Title: Gaussian lower bounds for the Boltzmann equation without cut-off
Date: July 1, 2021, 11:30 am ET
Speaker: Luis Silvestre, University of Chicago

Abstract: The Boltzmann equation models the evolution of densities of particles in a gas. Its global well posedness is a major open problem, facing comparable difficulties as similar questions for equations in fluids. With current techniques, we cannot rule out the possibility of a spontaneous emergence of a singularity in the form of infinite mass or energy density concentrating at some point in space. This work is part of a series of a priori estimates for the inhomogeneous non-cutoff Boltzmann equation that are conditional to bounds on macroscopic quantities. We establish a 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. In the talk, we will discuss how this lower bound fits in the larger program of conditional estimates.

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