Almost global existence for some nonlinear Schrödinger equations on $\mathbb{T}^d$ in low regularity

Bernier, Joackim; Grébert, Benoît

doi:10.5802/aif.3706

Almost global existence for some nonlinear Schrödinger equations on

𝕋^{d}

in low regularity

Bernier, Joackim ¹; Grébert, Benoît ¹

¹ Nantes Université, CNRS, Laboratoire de Mathématiques Jean Leray, LMJL, F-44000 Nantes (France)

Annales de l'Institut Fourier, Online first, 44 p.

Abstract
Résumé

We are interested in the long time behavior of solutions of the nonlinear Schrödinger equations on the $d$ -dimensional torus in low regularity, i.e. for small initial data in the Sobolev space $H^{s_{0}} (𝕋^{d})$ with $s_{0} > d / 2$ . We prove that, even in this context of low regularity, the $H^{s}$ -norms, $s \geq 0$ , remain under control during times, $T_{ε} = exp (\frac{{| log ε |}^{2}}{4 log | log ε |})$ , exponential with respect to the initial size of the initial datum in $H^{s_{0}}$ , ${∥ u (0) ∥}_{H^{s_{0}}} = ε$ . For this, we add to the linear part of the equation a random Fourier multiplier in $ℓ^{\infty} (ℤ^{d})$ and show our stability result for almost any realization of this multiplier. In particular, with such Fourier multipliers, we obtain the almost global well posedness of the nonlinear Schrödinger equations in $H^{s_{0}} (𝕋^{d})$ for any $s_{0} > d / 2$ and any $d \geq 1$ .

On considère le comportement en temps longs des solutions des équations de Schrödinger non-linéaires sur le tore de dimension $d$ en faible régularité, i.e. pour de petites conditions initiales dans l’espace de Sobolev $H^{s_{0}} (𝕋^{d})$ avec $s_{0} > d / 2$ . Même dans ce contexte de faible régularité, on contrôle la croissance des normes $H^{s}$ , $s \geq 0$ , pendant des temps $T_{ε} = exp (\frac{{| log ε |}^{2}}{4 log | log ε |})$ exponentiellement longs par rapport à la taille des données initiales dans $H^{s_{0}}$ , ${∥ u (0) ∥}_{H^{s_{0}}} = ε$ . Pour y parvenir, on ajoute à la partie linéaire de l’équation un multiplicateur de Fourier aléatoire dans $ℓ^{\infty} (ℤ^{d})$ et on montre le résultat de stabilité pour presque toute réalisation de ce multiplicateur. En particulier, avec de tels multiplicateurs de Fourier, on prouve l’existence presque globale des solutions des équations de Schrödinger non-linéaires dans $H^{s_{0}} (𝕋^{d})$ pour n’importe quel $s_{0} > d / 2$ et $d \geq 1$ .

Received: 2022-03-26
Revised: 2023-10-06
Accepted: 2023-11-30
Online First: 2025-02-10

DOI: 10.5802/aif.3706

Classification: 35Q55, 37K45, 37K55
Keywords: Birkhoff normal forms, low regularity, NLS equation
Mots-clés : Formes normales de Birkhoff, faible régularité, équations de Schrödinger non-linéaires

Author's affiliations:

Bernier, Joackim ¹; Grébert, Benoît ¹

¹ Nantes Université, CNRS, Laboratoire de Mathématiques Jean Leray, LMJL, F-44000 Nantes (France)

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     title = {Almost global existence for some nonlinear {Schr\"odinger} equations on $\mathbb{T}^d$ in low regularity},
     journal = {Annales de l'Institut Fourier},
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     note = {Online first},
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Bernier, Joackim; Grébert, Benoît. Almost global existence for some nonlinear Schrödinger equations on $\mathbb{T}^d$ in low regularity. Annales de l'Institut Fourier, Online first, 44 p.

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