# Gaussian lower bounds for the Boltzmann equation without cut-off

## TL;DR

This paper proves Gaussian lower bounds for solutions to the non-cutoff Boltzmann equation with hard and moderately soft potentials, under bounded hydrodynamic quantities, resolving a long-standing open problem.

## Contribution

It establishes the first Gaussian lower bounds for non-cutoff Boltzmann solutions under minimal assumptions, extending previous results to long-range interactions.

## Key findings

- Gaussian lower bounds are proven for solutions with non-cutoff kernels.
- Results apply to hard and moderately soft potentials with bounded hydrodynamic quantities.
- The work completes the understanding of positivity properties for the Boltzmann equation without cutoff.

## Abstract

The study of positivity of solutions to the Boltzmann equation goes back to Carleman (1933), and the initial argument of Carleman was developed byPulvirenti-Wennberg (1997), the second author and Briant (2015). The appearance of a lower bound with Gaussian decay had however remained an open question for long-range interactions (the so-called non-cutoff collision kernels). We answer this question and establish such Gaussian lower bound for solutions to the Boltzmann equation without cutoff, in the case of hard and moderately soft potentials, with spatial periodic conditions, and under the sole assumption that hydrodynamic quantities (local mass, local energy and local entropy density) remain bounded. The paper is mostly self-contained, apart from the uniform upper bound on the solution established by the third author (2016).