LONG-TIME EXISTENCE FOR SEMI-LINEAR BEAM EQUATIONS ON IRRATIONAL TORI - Laboratoire Jacques-Louis Lions Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2020

LONG-TIME EXISTENCE FOR SEMI-LINEAR BEAM EQUATIONS ON IRRATIONAL TORI

Résumé

We consider the semi-linear beam equation on the d dimensional irrational torus with smooth nonlinearity of order n − 1 with n ≥ 3 and d ≥ 2. If ε ≪ 1 is the size of the initial datum, we prove that the lifespan Tε of solutions is O(ε −A(n−2) −) where A ≡ A(d, n) = 1 + 3 d−1 when n is even and A = 1 + 3 d−1 + max(4−d d−1 , 0) when n is odd. For instance for d = 2 and n = 3 (quadratic nonlinearity) we obtain Tε = O(ε −6 −), much better than O(ε −1), the time given by the local existence theory. The irrationality of the torus makes the set of differences between two eigenvalues of √ ∆ 2 + 1 accumulate to zero, facilitating the exchange between the high Fourier modes and complicating the control of the solutions over long times. Our result is obtained by combining a Birkhoff normal form step and a modified energy step.
Fichier principal
Vignette du fichier
BeamOnT2-cor.pdf (495.56 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-02987274 , version 1 (03-11-2020)

Identifiants

Citer

Joackim Bernier, Roberto Feola, Benoît Grébert, Felice Iandoli. LONG-TIME EXISTENCE FOR SEMI-LINEAR BEAM EQUATIONS ON IRRATIONAL TORI. 2020. ⟨hal-02987274⟩
43 Consultations
50 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More