Journal
JOURNAL DE L ECOLE POLYTECHNIQUE-MATHEMATIQUES
Volume 7, Issue -, Pages 143-184Publisher
ECOLE POLYTECHNIQUE
DOI: 10.5802/jep.113
Keywords
Boltzmann equation; non-cutoff; grazing collisions; regularity; decay; maximum principle; a priori solutions
Categories
Funding
- NSF [DMS-1254332, DMS-1362525]
- ERC grant MAFRAN
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We consider solutions f = f(t, x, v) to the full (spatially inhomogeneous) Boltzmann equation with periodic spatial conditions x is an element of T-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 integral f dv and local energy integral f vertical bar v vertical bar(2) dv and local entropy integral f ln f dv, 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 + vertical bar v vertical bar)(q), q >= 0. In the case of hard potentials, we also prove appearance of these moments for all q >= 0. 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 +infinity, conditionally to the bounds on the hydrodynamic fields being uniform.
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