Decay estimates for large velocities in the Boltzmann equation without cutoff
Journal de l’École polytechnique — Mathématiques, Volume 7 (2020), pp. 143-183.

We consider solutions f=f(t,x,v) to the full (spatially inhomogeneous) Boltzmann equation with periodic spatial conditions x𝕋 d , for hard and moderately soft potentials without the angular cutoff assumption, and under the a priori assumption that the main hydrodynamic fields, namely the local mass fdv and local energy f|v| 2 dv and local entropy flnfdv, are controlled along time. We establish quantitative estimates of propagation in time of “pointwise polynomial moments”, i.e., sup x,v f(t,x,v)(1+|v|) q , q0. In the case of hard potentials, we also prove appearance of these moments for all q0. In the case of moderately soft potentials, we prove the appearance of low-order pointwise moments. All these conditional bounds are uniform as t goes to +, conditionally to the bounds on the hydrodynamic fields being uniform.

Cet article considère des solutions a priori f=f(t,x,v) de l’équation de Boltzmann sans hypothèse d’homogénéité spatiale et avec conditions périodiques x𝕋 d , pour des interactions de type potentiels durs ou modérément mous sans troncature angulaire. Sous l’hypothèse a priori que les champs hydrodynamiques associés à la solution : masse locale fdv, énergie locale f|v| 2 dv, entropie locale flnfdv, restent bornés au cours du temps, nous montrons des bornes sur les « moments polynomiaux ponctuels » sup x,v f(t,x,v)(1+|v| q ), q0. Ces moments sont propagés dans le cas des potentiels modérément mous, et apparaissent dans le cas des potentiels durs. Dans le cas des potentiels modérément mous, nous montrons également l’apparition de moments ponctuels d’ordre bas. Toutes ces bornes conditionnelles sont uniformes en temps grand, dès lors que les bornes sur les champs hydrodynamiques sont elles-mêmes uniformes en temps grand.

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DOI: 10.5802/jep.113
Classification: 35Q82,  76P05,  35Q20
Keywords: Boltzmann equation, non-cutoff, grazing collisions, regularity, decay, maximum principle, a priori solutions
Cyril Imbert 1; Clément Mouhot 2; Luis Silvestre 3

1 CNRS & Department of Mathematics and Applications, École Normale Supérieure (Paris) 45 rue d’Ulm, 75005 Paris, France
2 University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce road, Cambridge CB3 0WA, UK
3 Mathematics Department, University of Chicago, Chicago, Illinois 60637, USA
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Cyril Imbert; Clément Mouhot; Luis Silvestre. Decay estimates for large velocities in the Boltzmann equation without cutoff. Journal de l’École polytechnique — Mathématiques, Volume 7 (2020), pp. 143-183. doi : 10.5802/jep.113. https://jep.centre-mersenne.org/articles/10.5802/jep.113/

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