Hilbert’s sixth problem and the failure of the Boltzmann to Euler limit

DOI10.1098/rsta.2017.0222zbMath1470.76017OpenAlexW2791389036WikidataQ52382218 ScholiaQ52382218MaRDI QIDQ5154190

Marshall Slemrod

Publication date: 4 October 2021

Published in: Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1098/rsta.2017.0222

zbMATH Keywords

Navier-Stokes equationsBoltzmann equationKorteweg equationsfluid mechanicskinetic equationscapillarity

Mathematics Subject Classification ID

Research exposition (monographs, survey articles) pertaining to statistical mechanics (82-02) Kinetic theory of gases in time-dependent statistical mechanics (82C40) Research exposition (monographs, survey articles) pertaining to fluid mechanics (76-02) Foundations of fluid mechanics (76A02)

Related Items

From Newton to Navier–Stokes, or how to connect fluid mechanics equations from microscopic to macroscopic scales ⋮ Linear hydrodynamics and stability of the discrete velocity Boltzmann equations ⋮ Derivation of regularized Grad's moment system from kinetic equations: modes, ghosts and non-Markov fluxes ⋮ Hilbert’s sixth problem: the endless road to rigour ⋮ Conservation laws in biology: Two new applications

Cites Work

• Unnamed Item
• Unnamed Item
• The Euler limit and initial layer of the nonlinear Boltzmann equation
• Instabilities in the Chapman-Enskog expansion and hyperbolic Burnett equations
• Macroscopic transport equations for rarefied gas flows. Approximation methods in kinetic theory.
• From Boltzmann's kinetic theory to Euler's equations
• Generalized Burnett hydrodynamics
• Hydrodynamic limits of the Boltzmann equation
• On the thermomechanics of interstitial working
• Dynamic phase transitions in a van der Waals fluid
• The Boltzmann equation and its applications
• Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation
• Admissibility criteria for propagating phase boundaries in a van der Waals fluid
• Thermal creep of a rarefied gas on the basis of non-linear Korteweg-theory
• Macroscopic description of steady and unsteady rarefaction effects in boundary value problems of gas dynamics
• Boundary singularity for thermal transpiration problem of the linearized Boltzmann equation
• From Boltzmann to Euler: Hilbert's 6th problem revisited
• Justification of limit for the Boltzmann equation related to Korteweg theory
• Chapman-Enskog $⇒$ viscosity-capillarity
• Macroscopic transport models for rarefied gas flows: a brief review
• The small dispersion limit of the Korteweg-de Vries equation. I
• The Problem with Hilbert’s 6th Problem
• Numerical analysis of the shear and thermal creep flows of a rarefied gas over a plane wall on the basis of the linearized Boltzmann equation for hard-sphere molecules
• Numerical analysis of the Poiseuille and thermal transpiration flows between two parallel plates on the basis of the Boltzmann equation for hard-sphere molecules
• Convergence of solution to nonlinear dispersive equations
• Structure and approximations of the chapman-enskog expansion for the linearized grad equations
• Hydrodynamics from Grad's equations: What can we learn from exact solutions?
• Free Energy of a Nonuniform System. I. Interfacial Free Energy
• Singular behaviour of a rarefied gas on a planar boundary
• Hilbert’s 6th Problem: exact and approximate hydrodynamic manifolds for kinetic equations
• A mathematical PDE perspective on the Chapman–Enskog expansion
• Admissibility of weak solutions for the compressible Euler equations, n ≥ 2
• On the kinetic theory of rarefied gases