Abstract
Using a weighted $H^s$-contraction mapping argument based on themacro-micro decomposition of Liu and Yu, we give an elementary proofof existence, with sharp rates of decay and distance from theChapman-Enskog approximation, of small-amplitude shock profiles ofthe Boltzmann equation with hard-sphere potential, recovering andslightly sharpening results obtained by Caflisch and Nicolaenkousing different techniques. A key technical point in both analysesis that the linearized collision operator $L$ is negative definiteon its range, not only in the standard square-root Maxwellianweighted norm for which it is self-adjoint, but also in norms withnearby weights. Exploring this issue further, we show that $L$ isnegative definite on its range in a much wider class of normsincluding norms with weights asymptotic nearly to a full Maxwellianrather than its square root. This yields sharp localization invelocity at near-Maxwellian rate, rather than the square-root rateobtained in previous analyses.
Highlights
In this paper, we study existence and structure of small-amplitude shock profiles (1.1)f (x, ξ, t) = f(x − st, ξ), lim z→±∞ f(z) = f±of the one-dimensional Boltzman equation (1.2)ft + ξ1∂xf = τ −1Q(f, f ), x, t ∈ R, where f (x, t, ξ) ∈ R denotes the distribution of velocities ξ ∈ R3 at point x, t, τ > 0 is the Knudsen number, and (1.3)Q(g, h) := g(ξ′)h(ξ∗′ ) − g(ξ)h(ξ∗) C(Ω, ξ − ξ∗)dΩdξ∗is the collision operator, with (1.4)ξ ∈ R3, ξ∗ ∈ R3, Ω ∈ S2, ξ′ = ξ + Ω · (ξ∗ − ξ) Ω ξ∗′ = ξ∗ − Ω · (ξ∗ − ξ) Ω.and various collision kernels C
Taking moments of (1.2) and applying definition (1.8), we find that the fluid variables obey the one-dimensional Euler equations ρt + ∂x(ρv1) = 0t + ∂x(v1ρv + pe1) = 0t + ∂x(v1(ρE + p)) = 0, e1 = (1, 0, 0)T the first standard basis element, where the new variable p = p(f ), denoting pressure, depends in general on higher, non-fluid-dynamical moments of f
Noting that endstates f± of (1.1) by (1.6) necessarily satisfy Q(f, f )± = 0, we find that they are Maxwellians f± = Mu±, and so the associated pressures p± = p(f±) are given by the ideal gas formula (1.11), recovering the standard fact that endstates of a Boltzmann shock (1.1) are Maxwellians with fluid-dynamical variables corresponding to fluid-dynamical shock waves of the Euler equations with monatomic ideal gas equation of state [G, CN]
Summary
We study existence and structure of small-amplitude shock profiles (1.1). For the hard sphere potential, positivity of profiles, and the improved estimate (1.18) were shown by Liu and Yu [LY] by a “macromicro decomposition” method in which fluid (macroscopic, or equilibrium) and transient (microscopic) effects are separated and estimated by different techniques This was used in [LY] to establish time-evolutionary stability of profiles with respect to perturbations of zero fluid-dynamical mass, u(x)dx = 0, and assuming the existence result of [CN], to establish positivity of Boltzmann profiles by the positive maximum principle for the Boltzmann equation (1.2) together with convergence to the Boltzmann profile of its own Maxwellian approximation: by definition, a perturbation of zero relative mass in fluiddynamical variables. It would be very interesting to continue along the same lines to obtain a complete nonlinear stability result as in [SX] or [MaZ1, MaZ2], with respect to general, not necessarily zero mass, perturbations
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