Concerning the pathological set in the context of probabilistic well-posedness

Chenmin Sun; Nikolay Tzvetkov

doi:10.5802/crmath.102

Partial Differential Equations

Concerning the pathological set in the context of probabilistic well-posedness

Chenmin Sun ¹; Nikolay Tzvetkov ¹

¹ Université de Cergy-Pontoise, Laboratoire de Mathématiques AGM, UMR 8088 du CNRS, 2 av. Adolphe Chauvin 95302 Cergy-Pontoise Cedex, France

Comptes Rendus. Mathématique, Volume 358 (2020) no. 9-10, pp. 989-999.

Abstract
Résumé

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.

Received: 2020-01-28
Accepted: 2020-07-27
Published online: 2021-01-05

DOI: 10.5802/crmath.102

Author's affiliations:

Chenmin Sun ¹; Nikolay Tzvetkov ¹

¹ Université de Cergy-Pontoise, Laboratoire de Mathématiques AGM, UMR 8088 du CNRS, 2 av. Adolphe Chauvin 95302 Cergy-Pontoise Cedex, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{CRMATH_2020__358_9-10_989_0,
     author = {Chenmin Sun and Nikolay Tzvetkov},
     title = {Concerning the pathological set in the context of probabilistic well-posedness},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {989--999},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {9-10},
     year = {2020},
     doi = {10.5802/crmath.102},
     language = {en},
}

TY  - JOUR
AU  - Chenmin Sun
AU  - Nikolay Tzvetkov
TI  - Concerning the pathological set in the context of probabilistic well-posedness
JO  - Comptes Rendus. Mathématique
PY  - 2020
SP  - 989
EP  - 999
VL  - 358
IS  - 9-10
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.102
LA  - en
ID  - CRMATH_2020__358_9-10_989_0
ER  -

%0 Journal Article
%A Chenmin Sun
%A Nikolay Tzvetkov
%T Concerning the pathological set in the context of probabilistic well-posedness
%J Comptes Rendus. Mathématique
%D 2020
%P 989-999
%V 358
%N 9-10
%I Académie des sciences, Paris
%R 10.5802/crmath.102
%G en
%F CRMATH_2020__358_9-10_989_0

Chenmin Sun; Nikolay Tzvetkov. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.102/

References
Cited by

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

[2] Árpad Bényi; Tahadiro Oh; Oana Pocovnicu On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on $ℝ^{d}$ , $d \geq 3$ , Trans. Am. Math. Soc., Volume 2 (2015), pp. 1-50 | DOI | Zbl

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

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

[7] James E. Colliander; Tahadiro Oh 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] Manoussos G. Grillakis 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] Mickaël Latocca Almost Sure Existence of Global Solutions for supercritical semilinear Wave Equations (2018) (https://arxiv.org/abs/1809.07061)

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

[11] Gilles Lebeau 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] Hans Lindblad 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] Jonas Lührmann; D. ana Mendelson Random data Cauchy theory for nonlinear wave equations of power-type on $ℛ^{3}$ , Commun. Partial Differ. Equations, Volume 39 (2014) no. 12, pp. 2262-2283 | DOI | MR | Zbl

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

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

[17] Chenmin Sun; Bo Xia 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] Nikolay Tzvetkov Random data wave equations, Lecture Notes in Mathematics, 2253, Springer, 2019, pp. 221-313 | MR

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

Cited by Sources:

Comments - Policy