Almost Global Solutions of Kirchhoff Equation

TL;DR

This paper proves almost global existence and stability of solutions to the Kirchhoff equation for small initial data, using rational normal form transformations in infinite-dimensional reversible vector fields.

Contribution

It introduces a novel approach to analyze the Kirchhoff equation by constructing rational normal forms for infinite-dimensional reversible vector fields, differing from Hamiltonian methods.

Findings

Solutions exist for times of order ^{( ext{log}\u03b5)^2} in Gevrey and analytic spaces.

Time of existence scales as psilon^{-r} in Sobolev spaces.

The method applies to almost any small initial data, demonstrating stability.

Abstract

This paper is concerned with the original Kirchhoff equation $$\left\{\begin{aligned} & \pa_{tt}u-\Big(1+\int_{0}^{\pi}|\pa_xu|^2 dx\Big)\pa_{xx}u=0, \\&u(t,0)=u(t,\pi)=0. \end{aligned}\right.$$ We obtain almost global existence and stability of solutions for almost any small initial data of size $\varepsilon$. In Sobolev spaces, the time of existence and stability is of order $\varepsilon^{-r}$ for arbitrary positive integer $r$. In Gevrey and analytic spaces, the time is of order $e^{\frac{|\ln\varepsilon|^2}{c\ln|\ln\varepsilon|}}$ with some positive constant $c$. To achieve these, we build rational normal form for infinite dimensional reversible vector fields without external parameters. We emphasize that for vector fields, the homological equation and the definition of rational normal form are significantly different from those for Hamiltonian functions.