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On a class of nonlinear BGK-type kinetic equations with density dependent collision rates

Josephine Evans, Daniel Morris, Havva Yoldaş

TL;DR

The paper analyzes a nonlinear BGK-type kinetic model with density-dependent collision rates and proves global well-posedness, propagation of Maxwellian bounds, and exponential relaxation to equilibrium via $L^2$-hypocoercivity. It also derives a quantitative diffusive limit under diffusive scaling, showing that as $\varepsilon \to 0$, solutions converge to a Maxwellian in velocity with density $\rho$ solving the nonlinear diffusion equation $\partial_t \rho = \nabla_x \cdot ( \rho^{-\ abla\alpha} \nabla_x \rho )$, encompassing both the porous medium and fast diffusion regimes. The analysis blends hypocoercivity and relative entropy methods to obtain rates and rigorously justify the hydrodynamic limit, including finite-time and long-time regimes. These results establish a rigorous link between kinetic models with density-dependent collisions and macroscopic nonlinear diffusion equations, with potential extensions to whole-space settings and cross-diffusion systems. The work advances understanding of how microscopic interaction rates shape macroscopic diffusion phenomena in complex systems.

Abstract

We consider a class of nonlinear, spatially inhomogeneous kinetic equations of BGK-type with density dependent collision rates. These equations share the same superlinearity as the Boltzmann equation, and fall into the class of run and tumble equations appearing in mathematical biology. We prove that the Cauchy problem is well-posed, and the solutions propagate Maxwellian bounds over time. Moreover, we show that the solutions approach to equilibrium with an exponential rate, known as a hypocoercivity result. Lastly, we derive a class of nonlinear diffusion equations as the hydrodynamic limit of the kinetic equations in the diffusive scaling, employing both hypocoercivity and relative entropy methods. The limit equations cover a wide range of nonlinear diffusion equations including both the porous medium and the fast diffusion equations.

On a class of nonlinear BGK-type kinetic equations with density dependent collision rates

TL;DR

The paper analyzes a nonlinear BGK-type kinetic model with density-dependent collision rates and proves global well-posedness, propagation of Maxwellian bounds, and exponential relaxation to equilibrium via -hypocoercivity. It also derives a quantitative diffusive limit under diffusive scaling, showing that as , solutions converge to a Maxwellian in velocity with density solving the nonlinear diffusion equation , encompassing both the porous medium and fast diffusion regimes. The analysis blends hypocoercivity and relative entropy methods to obtain rates and rigorously justify the hydrodynamic limit, including finite-time and long-time regimes. These results establish a rigorous link between kinetic models with density-dependent collisions and macroscopic nonlinear diffusion equations, with potential extensions to whole-space settings and cross-diffusion systems. The work advances understanding of how microscopic interaction rates shape macroscopic diffusion phenomena in complex systems.

Abstract

We consider a class of nonlinear, spatially inhomogeneous kinetic equations of BGK-type with density dependent collision rates. These equations share the same superlinearity as the Boltzmann equation, and fall into the class of run and tumble equations appearing in mathematical biology. We prove that the Cauchy problem is well-posed, and the solutions propagate Maxwellian bounds over time. Moreover, we show that the solutions approach to equilibrium with an exponential rate, known as a hypocoercivity result. Lastly, we derive a class of nonlinear diffusion equations as the hydrodynamic limit of the kinetic equations in the diffusive scaling, employing both hypocoercivity and relative entropy methods. The limit equations cover a wide range of nonlinear diffusion equations including both the porous medium and the fast diffusion equations.

Paper Structure

This paper contains 19 sections, 10 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Outline of paper.
  3. The models
  4. The kinetic equations.
  5. The limit equations.
  6. Formal diffusive limit.
  7. Motivations and open questions
  8. State of the art
  9. Hydrodynamic limits in kinetic theory.
  10. Particle approximations to the porous medium or fast diffusion equations.
  11. Preliminary and main results
  12. Main results
  13. Preliminary results on hydrodynamic quantities
  14. Well-posedness
  15. Long-time behaviour
  16. ...and 4 more sections

Key Result

Theorem 2.1

Let $\varepsilon>0, A>1, \alpha\in\mathbb{R}$, and suppose that $f_{\varepsilon, \mathrm{in}}(x,v)\in L^{1}(\mathbb{T}^{d}\times\mathbb{R}^{d})$ satisfy $A^{-1}\mathcal{M} (v)\leq f_{\varepsilon, \mathrm{in}}(x,v)\leq A \mathcal{M} (v)$. Then problem eq:mainkinetic with initial data $f_{\varepsilon,

Theorems & Definitions (21)

  • Theorem 2.1: Well-posedness, propagation of bounds
  • Theorem 2.2: Long-time behaviour
  • Theorem 2.3: Diffusion asymptotics
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Remark 2.7
  • Remark 2.8
  • ...and 11 more