Quasi-periodic solutions for the forced Kirchhoff equation on $\mathbb{T}^d$

TL;DR

This paper establishes the existence of small-amplitude quasi-periodic solutions for a high-dimensional, forced Kirchhoff equation, marking a novel achievement for quasi-linear PDEs in multiple dimensions.

Contribution

It introduces a Nash-Moser scheme combined with multiscale analysis to prove quasi-periodic solutions for a complex, high-dimensional quasi-linear PDE, a first in this field.

Findings

Existence of small-amplitude quasi-periodic solutions in Sobolev spaces.

First such result for high-dimensional quasi-linear equations.

Development of a novel analytical framework combining Nash-Moser and multiscale methods.

Abstract

In this paper we prove the existence of small-amplitude quasi-periodic solutions with Sobolev regularity, for the $d$-dimensional forced Kirchhoff equation with periodic boundary conditions. This is the first result of this type for a quasi-linear equations in high dimension. The proof is based on a Nash-Moser scheme in Sobolev class and a regularization procedure combined with a multiscale analysis in order to solve the linearized problem at any approximate solution.