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Journal Articles SIAM Journal on Mathematical Analysis Year : 2020

Gaussian lower bounds for the Boltzmann equation without cut-off

Abstract

The study of positivity of solutions to the Boltzmann equation goes back to [T. Carleman, Acta Math., 60 (1933), pp. 91--146], and the initial argument of Carleman was developed in [A. Pulvirenti and B. Wennberg, Comm. Math. Phys., 183 (1997), pp. 145--160; C. Mouhot, Comm. Partial Differential Equations, 30 (2005), pp. 881--917; M. Briant, Arch. Ration. Mech. Anal., 218 (2015), pp. 985--1041; M. Briant, Kinet. Relat. Models, 8 (2015), pp. 281--308] but the appearance of a lower bound with Gaussian decay had remained an open question for long-range interactions (the so-called noncutoff collision kernels). We answer this question and establish such 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. The paper is mostly self-contained, apart from the $L^\infty$ upper bound and weak Harnack inequality on the solution established, respectively in [L. Silvestre, Comm. Math. Phys., 348 (2016), pp. 69--100; C. Imbert, C. Mouhot, and L. Silvestre, J. Éc. polytech. Math., 7 (2020), pp. 143--184.; C. Imbert and L. Silvestre, J. Eur. Math. Soc. (JEMS), 22 (2020), pp. 507--592].
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Dates and versions

hal-02078069 , version 1 (25-03-2019)
hal-02078069 , version 2 (04-03-2020)

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Cyril Imbert, Clément Mouhot, Luis Silvestre. Gaussian lower bounds for the Boltzmann equation without cut-off. SIAM Journal on Mathematical Analysis, 2020, 52 (3), pp.2930-2944. ⟨10.1137/19M1252375⟩. ⟨hal-02078069v2⟩
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