Quantitative phase mixing for Hamiltonians with trapping

TL;DR

This paper establishes quantitative decay estimates for solutions to linear transport equations driven by Hamiltonians with trapped trajectories, covering applications in plasma physics, fluid flows, and gravitational systems.

Contribution

It provides the first rigorous decay estimates for phase mixing in Hamiltonian systems with trapping and elliptic stagnation points, including the challenging 1+1-dimensional case.

Findings

Decay estimates for macroscopic quantities in trapped Hamiltonian systems

Analysis applicable to gravitational Vlasov-Poisson equilibria

Results relevant for plasma physics and fluid dynamics

Abstract

We prove quantitative decay estimates of macroscopic quantities generated by the solutions to linear transport equations driven by a general family of Hamiltonians. The associated particle trajectories are all trapped in a compact region of phase-space and feature a non-degenerate elliptic stagnation point. The analysis covers a large class of Hamiltonians generated by the radially symmetric compactly supported equilibria of the gravitational Vlasov-Poisson system. Working in radial symmetry, our analysis features both the 1+2-dimensional case and the harder 1+1-dimensional case, where all the particles have the same value of the modulus of angular momentum. The latter case is also of importance in both the plasma physics case and two dimensional incompressible fluid flows.