Rescaled Objective Solutions of Fokker-Planck and Boltzmann equations

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.

arXiv comment25 pages; v2: improved presentation, conclusion added

References in corpus (4)

• Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials
• Asymptotic analysis for the generalized langevin equation
• Self-similar profiles for homoenergetic solutions of the Boltzmann equation: particle velocity distribution and entropy
• Celebrating Cercignani's conjecture for the Boltzmann equation

Cited by in corpus (5)

• Long time asymptotics for homoenergetic solutions of the Boltzmann equation. Hyperbolic-dominated case
• Self-similar asymptotics for a modified Maxwell-Boltzmann equation in systems subject to deformations
• The Boltzmann equation for uniform shear flow
• Self-similar profiles for homoenergetic solutions of the Boltzmann equation for non-cutoff Maxwell molecules
• Longtime behavior for homoenergetic solutions in the collision dominated regime for hard potentials