On the existence time for the Kirchhoff equation with periodic boundary conditions

TL;DR

This paper establishes a lower bound on the existence time for solutions to the Kirchhoff equation with periodic boundary conditions, improving previous bounds by employing a normal form transformation and diagonalization techniques.

Contribution

It introduces a novel approach using nonlinear diagonalization and normal form transformations to extend the existence time for the Kirchhoff equation.

Findings

Lower bound of  for existence time with small initial data

Improved upon the standard  bound from local theory

Method relies on normal form and nonlinear diagonalization

Abstract

We consider the Cauchy problem for the Kirchhoff equation on $\mathbb{T}^d$ with initial data of small amplitude $\varepsilon$ in Sobolev class. We prove a lower bound $\varepsilon^{-4}$ for the existence time, which improves the bound $\varepsilon^{-2}$ given by the standard local theory. The proof relies on a normal form transformation, preceded by a nonlinear transformation that diagonalizes the operator at the highest order, which is needed because of the quasilinear nature of the equation.