Probabilistic local Cauchy theory of the cubic nonlinear wave equation in negative Sobolev spaces

Oh, Tadahiro; Pocovnicu, Oana; Tzvetkov, Nikolay

doi:10.5802/aif.3454

Probabilistic local Cauchy theory of the cubic nonlinear wave equation in negative Sobolev spaces
[Le caractère bien posé probabiliste de l’équation des ondes cubique dans des espaces de Sobolev d’indices négatifs]

Oh, Tadahiro ¹ ; Pocovnicu, Oana ² ; Tzvetkov, Nikolay ³

¹ School of Mathematics The University of Edinburgh and The Maxwell Institute for the Mathematical Sciences James Clerk Maxwell Building The King’s Buildings Peter Guthrie Tait Road Edinburgh EH9 3FD (United Kingdom)
² Department of Mathematics, Heriot-Watt University and The Maxwell Institute for the Mathematical Sciences, Edinburgh, EH14 4AS (United Kingdom)
³ CY Cergy Paris University Cergy-Pontoise F-95000 UMR 8088 du CNRS (France)

Annales de l'Institut Fourier, Tome 72 (2022) no. 2, pp. 771-830.

Résumé
Abstract

On étudie l’équation des ondes non linéaire cubique (NLW) en dimension 3 avec une donnée initiale aléatoire en-dessous de $L^{2} (𝕋^{3})$ . En considérant le développement d’ordre 2 en termes de la solution aléatoire linéaire, on prouve le caractère presque sûrement localement bien posé de NLW renormalisée dans les espaces de Sobolev d’indices négatifs. On montre aussi un nouveau résultat d’instabilité pour l’équation NLW cubique défocalisante sans renormalisation dans les espaces de Sobolev d’indices négatifs, dans l’esprit du caractère non trivial dans l’étude des équations aux dérivées partielles stochastiques. Plus précisément, en étudiant NLW non renormalisée avec des données initiales régulières déterministes plus une donnée initiale aléatoire tronquée, on montre que, dès que la troncature est supprimée, les solutions tendent vers $0$ au sens des distributions pour toute donnée initiale déterministe.

We study the three-dimensional cubic nonlinear wave equation (NLW) with random initial data below $L^{2} (𝕋^{3})$ . By considering the second order expansion in terms of the random linear solution, we prove almost sure local well-posedness of the renormalized NLW in negative Sobolev spaces. We also prove a new instability result for the defocusing cubic NLW without renormalization in negative Sobolev spaces, which is in the spirit of the so-called triviality in the study of stochastic partial differential equations. More precisely, by studying (un-renormalized) NLW with given smooth deterministic initial data plus a certain truncated random initial data, we show that, as the truncation is removed, the solutions converge to $0$ in the distributional sense for any deterministic initial data.

Reçu le : 2019-04-09
Révisé le : 2020-10-04
Accepté le : 2020-10-28
Publié le : 2022-07-07

Zbl

DOI : 10.5802/aif.3454

Classification : 35L71
Keywords: nonlinear wave equation, Gaussian measure, local well-posedness, renormalization, triviality
Mot clés : équation des ondes non linéaire, mesures gaussiennes, caractère bien posé, renormalisation, trivialité

Affiliations des auteurs :

Oh, Tadahiro ¹ ; Pocovnicu, Oana ² ; Tzvetkov, Nikolay ³

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Probabilistic local {Cauchy} theory of the cubic nonlinear wave equation in negative {Sobolev} spaces},
     journal = {Annales de l'Institut Fourier},
     pages = {771--830},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Oh, Tadahiro; Pocovnicu, Oana; Tzvetkov, Nikolay. Probabilistic local Cauchy theory of the cubic nonlinear wave equation in negative Sobolev spaces. Annales de l'Institut Fourier, Tome 72 (2022) no. 2, pp. 771-830. doi : 10.5802/aif.3454. https://aif.centre-mersenne.org/articles/10.5802/aif.3454/

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