Rescaled Objective Solutions of Fokker-Planck and Boltzmann equations

TL;DR

This paper investigates the long-time behavior of symmetric solutions to the nonlinear Boltzmann and Fokker-Planck equations under anisotropic rescaling, revealing differences in stationary solutions and convergence rates.

Contribution

It introduces an anisotropic rescaling method that conserves energy and analyzes the long-term behavior, showing the absence of Maxwellian stationary solutions for Boltzmann and algebraic convergence for Fokker-Planck.

Findings

Rescaled Boltzmann equation has no Maxwellian stationary solutions.

Rescaled Fokker-Planck solutions converge to Maxwellian.

Convergence for Fokker-Planck is algebraic, not exponential.

Abstract

We study the long-time behavior of symmetric solutions of the nonlinear Boltzmann equation and a closely related nonlinear Fokker-Planck equation. If the symmetry of the solutions corresponds to shear flows, the existence of stationary solutions can be ruled out because the energy is not conserved. After anisotropic rescaling both equations conserve the energy. We show that the rescaled Boltzmann equation does not admit stationary densities of Maxwellian type (exponentially decaying). For the rescaled Fokker-Planck equation we demonstrate that all solutions converge to a Maxwellian in the long-time limit, however the convergence rate is only algebraic, not exponential.