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 ∫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 , 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 +∞, conditionally to the bounds on the hydrodynamic fields being uniform.
Highlights
(b) Weakening the statement: more regularity or decay could be assumed on the initial data, as long as it is propagated conditionally to the hydrodynamic bounds assumed on the solution
– In the case of spatially homogeneous moderately soft potentials with cutoff, Desvillettes [21] proved for γ ∈ (−1, 0) that initially bounded polynomial moments grow at most linearly with time and it is explained in [52] that the method applies to γ ∈ [−2, 0)
We prove the existence of a first contact point (t0, x0, v0) such that f (t0, x0, v0) = g(t0, v0), and search for a contradiction at this point
Summary
Fluctuations around steady state, on a large scale, follow incompressible Navier-Stokes equations under the appropriate limit This probability density of particles is a non-negative function f = f (t, x, v) defined on a given time interval I ∈ R and (x, v) ∈ Rd×Rd and it solves the integro-differential Boltzmann equation (1.1). For all long-range interactions, that is all interactions apart from the hard spheres model, it is singular at θ ∼ 0, i.e., small deviation angles that correspond to grazing collisions. Keeping this singularity dictated by physics in the mathematical analysis has come to be known quite oddly as a non-cutoff assumption. The singularity of b at grazing collisions θ ∼ 0 is the legacy of long-range interactions
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