# Uniform estimates on the Fisher information for solutions to Boltzmann and Landau equations

## TL;DR

This paper establishes uniform propagation of Fisher information for solutions to the Boltzmann and Landau equations under minimal regularity assumptions, using explicit bounds and diffusion properties.

## Contribution

It provides new uniform estimates on Fisher information for these equations, enhancing understanding of solution regularity and tail behavior.

## Key findings

- Uniform Fisher information propagation for Boltzmann and Landau solutions
- Explicit pointwise lower bounds on Boltzmann solutions
- Emergence and propagation of exponential tails for gradients

## Abstract

In this note we prove that, under some minimal regularity assumptions on the initial datum, solutions to the spatially homogenous Boltzmann and Landau equations for hard potentials uniformly propagate the Fisher information. The proof of such a result is based upon some explicit pointwise lower bound on solutions to Boltzmann equation and strong diffusion properties for the Landau equation. We include an application of this result related to emergence and propagation of exponential tails for the solution's gradient. These results complement estimates provided in the literature.