Gaussian lower bounds for the Boltzmann equation without cut-off

Abstract

The study of positivity of solutions to the Boltzmann equation goes back to Carleman (1933), and the initial argument of Carleman was developed by Pulvirenti-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). Contents 1. Introduction 1 2. Preliminaries 5 3. Lower bounds 9 References 13

Self-generating lower bounds and continuation for the Boltzmann equation

“…(a) Continuation. The recent result of Imbert-Silvestre [13] (which finished a long program of the two authors and Mouhot, see [21,12,16,11]) showed that solutions to (1.1) can be continued for as long as the mass, energy, and entropy densities of f are bounded above, and the mass density is bounded below, when γ + 2s ∈ [0, 2]. Our main theorem implies CH was partially supported by NSF grant DMS-2003110.…”

“…Fournier [7] proved strict positivity for the homogeneous, non-cutoff case, and Mouhot [18] derived the first quantitative lower bounds for the inhomogeneous equation (with periodic boundary conditions), obtaining Gaussian decay in the cutoff case, and exponential decay in the non-cutoff case. The more recent work of Imbert-Mouhot-Silvestre [11] is the first to prove the optimal Gaussian asymptotics for the non-cutoff equation. The current article borrows some techniques from [11].…”

“…The more recent work of Imbert-Mouhot-Silvestre [11] is the first to prove the optimal Gaussian asymptotics for the non-cutoff equation. The current article borrows some techniques from [11].…”

“…Steps 1 and 3 are achieved by a barrier argument that propagates lower bounds along characteristics of the free transport equation (Lemma 3.1), and Step 2 is accomplished by adapting the strategy of [11] to handle lower bounds that are not uniform in x (see Lemma 3.2).…”

“…As an application, we improve the weakest known continuation criterion for large-data solutions, by removing the assumptions of mass bounded below and entropy bounded above.where b is uniformly positive and bounded. * :At this point, we may quote verbatim the calculations of [11, Lemma 3.4] to obtain…”