Partial Differential Equations
Concerning the pathological set in the context of probabilistic well-posedness
Comptes Rendus. Mathématique, Volume 358 (2020) no. 9-10, pp. 989-999.

We prove a complementary result to the probabilistic well-posedness for the nonlinear wave equation. More precisely, we show that there is a dense set S of the Sobolev space of super-critical regularity such that (in sharp contrast with the probabilistic well-posedness results) the family of global smooth solutions, generated by the convolution with some approximate identity of the elements of S, does not converge in the space of super-critical Sobolev regularity.

On démontre un résultat complémentaire à ceux manifestant le caractère bien posé probabiliste de l’équation des ondes avec des données initiales de régularité de Sobolev super critique par rapport au changement d’échelle laissant invariant l’équation.

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Published online:
DOI: 10.5802/crmath.102
Sun, Chenmin 1; Tzvetkov, Nikolay 1

1 Université de Cergy-Pontoise, Laboratoire de Mathématiques AGM, UMR 8088 du CNRS, 2 av. Adolphe Chauvin 95302 Cergy-Pontoise Cedex, France
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Sun, Chenmin; Tzvetkov, Nikolay. Concerning the pathological set in the context of probabilistic well-posedness. Comptes Rendus. Mathématique, Volume 358 (2020) no. 9-10, pp. 989-999. doi : 10.5802/crmath.102. http://www.numdam.org/articles/10.5802/crmath.102/

[1] Alazard, Thomas; Carles, Rémi Loss of regularity for supercritical nonlinear Schrödinger equations, Math. Ann., Volume 343 (2009) no. 2, pp. 397-420 | DOI | Zbl

[2] Bényi, Árpad; Oh, Tahadiro; Pocovnicu, Oana On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on d , d3, Trans. Am. Math. Soc., Volume 2 (2015), pp. 1-50 | DOI | Zbl

[3] Bourgain, Jean Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Commun. Math. Phys., Volume 176 (1996) no. 2, pp. 421-445 | DOI | Zbl

[4] Burq, Nicolas; Tzvetkov, Nikolay Random data Cauchy theory for supercritical wave equations I: local theory, Invent. Math., Volume 173 (2008) no. 3, pp. 449-475 | DOI | MR | Zbl

[5] Burq, Nicolas; Tzvetkov, Nikolay Probabilistic well-posedness for the cubic wave equation, J. Eur. Math. Soc. (JEMS), Volume 16 (2014) no. 1, pp. 1-30 | DOI | MR | Zbl

[6] Christ, Michael; Colliander, James; Tao, Terence Ill-posedness for nonlinear Schrödinger and wave equations (2003) (https://arxiv.org/abs/math/0311048)

[7] Colliander, James E.; Oh, Tahadiro Almost sure local well-posedness of the cubic NLS below L 2 , Duke Math. J., Volume 161 (2012) no. 3, pp. 367-414 | DOI | Zbl

[8] Grillakis, Manoussos G. Regularity and asymptotic behaviour of the wave equation with a critical non linearity, Ann. Math., Volume 132 (1990) no. 3, pp. 485-509 | DOI | MR | Zbl

[9] Latocca, Mickaël Almost Sure Existence of Global Solutions for supercritical semilinear Wave Equations (2018) (https://arxiv.org/abs/1809.07061)

[10] Lebeau, Gilles Nonlinear optic and supercritical wave equation, Bull. Soc. R. Sci. Liège, Volume 70 (2001) no. 4-6, pp. 267-306 | Zbl

[11] Lebeau, Gilles Perte de régularité pour l’équation d’ondes sur-critiques, Bull. Soc. Math. Fr., Volume 133 (2005) no. 1, pp. 145-157 | DOI | Zbl

[12] Lindblad, Hans A sharp counterexample to the local existence of low-regularity solutions to nonlinear wave equations, Duke Math. J., Volume 72 (1993) no. 2, pp. 503-539 | DOI | MR | Zbl

[13] Lührmann, Jonas; Mendelson, D. ana Random data Cauchy theory for nonlinear wave equations of power-type on 3 , Commun. Partial Differ. Equations, Volume 39 (2014) no. 12, pp. 2262-2283 | DOI | MR | Zbl

[14] Oh, Tahadiro; Pocovnicu, Oana Probabilistic global well-posedness of the energy-critical defocusing quintic nonlinear wave equation on 3 , J. Math. Pures Appl., Volume 105 (2016) no. 3, pp. 342-366 | MR | Zbl

[15] Shatah, Jalal; Struwe, Michael Well-posedness in the energy space for semilinear wave equation with critical growth, Int. Math. Res. Not., Volume 1094 (1994) no. 7, pp. 303-309 | DOI | MR | Zbl

[16] Sun, Chenmin; Tzvetkov, Nikolay New examples of probabilistic well-posedness for nonlinear wave equations, J. Funct. Anal., Volume 278 (2020) no. 2, 108322, 47 pages | MR | Zbl

[17] Sun, Chenmin; Xia, Bo Probabilistic well-posedness for supercritical wave equations with periodic boundary condition on dimension three, Ill. J. Math., Volume 60 (2016) no. 2, pp. 481-503 | MR | Zbl

[18] Tzvetkov, Nikolay Random data wave equations, Lecture Notes in Mathematics, 2253, Springer, 2019, pp. 221-313 | MR

[19] Xia, Bo Equations aux dérivées partielles et aléa, Ph. D. Thesis, Université Paris-Sud, (France) (2016)

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