Download Hydrodynamic Limits of the Boltzmann Equation by Laure Saint-Raymond PDF
By Laure Saint-Raymond
The goal of this publication is to provide a few mathematical effects describing the transition from kinetic concept, and, extra accurately, from the Boltzmann equation for ideal gases to hydrodynamics.
Different fluid asymptotics might be investigated, beginning continuously from recommendations of the Boltzmann equation that are in basic terms assumed to fulfill the estimates coming from physics, particularly a few bounds on mass, strength and entropy.
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Extra resources for Hydrodynamic Limits of the Boltzmann Equation
14) associated with the Boltzmann equation, and the compressible Euler system. Indeed, as Kn → 0, solutions of the Boltzmann equation behave as local Maxwellians, namely f (t, x, v) ∼ R(t, x) |v − U (t, x)|2 exp − 2T (t, x) (2πT (t, x))3/2 for some R(t, x), T (t, x) > 0 and U (t, x) ∈ R3 . 19) which are the equations of hydrodynamics for perfect gases, satisfying in particular the state relation P = RT Id. That there is no excluded volume in this state relation is strongly linked with the Boltzmann-Grad scaling assumption N d3 << lo3 , which expresses the fact that the volume occupied by the particles is negligible compared with the volume of the domain.
E. on R+ ×Ω × R3 . 48) The idea is then to take limits in M (St∂t + v · ∇x ) log 1 + fδ M = − 1 Q+ δ − Qδ , Kn 1 + fδ /M fδ|t=0 = Γδ (fin ). e. 48). e. 49) 40 2 The Boltzmann Equation and its Formal Hydrodynamic Limits and thus in L1loc (R+ ×Ω × R3 ) by Lebesgue’s theorem. • The convergence of the gain term is more complicated to establish. e. 44) obtained in Step 2, and the Product Limit theorem. 44) and the Product Limit theorem, we establish that, for all λ > 0, Q+ (fn , fn ) 1 + λ fn∗ dv∗ Q+ (f, f ) weakly in L1loc (R+ ×Ω × R3 ).
8) is uniformly bounded in L2loc (dt, L2 (M (v · n(x))+ dσx dv)). Proof. Denoting by · ∂Ω g the average of any quantity defined on the boundary ∂Ω = √ M g|∂Ω 2π(v · n(x))+ dv, we get αn Ma2n Stn t h 0 ∂Ω fn|∂Ω − M M −h fn − M M √ dσx ds ≤ C 2π. ∂Ω ∂Ω By Taylor’s formula, fn − M fn − M −h M M ∂Ω ∂Ω 2 fn|∂Ω fn|∂Ω 1 1 fn fn = − h τ + (1 − τ ) 2 0 M M ∂Ω M M h −1 ∂Ω because the term of first order cancels fn|∂Ω fn − M M h ∂Ω fn M = 0. 9) Besides we have ηˆn = 2 Ma n αn 1Σ Stn + fn|∂Ω − M = 2 Ma n αn 1Σ Stn + fn|∂Ω fn − M M fn M ∂Ω ∂Ω fn|∂Ω + M fn M −1 ∂Ω and h τ fn|∂Ω fn + (1 − τ ) M M −1 = ∂Ω 1 τ fn|∂Ω /M + (1 − τ ) fn /M .