2013/10/03 by A. N. Gorban, Alexander Gorban, I. Karlin +1 · 3 citations
Mathematics · Physics and Astronomy · #Boltzmann constant #Boltzmann equation #Boltzmann's entropy formula #Dissipative system #Gas Dynamics and Kinetic Theory #Invariant (physics) #Invariant manifold #Lattice Boltzmann methods #Limit (mathematics) #Manifold (fluid mechanics) #Nonlinear Waves and Solitons #Statistical Mechanics and Entropy #math-ph #math.MP
paper · pdf · doi:10.1090/s0273-0979-2013-01439-3
published as Bull. Amer. Math. Soc., 51(2), 2014, 186-246, Posted online: November 20, 2013 · 58 pages, 8 Figures (v2: Technical improvement of eps files for better compatibility)
arxiv created 2013/10/03 · openalex publication_date 2013/11/20 · arxiv updated 2014/06/05 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
The problem of the derivation of hydrodynamics from the Boltzmann equation and related dissipative systems is formulated as the problem of a slow invariant manifold in the space of distributions. We review a few instances where such hydrodynamic manifolds were found analytically both as the result of summation of the Chapman–Enskog asymptotic expansion and by the direct solution of the invariance equation. These model cases, comprising Grad’s moment systems, both linear and nonlinear, are studied in depth in order to gain understanding of what can be expected for the Boltzmann equation. Particularly, the dispersive dominance and saturation of dissipation rate of the exact hydrodynamics in the short-wave limit and the viscosity modification at high divergence of the flow velocity are indicated as severe obstacles to the resolution of Hilbert’s 6th Problem. Furthermore, we review the derivation of the approximate hydrodynamic manifold for the Boltzmann equation using Newton’s iteration and avoiding smallness parameters, and compare this to the exact solutions. Additionally, we discuss the problem of projection of the Boltzmann equation onto the approximate hydrodynamic invariant manifold using entropy concepts. Finally, a set of hypotheses is put forward where we describe open questions and set a horizon for what can be derived exactly or proven about the hydrodynamic manifolds for the Boltzmann equation in the future.