Decay estimates for large velocities in the Boltzmann equation without cutoff

Imbert, Cyril; Mouhot, Clément; Silvestre, Luis

Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 143-183.

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 $x\in {𝕋}^d$, 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 $\int f\mathrm{d}v$, énergie locale ${\int f\left|v\right|}^2\mathrm{d}v$, entropie locale $\int flnf\mathrm{d}v$, restent bornés au cours du temps, nous montrons des bornes sur les « moments polynomiaux ponctuels » ${sup}_{x,v}{f(t,x,v)(1+|v|}^q)$, $q\ge 0$. 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.

We consider solutions $f=f(t,x,v)$ to the full (spatially inhomogeneous) Boltzmann equation with periodic spatial conditions $x\in {𝕋}^d$, 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 $\int f\mathrm{d}v$ and local energy ${\int f\left|v\right|}^2\mathrm{d}v$ and local entropy $\int flnf\mathrm{d}v$, are controlled along time. We establish quantitative estimates of propagation in time of “pointwise polynomial moments”, i.e., ${sup}_{x,v}f(t,x,v){(1+|v\left|\right)}^q$, $q\ge 0$. In the case of hard potentials, we also prove appearance of these moments for all $q\ge 0$. 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 $+\infty$, conditionally to the bounds on the hydrodynamic fields being uniform.

Classification : 35Q82, 76P05, 35Q20
Keywords: Boltzmann equation, non-cutoff, grazing collisions, regularity, decay, maximum principle, a priori solutions
Mot clés : Équation de Boltzmann, sans troncature angulaire, collisions rasantes, régularité, décroissance, principe du maximum, solutions a priori

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