Quantitative phase mixing for Hamiltonians with trapping
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 phasespace and feature a nondegenerate elliptic stagnation point. The analysis covers a large class of Hamiltonians generated by the radially symmetric compactly supported equilibria of the gravitational VlasovPoisson system. Working in radial symmetry, our analysis features both the 1+2dimensional case and the harder 1+1dimensional 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2024
 DOI:
 10.48550/arXiv.2405.17153
 arXiv:
 arXiv:2405.17153
 Bibcode:
 2024arXiv240517153H
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematical Physics;
 Mathematics  Dynamical Systems
 EPrint:
 60 pages, added the decay of the macroscopic density to Theorem 1.7