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On the fractional diffusion for the linear Boltzmann equation with drift and general cross-section

Dahmane Dechicha

TL;DR

This work establishes a fractional diffusion limit for the linear Boltzmann equation with heavy-tail equilibrium and a cross-section that depends on space and degenerates at high velocities, without assuming symmetry. By adapting the auxiliary problem method to this non-symmetric setting and incorporating a drift term j^ε_F, the authors show that, under the scaling θ(ε) = ε^γ with γ = (α−β)/(1−β), the density ρ(x,t) satisfies a macroscopic equation ∂_t ρ + κ L(ρ) = 0, where L is a spatial elliptic operator of order γ defined by a weighted PV integral. The operator L is expressed with a kernel η(x,y) that remains bounded and self-adjoint, reducing to a multiple of the fractional Laplacian when σ and ν are x-independent. The result extends prior symmetric-case diffusion limits, demonstrating that anomalous diffusion can arise in broad kinetic settings with non-symmetric cross-sections and degenerating collision frequencies. This has implications for modeling transport in media with heavy-tailed equilibria and velocity-dependent interactions, where fractional diffusion governs macroscopic behavior.

Abstract

This paper is devoted to the hydrodynamic limit for the linear Boltzmann equation, in the case of a heavy tail equilibrium and a cross section which depends on the space variable and which degenerates for large velocities, without symmetry assumptions. For an appropriate time scale, a macroscopic equation with an elliptic operator, which is equivalent to the fractional Laplacian, is obtained. This problem has been addressed in [Mellet, Mischler and Mouhot, ARMA, 2011] for a space-independent cross section, using a Fourier-Laplace transform, where a fractional diffusion equation has been obtained, and revisited in [Mellet, Indiana Univ., 2010] for a space-dependent but a bounded cross section, using the moments method by introducing an auxiliary problem. In this work, we will adapt the latter method to generalize both results.

On the fractional diffusion for the linear Boltzmann equation with drift and general cross-section

TL;DR

This work establishes a fractional diffusion limit for the linear Boltzmann equation with heavy-tail equilibrium and a cross-section that depends on space and degenerates at high velocities, without assuming symmetry. By adapting the auxiliary problem method to this non-symmetric setting and incorporating a drift term j^ε_F, the authors show that, under the scaling θ(ε) = ε^γ with γ = (α−β)/(1−β), the density ρ(x,t) satisfies a macroscopic equation ∂_t ρ + κ L(ρ) = 0, where L is a spatial elliptic operator of order γ defined by a weighted PV integral. The operator L is expressed with a kernel η(x,y) that remains bounded and self-adjoint, reducing to a multiple of the fractional Laplacian when σ and ν are x-independent. The result extends prior symmetric-case diffusion limits, demonstrating that anomalous diffusion can arise in broad kinetic settings with non-symmetric cross-sections and degenerating collision frequencies. This has implications for modeling transport in media with heavy-tailed equilibria and velocity-dependent interactions, where fractional diffusion governs macroscopic behavior.

Abstract

This paper is devoted to the hydrodynamic limit for the linear Boltzmann equation, in the case of a heavy tail equilibrium and a cross section which depends on the space variable and which degenerates for large velocities, without symmetry assumptions. For an appropriate time scale, a macroscopic equation with an elliptic operator, which is equivalent to the fractional Laplacian, is obtained. This problem has been addressed in [Mellet, Mischler and Mouhot, ARMA, 2011] for a space-independent cross section, using a Fourier-Laplace transform, where a fractional diffusion equation has been obtained, and revisited in [Mellet, Indiana Univ., 2010] for a space-dependent but a bounded cross section, using the moments method by introducing an auxiliary problem. In this work, we will adapt the latter method to generalize both results.

Paper Structure

This paper contains 10 sections, 8 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Setting of the problem
  3. Main result
  4. Preliminaries results
  5. Properties of the operator $Q$
  6. Compactness lemma
  7. The auxiliary problem
  8. Proof of the main theorem
  9. First part of the limiting process
  10. Obtention of the limiting operator

Key Result

Theorem 1.3

Assume (A1-A2) and (B1-B2-B3) with $\alpha > 0$ and $\beta < \min\{\alpha;2-\alpha\}$. We define $\gamma := \frac{\alpha - \beta}{1 - \beta}$, and the drift term $j^\varepsilon_F$ by Let $f^\varepsilon(t,x,v)$ be the solution of BL-eps with $\theta(\varepsilon)=\varepsilon^\gamma$ and non-negative $f_0 \in L^2_{F ^{-1}}(\mathbb{R}^{2d})$. Then, $f^\varepsilon$ converges weakly star in $L^\infty(0

Theorems & Definitions (17)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1.3: Fractional diffusion limit for the linear Boltzmann equation
  • Remark 4
  • Remark 5
  • Proposition 2.1
  • Remark 6
  • Remark 7
  • Lemma 2.2
  • ...and 7 more