Fractional diffusion limit for collisional kinetic equations

Abstract

This paper is devoted to diffusion limits of linear Boltzmann equations. When the equilibrium distribution function is Maxwellian distribution, it is well known that for an appropriate time scale, the small mean free path limit gives rise to a diffusion equation. In this paper, we consider situations in which the equilibrium distribution function is a heavy-tailed distribution with infinite variance. We then show that for an appropriate time scale, the small mean free path limit gives rise to a fractional diffusion equation.

References in corpus (2)

• Limit theorems for additive functionals of a Markov chain
• Boltzmann equations for mixtures of Maxwell gases: exact solutions and power like tails

Cited by in corpus (29)

• Limit theorems for additive functionals of a Markov chain
• Superdiffusion of energy in a chain of harmonic oscillators with noise
• Anomalous heat transport in one dimensional systems: a description using non-local fractional-type diffusion equation
• 3/4 Fractional superdiffusion of energy in a system of harmonic oscillators perturbed by a conservative noise
• Hypocoercivity and diffusion limit of a finite volume scheme for linear kinetic equations
• Central limit theorems for additive functionals of ergodic Markov diffusions processes
• Hypocoercivity and sub-exponential local equilibria
• Anomalous energy transport in FPU-$\beta$ chain
• The fractional diffusion limit of a kinetic model with biochemical pathway
• Fractional Fokker-Planck Equation with General Confinement Force
• Convergence of a kinetic equation to a fractional diffusion equation
• The time-fractional radiative transport equation -- Continuous-time random walk, diffusion approximation, and Legendre-polynomial expansion
• Quantitative fluid approximation in transport theory: a unified approach
• Fractional Diffusion limit of a kinetic equation with Diffusive boundary conditions in the upper-half space
• $L^$p Estimates For Degenerate Non-Local Kolmogorov Operators
• Long time, large scale limit of the Wigner transform for a system of linear oscillators in one dimension
• A gradient flow approach to linear Boltzmann equations
• Fluctuations in a kinetic transport model for quantum friction
• Fractional diffusion limit for a stochastic kinetic equation
• Fractional diffusion limit for collisional kinetic equations: A moments method
• Motility switching and front-back synchronisation in polarized cells
• Superdiffusion transition for a phonon Boltzmann equation
• Strong solutions to the inhomogeneous Navier-Stokes-BGK system
• Fractional diffusion for Fokker-Planck equation with heavy tail equilibrium: an à la Koch spectral method in any dimension
• Stationary Flows of the ES-BGK model with the correct Prandtl number
• On Rosenau-Type Approximations to Fractional Diffusion Equations
• Global regularity of 2D tropical climate model with zero thermal diffusion
• Large Deviations for Stochastic equations in Hilbert Spaces with non-Lipschitz drift
• A doubly nonlinear evolution problem involving the fractional p-Laplacian