Acoustic limit for the Boltzmann equation in optimal scaling

Abstract

Based on a recent L 2 -L 1 framework, we establish the acoustic limit of the Boltzmann equation for general collision kernels. The scaling of the fluctuations with respect to the Knudsen number is optimal. Our approach is based on a new analysis of the compressible Euler limit of the Boltzmann equation, as well as refined estimates of Euler and acoustic solutions.

Incompressible hydrodynamic approximation with viscous heating to the Boltzmann equation

“…Recently, a new model called the INSF system with viscous heating was derived by Bardos-Levermore-Ukai-Yang [4]. The aim of the present paper is to justify such an incompressible hydrodynamic approximation to the Boltzmann equation in a periodic box via an L 2 − L ∞ method developed in [12,13,21,22,23,24]. We now outline a few key points of the paper which are distinct to some extent with the previous work by Bardos-Levermore-Ukai-Yang [4]:…”

Diffusive limit for a Boltzmann-like equation with non-conserved momentum

“…The other terms which have to be dealt with contain a polynomial of degree three in v which gives troubles for large velocities. To this end, following [10], we introduce a global Maxwellian µ T with temperature given by the min T (x, t) assumed strictly positive and bound the high velocity tail of f in terms of the L ∞ norm of…”

“…As we shall see, the spectral properties are crucial in our proof, so we need to deal with this extra term. We will follow the approach in [10] and [7]. Instead of repeating all the proofs in these papers, we only outline and give explicit proofs when the previous approach has to be modified to adapt to the case we study.…”

Global Hilbert Expansion for the Vlasov-Poisson-Boltzmann System

“…We only requrie k ≥ 3 for the expansion in (1.7). This requirement is the same as the case of the Hilbert expansion for the Boltzmann equation [22,23,32] and more relaxed than that for the Vlasov-Poisson-Boltzmann system [21]. Moreover, our uniform estimates lead to the relativistic Euler-Maxwell limit:…”

“…Similar result was also obtained in [21] for the Hilbert expansion of the Vlasov-Poisson-Boltzmann system. Due to shock formations in the pure compressible Euler flow, as illustrated in [31,5], corresponding Hilbert expansion, acoustic limit for the Boltzmann equation or relativistic Boltzmann equation [4,22,23,32] is only valid local in time. Different from the pure compressible Euler fluids where shock waves may develop even for smooth irrotational initial data with small amplitude, the electromagnetic interaction in the two-fluid models [16,19,20] could create stronger dispersive effects, enhance linear decay rates, and prevent formation of shock waves with small amplitude.…”