Decay estimates for large velocities in the Boltzmann equation without cutoff

Abstract

We consider solutions f = f (t, x, v) to the full (spatially inhomogeneous) Boltzmann equation with periodic spatial conditions x ∈ 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 v f (t, x, v) and local energy v f (t, x, v)|v| 2 and local entropy v f (t, x, v) ln f (t, x, 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|) 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.

Global Mild Solutions of the Landau and Non‐Cutoff Boltzmann Equations

“…The De Giorgi's method applied to the Landau equation has been recently developed in Golse-Imbert-Mouhot-Vasseur [39]. Notice that for the Boltzmann equation without cutoff, the regularity issue has also been studied in many recent works, for instance, Silvestre [72], Imbert-Silvestre [56], Imbert-Mouhot-Silvestre [55], and Chen-Hu-Li-Zhan [20]. See also a recent survey by Mouhot [66] and references therein.…”

Gaussian Lower Bounds for the Boltzmann Equation without Cutoff

“…The following L ∞ bound is one of the main results in [30]. We state the slightly refined version in [22,Theorem 4.1], which includes in particular the limit case γ + 2s = 0.…”

“…Note that this result holds for γ + 2s < 0 conditionally to the L ∞ bound. However this L ∞ bound is proved only when γ + 2s ∈ [0, 2] (see [30,22]).…”

“…Previous results of conditional regularity include, for the Boltzmann equation and under the conditions above, the proof of L ∞ bounds in [30], the proof of a weak Harnack inequality and Hölder continuity in [24], the proof of polynomially decaying upper bounds in [22], and the proof of Schauder estimates to bootstrap higher regularity estimates in [23]. In the case of the closely related Landau equation, which is a nonlinear diffusive approximation of the Boltzmann equation, the L ∞ bound was proved in [31,17], the Harnack inequality and Hölder continuity were obtained in [32,17], decay estimates were obtained in [14], and Schauder estimates were established in [20] (see also [21]).…”

“…In [26], a priori assumptions of smoothness of the solution are used to remove a neighbourhood around θ = 0 in the collision integral and treat it as an error term. Here instead we use coercivity and sign properties of this singular part of the collision operators, as developed and used in [30,23,22]; because of the fractional derivative involved we use also a barrier method to justify the argument; it is inspired from [29], and was recently applied to the Boltzmann equation in [30,22]. 1.6.…”