Dynamics of nonlinear Klein–Gordon equations in low regularity on S2\mathbb{S}^2

  • Joackim Bernier

    Université de Nantes, France
  • Benoît Grébert

    Université de Nantes, France
  • Gabriel Rivière

    Université de Nantes, France; Institut Universitaire de France, Paris, France
Dynamics of nonlinear Klein–Gordon equations in low regularity on $\mathbb{S}^2$ cover
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Abstract

We describe the long-time behavior of small nonsmooth solutions to the nonlinear Klein–Gordon equations on the sphere S2\mathbb{S}^2. More precisely, we prove that the low harmonic energies (also called super-actions) are almost preserved for times of order εr\varepsilon^{-r}, where r1r \gg 1 is an arbitrarily large number and ε1\varepsilon \ll 1 is the norm of the initial datum in the energy space H1×L2H^1\times L^2. Roughly speaking, it means that, in order to exchange energy, modes have to oscillate at the same frequency. The proof relies on new multilinear estimates on Hamiltonian vector fields to put the system in Birkhoff normal form. They are derived from new probabilistic bounds on products of Laplace eigenfunctions that we obtain using Levy’s concentration inequality.

Cite this article

Joackim Bernier, Benoît Grébert, Gabriel Rivière, Dynamics of nonlinear Klein–Gordon equations in low regularity on S2\mathbb{S}^2. Ann. Inst. H. Poincaré Anal. Non Linéaire (2022),

DOI 10.4171/AIHPC/55