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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).
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Contributor : Clément Mouhot <>
Submitted on : Wednesday, March 4, 2020 - 10:12:08 AM
Last modification on : Tuesday, September 22, 2020 - 3:50:16 AM
Long-term archiving on: : Friday, June 5, 2020 - 1:24:21 PM


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  • HAL Id : hal-02078069, version 2
  • ARXIV : 1903.11278



Cyril Imbert, Clément Mouhot, Luis Silvestre. Gaussian lower bounds for the Boltzmann equation without cut-off. 2020. ⟨hal-02078069v2⟩



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