# Diffusive limit for a Boltzmann-like equation with non-conserved momentum

## TL;DR

This paper proves that a kinetic model with non-conserved momentum converges to a diffusive limit described by coupled diffusion equations for density and temperature, with explicit error estimates.

## Contribution

It establishes the diffusive limit for a Boltzmann-like equation with non-conserved momentum, including rigorous error bounds.

## Key findings

- The rescaled distribution converges to a Maxwellian with density and temperature solving coupled diffusion equations.
- Explicit error estimates in $L^2_{x,v}$ norm are provided.
- The model accounts for collisions with fixed scatterers and hard-sphere interactions.

## Abstract

We consider a kinetic model whose evolution is described by a Boltzmann-like equation for the one-particle phase space distribution $f(x,v,t)$. There are hard-sphere collisions between the particles as well as collisions with randomly fixed scatterers. As a result, this evolution does not conserve momentum but only mass and energy. We prove that the diffusively rescaled $f^\varepsilon(x,v,t)=f(\varepsilon^{-1}x,v,\varepsilon^{-2}t)$, as $\varepsilon\to 0$ tends to a Maxwellian $M_{\rho, 0, T}=\frac{\rho}{(2\pi T)^{3/2}}\exp[{-\frac{|v|^2}{2T}}]$, where $\rho$ and $T$ are solutions of coupled diffusion equations and estimate the error in $L^2_{x,v}$.