On the Normal Form of the Kirchhoff Equation

Pietro Baldi,Emanuele Haus

doi:10.1007/s10884-020-09904-w

Abstract

Consider the Kirchhoff equation ∂ttu-Δu(1+∫Td|∇u|2)=0\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\partial _{tt} u - \\Delta u \\Big ( 1 + \\int _{\\mathbb {T}^d} |\\nabla u|^2 \\Big ) = 0 \\end{aligned}$$\\end{document}on the d-dimensional torus mathbb {T}^d. In a previous paper we proved that, after a first step of quasilinear normal form, the resonant cubic terms show an integrable behavior, namely they give no contribution to the energy estimates. This leads to the question whether the same structure also emerges at the next steps of normal form. In this paper, we perform the second step and give a negative answer to the previous question: the quintic resonant terms give a nonzero contribution to the energy estimates. This is not only a formal calculation, as we prove that the normal form transformation is bounded between Sobolev spaces.

Highlights

We consider the Kirchhoff equation on the d-dimensional torus Td, T := R/2π Z∂tt u − u 1 + |∇u|2 d x = 0
In [4], we performed one step of quasilinear normal form and established a longer existence time, of the order of ε−4; all the cubic terms giving a nontrivial contribution to the energy estimates are erased by the normal form
Remark 1.4 There is a certain similarity between our computation and the one performed by Craig and Worfolk [14] for the normal form of gravity water waves

Summary

Introduction

We consider the Kirchhoff equation on the d-dimensional torus Td , T := R/2π Z (periodic boundary conditions). In [4], we performed one step of quasilinear normal form and established a longer existence time, of the order of ε−4; all the cubic terms giving a nontrivial contribution to the energy estimates are erased by the normal form. Normal form can somehow be used to construct “weakly turbulent” solutions pushing energy from low to high Fourier modes, in the spirit of the works [11,22,23,24,25] for the semilinear Schrödinger equations on T2 Proving existence of such solutions may be a very hard task, but one may at least hope to use the normal form that we compute in this paper to detect some genuinely nonlinear behavior of the flow, over long time-scales (as in [20,27]) or even for all times (as in [26])

Main Result

Related Literature

Linear Transformations

Diagonalization of the Order One

Normal Form