From the Boltzmann equation to the incompressible Navier-Stokes equations on the torus: a quantitative error estimate
type: Item
instance of: scholarly article
title: From the Boltzmann equation to the incompressible Navier-Stokes equations on the torus: a quantitative error estimate
zbMATH Open document ID: 1326.35220
published in: Journal of Differential Equations
publication date: 2015-09-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/1207.4379
review text: The author is interested in the Boltzmann equation as the Knudsen number tends to 0. After some manipulations, this leads to the equation \[ \partial_th_{\epsilon}+\frac{1}{\epsilon}\nu\cdot\nabla_xh_{\epsilon}=\frac{1}{\epsilon^2}L(h_{\epsilon})+\frac{1}{\epsilon}\Gamma(h_{\epsilon},h\epsilon), \tag{f} \] where \(L(h)\) and \(\Gamma(g,h)\) are certain functions. The purpose of this paper is to study the existence of solutions and exponential decay for ({f}). Thus the limit Knudsen number tends to 0 corresponds to \(\epsilon\to 0\). The main results are {\parindent=6mm \begin{itemize} \item[-] A proof by induction on \(s\), the order of Sobolev spaces, for the norm to the solution of ({f}). \item [-] A proof that the solution of an equation similar to the linearized ({f}) satisfies a Sobolev space bound. \item [-] An exponential decay theorem for the previous equation, again proved by induction on \(s\). \item [-] A proof of strong convergences for some variables, which shows that these variables solve the incompressible Navier-Stokes and Boussinesq equation. \end{itemize}} The proofs use Gronvall, Cauchy-Schwarz and Poincaré type inequalities. In an appendix, the assumptions are validated.
reviewed by: Thomas Ernst
Mathematics Subject Classification ID: 35Q20
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 76D05
zbMATH DE Number: 6484234
zbMATH Keywords: Boltzmann equation on the torus
zbMATH Keywords: explicit trend to equilibrium
zbMATH Keywords: incompressible Navier-Stokes hydrodynamical limit
zbMATH Keywords: Knudsen number
zbMATH Keywords: hypocoercivity
zbMATH Keywords: kinetic models
author: Marc Briant
MaRDI profile type: Publication
OpenAlex ID: W1579672942
arXiv ID: 1207.4379
cites work: THE BOLTZMANN EQUATION WITHOUT ANGULAR CUTOFF IN THE WHOLE SPACE: II, GLOBAL EXISTENCE FOR HARD POTENTIAL
cites work: Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials
cites work: THE CLASSICAL INCOMPRESSIBLE NAVIER-STOKES LIMIT OF THE BOLTZMANN EQUATION
cites work: Q4748596
cites work: Equilibration Rate for the Linear Inhomogeneous Relaxation-Time Boltzmann Equation for Charged Particles
cites work: The Boltzmann equation and its applications
cites work: The mathematical theory of dilute gases
cites work: The Boltzmann equation near Maxwellian in the whole space
cites work: Incompressible navier-stokes and euler limits of the boltzmann equation
cites work: On the Cauchy problem for Boltzmann equations: Global existence and weak stability
cites work: The first and second fluid approximation to the linearized Boltzmann equation
cites work: From Newton to Boltzmann: hard spheres and short-range potentials
cites work: The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels
cites work: The incompressible Navier-Stokes limit of the Boltzmann equation for hard cutoff potentials
cites work: Q5521434
cites work: The Landau equation in a periodic box
cites work: The Vlasov‐Poisson‐Boltzmann system near Maxwellians
cites work: Classical solutions to the Boltzmann equation for molecules with an angular cutoff
cites work: Boltzmann diffusive limit beyond the Navier‐Stokes approximation
cites work: Perturbation theory for linear operators.
cites work: Time evolution of large classical systems
cites work: From the Boltzmann equation to an incompressible Navier-Stokes-Fourier system
cites work: Q4893762
cites work: Vorticity and incompressible flow
cites work: Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids
cites work: Explicit Coercivity Estimates for the Linearized Boltzmann and Landau Operators
cites work: Q3629957
cites work: Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus
cites work: Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff
cites work: Convergence to Global Equilibrium for a Kinetic Fermion Model
cites work: Global solutions to the initial value problem for the nonlinear Boltzmann equation
cites work: Convergence to equilibria for compressible Navier-Stokes equations with large data
cites work: Hydrodynamic limits of the Boltzmann equation
cites work: On the interior regularity of weak solutions of the Navier-Stokes equations
cites work: Q5723392
cites work: Navier-Stokes equations and turbulence
cites work: On the existence of global solutions of mixed problem for non-linear Boltzmann equation
cites work: The incompressible limit and the initial layer of the compressible Euler equation
cites work: Q4422852
cites work: Q4409400
cites work: Global classical solutions of the Boltzmann equation near Maxwellians
DOI: 10.1016/J.JDE.2015.07.022
Recommended article: The Boltzmann equation with an external force on the torus: incompressible Navier-Stokes-Fourier hydrodynamical limit
Recommended article: From the Boltzmann equation to an incompressible Navier-Stokes-Fourier system
Recommended article: Incompressible navier-stokes and euler limits of the boltzmann equation
Recommended article: Q5432021
Recommended article: On the convergence of smooth solutions from Boltzmann to Navier-Stokes
Recommended article: On the Convergence from Boltzmann to Navier–Stokes–Fourier for General Initial Data
Recommended article: The incompressible Navier-Stokes limit of the Boltzmann equation for hard cutoff potentials
Recommended article: Compressible Navier-Stokes approximation for the Boltzmann equation in bounded domains
Recommended article: On the Navier–Stokes limit of the Boltzmann equation
Recommended article: The incompressible Navier-Stokes-Fourier limit from Boltzmann-Fermi-Dirac equation
description: scientific article