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A Kirchhoff equation with infinite conservation laws

Chiara Boiti, Renato Manfrin

TL;DR

This work proves that the Kirchhoff-Pokhozhaev equation $u_{tt}-\big(a \|\nabla u\|^2 + b\big)^{-2} \Delta u = 0$ admits conservation laws of all orders $k\ge 3$ under suitable regularity. The authors transform the problem to a Liouville-type equation via partial Fourier transform, construct a family of quadratic forms ${\cal E}_k(\xi,t)$ with carefully chosen polynomial coefficients in $q$ and its derivatives, and derive time-invariant functionals $I_k$ by combining $E_k$ with lower-order corrections $Q_k$. A recursive, polynomial-structure analysis using auxiliary functions $G_i$ demonstrates solvability and yields invariants of order up to $k-1$, culminating in a rigorous proof of conservation laws of all orders. The results enhance understanding of the equation’s conserved quantities and may inform global solvability and stability analyses for Kirchhoff-type hyperbolic equations.

Abstract

We show here that the quasilinear Kirchhoff-Pokhozhaev equation $$u_{tt}-\big(a\int_{\mathbb{R}^n} |\nabla u |^2 dx + b \big)^{-2} Δu = 0,$$ with $a\neq0$, admits conservation laws of all orders.

A Kirchhoff equation with infinite conservation laws

TL;DR

This work proves that the Kirchhoff-Pokhozhaev equation admits conservation laws of all orders under suitable regularity. The authors transform the problem to a Liouville-type equation via partial Fourier transform, construct a family of quadratic forms with carefully chosen polynomial coefficients in and its derivatives, and derive time-invariant functionals by combining with lower-order corrections . A recursive, polynomial-structure analysis using auxiliary functions demonstrates solvability and yields invariants of order up to , culminating in a rigorous proof of conservation laws of all orders. The results enhance understanding of the equation’s conserved quantities and may inform global solvability and stability analyses for Kirchhoff-type hyperbolic equations.

Abstract

We show here that the quasilinear Kirchhoff-Pokhozhaev equation with , admits conservation laws of all orders.
Paper Structure (5 sections, 9 theorems, 105 equations)

This paper contains 5 sections, 9 theorems, 105 equations.

Table of Contents

  1. Introduction and results
  2. A quadratic form for a Liouville type equation
  3. The polynomial structure of $\alpha_i, \beta_i, \gamma_i$.
  4. The quadratic form for the Kirchhoff-Pokhozhaev equation
  5. Proof of Theorem \ref{['T1']}

Key Result

Theorem 1.3

Let $k\in \mathbb N$, $k\ge 3$. There exists a functional $\, J_{k}= J_k(u)$, of order $\le k-1$, such that, given a function which satisfies reg2, if $u$ is a solution of KP then remains constant in $[0,T)$.

Theorems & Definitions (22)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Definition 2.1
  • Proposition 2.2
  • Definition 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 12 more