Self-generating lower bounds and continuation for the Boltzmann equation
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Calculus of Variations
Self-generating lower bounds and continuation for the Boltzmann equation Christopher Henderson1 · Stanley Snelson2 · Andrei Tarfulea3 Received: 31 May 2020 / Accepted: 31 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract For the spatially inhomogeneous, non-cutoff Boltzmann equation posed in the whole space R3x , we establish pointwise lower bounds that appear instantaneously even if the initial data contains vacuum regions. Our lower bounds depend only on the initial data and upper bounds for the mass and energy densities of the solution. As an application, we improve the weakest known continuation criterion for large-data solutions, by removing the assumptions of mass bounded below and entropy bounded above. Mathematics Subject Classification 35Q20 · 35B60 · 35B65
1 Introduction The Boltzmann equation is a kinetic equation arising in statistical physics. Its solution f (t, x, v) ≥ 0 models the density of particles of a diffuse gas at time t ∈ [0, T ], at location x ∈ R3 , and with velocity v ∈ R3 . The equation reads ∂t f + v · ∇x f = Q( f , f ) =
R3
S2
B(v − v∗ , σ ) f (v∗ ) f (v ) − f (v∗ ) f (v) dσ dv∗ , (1.1)
Communicated by Y. Giga.
B
Stanley Snelson [email protected] Christopher Henderson [email protected] Andrei Tarfulea [email protected]
1
Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA
2
Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA
3
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA 0123456789().: V,-vol
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C. Henderson et al.
where v and v∗ are post-collisional velocities, and v and v∗ are pre-collisional velocities, given by v + v∗ v + v∗ |v − v∗ | |v − v∗ | +σ and v∗ = −σ . 2 2 2 2 In this article, we focus on the non-cutoff version of (1.1) that includes the physically realistic singularity at grazing collisions. The collision kernel is given by v =
˜ θ), B(v − v∗ , σ ) = |v − v∗ |γ θ −2−2s b(cos
where cos θ = σ ·
v − v∗ , γ > −3, s ∈ (0, 1), |v − v∗ |
and b˜ a positive bounded function. The main purpose of this article is to prove that pointwise lower bounds for f appear instantaneously, under rather weak assumptions on both the solution and the initial data. We make no a priori assumption of positive mass, except at t = 0, where uniform positivity in some small ball in (x, v) space is required (but otherwise vacuum regions may exist). The constants in our lower bounds depend only on the initial data and zeroth-order norms of the solution (see (1.2) below). On physical grounds, gases modeled by (1.1) should be expected to fill vacuum regions instantaneously, so it is important to establish this property under as few assumptions as possible. On a mathematical level, lower bounds for f grant nice coercivity properties to the collision operator Q( f , f ), which are a key ingredient of the regularity and existence theory for (1.1) (see, e.g., [14]). Two specific applications we
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