces with Negative Regularity, Mathematics, Volume 9 (2021) no. 23, p. 3145 | DOI:10.3390/math9233145
Elgindi, Tarek M.; Masmoudi, Nader $L^{\infty}$ ill-posedness for a class of equations arising in hydrodynamics, Archive for Rational Mechanics and Analysis, Volume 235 (2020) no. 3, pp. 1979-2025 | DOI:10.1007/s00205-019-01457-7 | Zbl:1507.35153
Sun, Chenmin; Tzvetkov, Nikolay Concerning the pathological set in the context of probabilistic well-posedness, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 358 (2020) no. 9-10, pp. 989-999 | DOI:10.5802/crmath.102 | Zbl:1456.35140
Bhimani, Divyang G.; Carles, Rémi Norm inflation for nonlinear Schrödinger equations in Fourier-Lebesgue and modulation spaces of negative regularity, The Journal of Fourier Analysis and Applications, Volume 26 (2020) no. 6, p. 33 (Id/No 78) | DOI:10.1007/s00041-020-09788-w | Zbl:1455.35232
Sun, Chenmin; Tzvetkov, Nikolay New examples of probabilistic well-posedness for nonlinear wave equations, Journal of Functional Analysis, Volume 278 (2020) no. 2, p. 47 (Id/No 108322) | DOI:10.1016/j.jfa.2019.108322 | Zbl:1437.35504
Ghanmi, Radhia; Saanouni, Tarek Well-posedness issues for some critical coupled non-linear Klein-Gordon equations, Communications on Pure and Applied Analysis, Volume 18 (2019) no. 2, pp. 603-623 | DOI:10.3934/cpaa.2019030 | Zbl:1404.35385
Grande, Ricardo Space-time fractional nonlinear Schrödinger equation, SIAM Journal on Mathematical Analysis, Volume 51 (2019) no. 5, pp. 4172-4212 | DOI:10.1137/19m1247140 | Zbl:1431.35173
Tzvetkov, Nikolay Random data wave equations, Singular random dynamics. Cetraro, Italy, August 22–26, 2016. Lecture notes given at the summer school., Cham: Springer; Florence: Fondazione CIME, 2019, pp. 221-313 | DOI:10.1007/978-3-030-29545-5_4 | Zbl:1498.60299
Saanouni, T. A note on strong instability of standing waves for some semilinear wave and heat equations, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 165 (2018) no. 1, pp. 53-68 | DOI:10.1017/s0305004117000226 | Zbl:1395.35022
Saanouni, Tarek Fourth-order damped wave equation with exponential growth nonlinearity, Annales Henri Poincaré, Volume 18 (2017) no. 1, pp. 345-374 | DOI:10.1007/s00023-016-0512-7 | Zbl:1379.35194
Mendelson, Dana Symplectic non-squeezing for the cubic nonlinear Klein-Gordon equation on $T^{3}$ , Journal of Functional Analysis, Volume 272 (2017) no. 7, pp. 3019-3092 | DOI:10.1016/j.jfa.2016.12.025 | Zbl:1370.35210
Pocovnicu, Oana Almost sure global well-posedness for the energy-critical defocusing nonlinear wave equation on $R^{d}$ , $d = 4$ and $5$ , Journal of the European Mathematical Society (JEMS), Volume 19 (2017) no. 8, pp. 2521-2575 | DOI:10.4171/jems/723 | Zbl:1375.35278
Tao, Terence Finite-time blowup for a supercritical defocusing nonlinear wave system, Analysis PDE, Volume 9 (2016) no. 8, p. 1999 | DOI:10.2140/apde.2016.9.1999
Choffrut, Antoine; Pocovnicu, Oana Ill-Posedness of the Cubic Nonlinear Half-Wave Equation and Other Fractional NLS on the Real Line, International Mathematics Research Notices (2016), p. rnw246 | DOI:10.1093/imrn/rnw246
Burq, Nicolas; Thomann, Laurent; Tzvetkov, Nikolay Global infinite energy solutions for the cubic wave equation, Bulletin de la Société Mathématique de France, Volume 143 (2015) no. 2, pp. 301-313 | DOI:10.24033/bsmf.2688 | Zbl:1320.35217
Saanouni, Tarek A blowing up wave equation with exponential type nonlinearity and arbitrary positive energy, Journal of Mathematical Analysis and Applications, Volume 421 (2015) no. 1, pp. 444-452 | DOI:10.1016/j.jmaa.2014.07.033 | Zbl:1297.35055
Todorova, Grozdena; Yordanov, Borislav On the regularizing effect of nonlinear damping in hyperbolic equations, Transactions of the American Mathematical Society, Volume 367 (2015) no. 7, pp. 5043-5058 | DOI:10.1090/s0002-9947-2015-06173-x | Zbl:1315.35049
Lührmann, Jonas; Mendelson, Dana Random Data Cauchy Theory for Nonlinear Wave Equations of Power-Type on ℝ3, Communications in Partial Differential Equations, Volume 39 (2014) no. 12, p. 2262 | DOI:10.1080/03605302.2014.933239
Saanouni, Tarek Global well-posedness of some high-order semilinear wave and Schrödinger type equations with exponential nonlinearity, Communications on Pure and Applied Analysis, Volume 13 (2014) no. 1, pp. 273-291 | DOI:10.3934/cpaa.2014.13.273 | Zbl:1291.35361
Majdoub, M.; Masmoudi, N. On Uniqueness for Supercritical Nonlinear Wave and Schrodinger Equations, International Mathematics Research Notices (2014) | DOI:10.1093/imrn/rnu002
