Global solutions to the Kirchhoff equation with spectral gap data in the energy space

TL;DR

This paper proves the existence of global solutions for the Kirchhoff equation with spectral gap initial data in the energy space, showing that any initial data can be decomposed into two such data, without linearization.

Contribution

It introduces spectral gap data as a new class of initial conditions ensuring global solutions to the Kirchhoff equation in the energy space.

Findings

Existence of global solutions for spectral gap initial data.

Any initial data in the energy space can be decomposed into two spectral gap data.

The proof avoids linearization due to low regularity issues.

Abstract

We prove that the classical hyperbolic Kirchhoff equation admits global-in-time solutions for some classes of initial data in the energy space. We also show that there are enough such solutions so that every initial datum in the energy space is the sum of two initial data for which a global-in-time solution exists. The proof relies on the notion of spectral gap data, namely initial data whose components vanish for large intervals of frequencies. We do not pass through the linearized equation, because it is not well-posed at this low level of regularity.