Weak universality results for a class of nonlinear wave equations

Sun, Chenmin; Tzvetkov, Nikolay; Xu, Weijun

doi:10.5802/aif.3715

Weak universality results for a class of nonlinear wave equations
[Résultats d’universalité faible pour une classe d’équations d’ondes non-linéaires]

Sun, Chenmin ¹ ; Tzvetkov, Nikolay ² ; Xu, Weijun ³

¹ CNRS and Université Paris Est Créteil, UMR 8050 du CNRS (France)
² École Normale Supérieure de Lyon, UMPA, UMR CNRS-ENSL 5669, 46, allée d’Italie, 69364, Lyon Cedex 07 (France)
³ Beijing International Center for Mathematical Research, Peking University, Beijing, 100871 (China)

Annales de l'Institut Fourier, Online first, 52 p.

Abstract (VO)
Résumé

We study the weak universality of the two-dimensional fractional nonlinear wave equation. For a sequence of Hamiltonians of high-degree potentials scaling to the fractional $Φ_{2}^{4}$ , we first establish a sufficient and almost necessary criteria for the convergence of invariant measures to the fractional $Φ_{2}^{4}$ . Then we prove the convergence result for the sequence of associated wave dynamics to the (renormalized) cubic wave equation. Our constraint on the fractional index is independent of the degree of the nonlinearity. This extends the result of Gubinelli–Koch–Oh [Renormalisation of the two-dimensional stochastic nonlinear wave equations, Trans. Amer. Math. Soc. 370 (2018)] to a situation where we do not have a local Cauchy theory with highly supercritical nonlinearities.

Nous étudions l’universalité faible de l’équation d’onde non linéaire fractionnaire en dimension deux. Pour une suite d’hamiltoniens de potentiels de haut degré convergeant vers le modèle $Φ_{2}^{4}$ fractionnaire, nous établissons d’abord un critère suffisant et presque nécessaire pour la convergence des mesures invariantes vers celle du modèle $Φ_{2}^{4}$ fractionnaire. Ensuite, nous démontrons le résultat de convergence pour la suite de dynamiques d’ondes associées vers l’équation d’onde cubique (renormalisée). Notre condition sur l’indice fractionnaire est indépendante du degré de la non-linéarité. Ceci étend un résultat de Gubinelli–Koch–Oh à une situation où nous n’avons pas de théorie de Cauchy locale avec des non-linéarités sur-critiques.

Reçu le : 2022-07-15
Révisé le : 2023-10-08
Accepté le : 2023-11-30
Première publication : 2025-06-17

DOI : 10.5802/aif.3715

Classification : 35L05, 35Q82, 60H15
Keywords: Weak Universality, nonlinear wave equations, Invariant measures
Mots-clés : Universalité faible, équation d’onde non-linéaire, mesures invariantes

Affiliations des auteurs :

Sun, Chenmin ¹ ; Tzvetkov, Nikolay ² ; Xu, Weijun ³

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Sun, Chenmin; Tzvetkov, Nikolay; Xu, Weijun. Weak universality results for a class of nonlinear wave equations. Annales de l'Institut Fourier, Online first, 52 p.

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