“LEANING TORI” An Hamiltonian Event under the Tower

Long time behavior of the solutions of NLW on the d-dimensional torus

speaker: Benoit Grébert (Université de Nantes)

abstract: We consider the non linear wave equation (NLW) on the d-dimensional torus $$u_{tt} - \Delta u + mu + f(u) =0\quad x\in\T^d$$ where $f=\partial_u F$ is analytic on a neighborhood of the origin and which is at least of order 2 at the origin. Let $u(t)$ be a solution corresponding to a small initial datum $u(0)\in H^s(\T^d)$. We prove that we control $u(t)_s$ that mix the $H^s$ norm of the $\eps^{-\beta(r)}$ lower Fourier modes of the solution $u$ and the energy norm of the remaining higher modes during long times of order $\eps^{-r}$. Our general strategy applies to any Hamiltonian PDEs whose linear frequencies satisfy only a first Melnikov condition. In particular it also applies to the Hamiltonian Boussinesq $abcd$ system and the Whitham-Boussinesq system in water waves theory. Joint work with Joackim Bernier and Erwan Faou.

timetable:
Mon 20 May, 14:30 - 15:20, Aula Dini
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