Sub-exponential stability for the Beam equation

TL;DR

This paper proves long-term stability for a class of nonlinear beam equations with Hamiltonian structure, using normal form techniques and Diophantine conditions, in Sobolev and smoother function spaces.

Contribution

It establishes exponential stability times for the beam equation in Sobolev and infinitely differentiable spaces, a novel result for degenerate equations with a single parameter.

Findings

Stability times are exponential in initial data size.

Results hold for almost all mass parameters.

First such stability result in Sobolev space for a degenerate equation.

Abstract

We consider a one-parameter family of beam equations with Hamiltonian non-linearity in one space dimension under periodic boundary conditions. In a unified functional framework we study the long time evolution of initial data in two categories of differentiability: (i) a subspace of Sobolev regularity, (ii) a subspace of infinitely many differentiable functions which is strictly contained in the Sobolev space but which strictly contains the Gevrey one. In both cases we prove exponential type times of stability. The result holds for almost all mass parameters and it is obtained by combining normal form techniques with a suitable Diophantine condition weaker than the one proposed by Bourgain. This is the first result of this kind in Sobolev regularity for a degenerate equation, where only one parameter is used to tune the linear frequencies of oscillations.