Saanouni, T. A note on the instability of a focusing nonlinear damped wave equation, Mathematical Methods in the Applied Sciences, Volume 37 (2014) no. 18, pp. 3064-3076 | DOI:10.1002/mma.3044 | Zbl:1309.35141
Radu, Petronela Strong solutions for semilinear wave equations with damping and source terms, Applicable Analysis, Volume 92 (2013) no. 4, p. 718 | DOI:10.1080/00036811.2011.633902
Bociu, Lorena; Radu, Petronela; Toundykov, Daniel Regular solutions of wave equations with super-critical sources and exponential-to-logarithmic damping, Evolution Equations Control Theory, Volume 2 (2013) no. 2, p. 255 | DOI:10.3934/eect.2013.2.255
SAANOUNI, T Blowing-up semilinear wave equation with exponential nonlinearity in two space dimensions, Proceedings - Mathematical Sciences, Volume 123 (2013) no. 3, p. 365 | DOI:10.1007/s12044-013-0132-9
Carles, Rémi Nonlinear Schrödinger equation and frequency saturation, Analysis PDE, Volume 5 (2012) no. 5, p. 1157 | DOI:10.2140/apde.2012.5.1157
Carles, Rémi; Dumas, Eric; Sparber, Christof Geometric optics and instability for NLS and Davey-Stewartson models, Journal of the European Mathematical Society (JEMS), Volume 14 (2012) no. 6, pp. 1885-1921 | DOI:10.4171/jems/350 | Zbl:1273.35248
Ibrahim, Slim; Majdoub, Mohamed; Masmoudi, Nader Well- and ill-posedness issues for energy supercritical waves, Analysis PDE, Volume 4 (2011) no. 2, p. 341 | DOI:10.2140/apde.2011.4.341
Ibrahim, S.; Jrad, R. Strichartz type estimates and the well-posedness of an energy critical 2D wave equation in a bounded domain, Journal of Differential Equations, Volume 250 (2011) no. 9, pp. 3740-3771 | DOI:10.1016/j.jde.2011.01.008 | Zbl:1218.35146
Brenner, Philip; Kumlin, Peter Nonlinear maps between Besov and Sobolev spaces, Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI, Volume 19 (2010) no. 1, pp. 105-120 | DOI:10.5802/afst.1238 | Zbl:1195.46018
Alazard, Thomas; Carles, Rémi Supercritical geometric optics for nonlinear Schrödinger equations, Archive for Rational Mechanics and Analysis, Volume 194 (2009) no. 1, pp. 315-347 | DOI:10.1007/s00205-008-0176-7 | Zbl:1179.35302
Ibrahim, Slim; Majdoub, Mohamed; Masmoudi, Nader; Nakanishi, Kenji Scattering for the two-dimensional energy-critical wave equation, Duke Mathematical Journal, Volume 150 (2009) no. 2, pp. 287-329 | DOI:10.1215/00127094-2009-053 | Zbl:1206.35175
Pausader, Benoit The cubic fourth-order Schrödinger equation, Journal of Functional Analysis, Volume 256 (2009) no. 8, pp. 2473-2517 | DOI:10.1016/j.jfa.2008.11.009 | Zbl:1171.35115
Ibrahim, Slim; Guyenne, Philippe Instability in supercritical nonlinear wave equations: theoretical results and symplectic integration, Mathematics and Computers in Simulation, Volume 80 (2009) no. 1, pp. 2-9 | DOI:10.1016/j.matcom.2009.06.023 | Zbl:1179.35190
Alazard, Thomas; Carles, Rémi Loss of regularity for supercritical nonlinear Schrödinger equations, Mathematische Annalen, Volume 343 (2009) no. 2, pp. 397-420 | DOI:10.1007/s00208-008-0276-6 | Zbl:1161.35047
Burq, Nicolas; Tzvetkov, Nikolay Random data Cauchy theory for supercritical wave equations I: Local theory, Inventiones Mathematicae, Volume 173 (2008) no. 3, pp. 449-475 | DOI:10.1007/s00222-008-0124-z | Zbl:1156.35062
Thomann, Laurent Instabilities for supercritical Schrödinger equations in analytic manifolds, Journal of Differential Equations, Volume 245 (2008) no. 1, pp. 249-280 | DOI:10.1016/j.jde.2007.12.001 | Zbl:1157.35107
Fang, Daoyuan; Wang, Chengbo Ill-posedness for semilinear wave equations with very low regularity, Mathematische Zeitschrift, Volume 259 (2008) no. 2, pp. 343-353 | DOI:10.1007/s00209-007-0228-y | Zbl:1185.35319
Carles, Rémi Geometric optics and instability for semi-classical Schrödinger equations, Archive for Rational Mechanics and Analysis, Volume 183 (2007) no. 3, pp. 525-553 | DOI:10.1007/s00205-006-0017-5 | Zbl:1134.35098
Ibrahim, Slim; Majdoub, Mohamed; Masmoudi, Nader Ill-posedness of $H^{1}$ -supercritical waves, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 345 (2007) no. 3, pp. 133-138 | DOI:10.1016/j.crma.2007.06.008 | Zbl:1127.35073
TAO, TERENCE GLOBAL REGULARITY FOR A LOGARITHMICALLY SUPERCRITICAL DEFOCUSING NONLINEAR WAVE EQUATION FOR SPHERICALLY SYMMETRIC DATA, Journal of Hyperbolic Differential Equations, Volume 04 (2007) no. 02, p. 259 | DOI:10.1142/s0219891607001124
Lucente, Sandra On a class of semilinear weakly hyperbolic equations, Annali dell'Università di Ferrara. Sezione VII. Scienze Matematiche, Volume 52 (2006) no. 2, pp. 317-335 | DOI:10.1007/s11565-006-0024-3 | Zbl:1138.35064

Cité par 49 documents. Sources : Crossref, zbMATH