On measures of accretion and dissipation for solutions of the Camassa-Holm equation

TL;DR

This paper studies the measures of dissipation and accretion in weak solutions of the Camassa-Holm equation, revealing their structural properties and differences between dissipative and non-dissipative solutions.

Contribution

It introduces a new representation formula for these measures and analyzes their structural features, including singularities and vanishing properties in dissipative solutions.

Findings

Measures of accretion vanish for dissipative solutions

New representation formula for dissipation and accretion measures

Structural analysis of measures reveals singularities with respect to Lebesgue measure

Abstract

We investigate the measures of dissipation and accretion related to the weak solutions of the Camassa-Holm equation. Demonstrating certain properties of nonunique characteristics, we prove a new representation formula for these measures and conclude about their structural features, such us singularity with respect to the Lebesgue measure. We apply these results to gain new insights into the structure of weak solutions, proving in particular that measures of accretion vanish for dissipative solutions of the Camassa-Holm equation.