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The conservative Camassa-Holm flow with step-like irregular initial data

Jonathan Eckhardt, Aleksey Kostenko

TL;DR

The paper develops a comprehensive inverse spectral transform framework for the conservative Camassa–Holm flow on the real line with step-like initial data by introducing a phase space $\mathcal{D}$ that encodes strong decay at $-\infty$ and mild growth at $+\infty$. It proves a bijective spectral mapping to a class of measures $\rho$ with a zero-gap around zero, and shows the evolution of $\rho$ is linear in time, which in turn yields global weak conservative solutions and a complete linearization of the dynamics. The authors discover new almost conservation laws linked to Besov-type norms, controlled by the spectral gap, and establish continuity of the spectral transform and density of multi-peakon profiles, enabling approximation arguments that extend the flow to the whole phase space. Positivity of the momentum recovers the classical Camassa–Holm flow without blow-up, while the framework also provides a detailed description of invariant sets and a broad family of almost conserved quantities, highlighting the deep spectral structure underlying this integrable system.

Abstract

We extend the inverse spectral transform for the conservative Camassa-Holm flow on the line to a class of initial data that requires strong decay at one endpoint but only mild boundedness-type conditions at the other endpoint. The latter condition appears to be close to optimal in a certain sense for the well-posedness of the conservative Camassa-Holm flow. As a byproduct of our approach, we also find a family of new (almost) conservation laws for the Camassa-Holm equation, which could not be deduced from its bi-Hamiltonian structure before and which are connected to certain Besov-type norms (however, in a rather involved way). These results appear to be new even under positivity assumptions on the corresponding momentum, in which case the conservative Camassa-Holm flow coincides with the classical Camassa-Holm flow and no blow-ups occur.

The conservative Camassa-Holm flow with step-like irregular initial data

TL;DR

The paper develops a comprehensive inverse spectral transform framework for the conservative Camassa–Holm flow on the real line with step-like initial data by introducing a phase space that encodes strong decay at and mild growth at . It proves a bijective spectral mapping to a class of measures with a zero-gap around zero, and shows the evolution of is linear in time, which in turn yields global weak conservative solutions and a complete linearization of the dynamics. The authors discover new almost conservation laws linked to Besov-type norms, controlled by the spectral gap, and establish continuity of the spectral transform and density of multi-peakon profiles, enabling approximation arguments that extend the flow to the whole phase space. Positivity of the momentum recovers the classical Camassa–Holm flow without blow-up, while the framework also provides a detailed description of invariant sets and a broad family of almost conserved quantities, highlighting the deep spectral structure underlying this integrable system.

Abstract

We extend the inverse spectral transform for the conservative Camassa-Holm flow on the line to a class of initial data that requires strong decay at one endpoint but only mild boundedness-type conditions at the other endpoint. The latter condition appears to be close to optimal in a certain sense for the well-posedness of the conservative Camassa-Holm flow. As a byproduct of our approach, we also find a family of new (almost) conservation laws for the Camassa-Holm equation, which could not be deduced from its bi-Hamiltonian structure before and which are connected to certain Besov-type norms (however, in a rather involved way). These results appear to be new even under positivity assumptions on the corresponding momentum, in which case the conservative Camassa-Holm flow coincides with the classical Camassa-Holm flow and no blow-ups occur.
Paper Structure (5 sections, 28 theorems, 161 equations)

This paper contains 5 sections, 28 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. The phase space
  3. The isospectral problem
  4. The spectral transform
  5. The conservative Camassa--Holm flow

Key Result

Lemma 2.1

If the pair $(u,\mu)$ belongs to $\mathcal{D}$, then the corresponding function $\tilde{\mathsf{w}}$ belongs to $L^2(0,\infty)$ and is real-valued almost everywhere, the corresponding measure $\tilde{\upsilon}$ is finite and

Theorems & Definitions (80)

  • Definition 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Definition 2.5
  • ...and 70 more