A Multiscale Camassa--Holm Equation

2025/07/07 by Holm, Darryl D, Singh, Maneesh Kumar · 1 citation
#FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2507.05018

Abstract

A system of equations for Multiscale Geodesic Flow (MGF) is introduced whose solutions illustrate the paradigm of whorls within whorls within whorls, introduced by L. F. Richardson in 1922 to describe the cascade of energy in fluid turbulence. Numerical simulations are given for MGF on S¹, where the MGF equation comprises a multiscale generalisation of the Camassa-Holm (CH) equation whose emergent singular solutions generalise the peakon solutions of the CH.

Citations

• Variational principles for fluid dynamics on rough paths
• Momentum Maps and Measure-valued Solutions (Peakons, Filaments and Sheets) for the EPDiff Equation
• The Three Dimensional Viscous Camassa-Holm Equations, and Their Relation to the Navier-Stokes Equations and Turbulence Theory
• The geometry and analysis of the averaged Euler equations and a new diffeomorphism group
• The Euler-Poincare Equations and Semidirect Products with Applications to Continuum Theories

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