Stationary solutions to the Boltzmann equation in the Hydrodynamic limit

Abstract

Despite its conceptual and practical importance, the rigorous derivation of the steady incompressible Navier-Stokes-Fourier system from the Boltzmann theory has been {an} outstanding {open problem} for general domains in 3D. We settle this open question in {the} affirmative, in the presence of a small external field and a small boundary temperature variation for the diffuse boundary condition. We employ a recent quantitative $L^2-L^{\infty }$ approach with new $L^6$ estimates for the hydrodynamic part $\mathbf{P}f$ of the distribution function. Our results also imply the validity of Fourier law in the hydrodynamical limit, and our method {leads to {asymptotical} stability of steady Boltzmann solutions as well as the derivation of the {unsteady} Navier-Stokes Fourier system}.

New $L^6$ bound of the hydrodynamic part is added, which allows direct convergence to Navier-Stokes-Fourier system in Theorem 1.1 and Theorem 1.3

References in corpus (2)

• The Boltzmann equation with specular boundary condition in convex domains
• Hilbert Expansion from the Boltzmann equation to relativistic Fluids

Cited by in corpus (34)

• Global mild solutions of the Landau and non-cutoff Boltzmann equations
• Global strong solutions of the Vlasov-Poisson-Boltzmann system in bounded domains
• Effects of soft interaction and non-isothermal boundary upon long-time dynamics of rarefied gas
• Hilbert expansion of the Boltzmann equation with specular boundary condition in half-space
• Hydrodynamic Limit of a Kinetic Gas Flow Past an Obstacle
• The initial boundary value problem for the Boltzmann equation with soft potential
• Decay of the Boltzmann equation with the specular boundary condition in non-convex cylindrical domains
• Local Well-posedness of Vlasov-Poisson-Boltzmann Equation with Generalized Diffuse Boundary Condition
• A Regularity Result for the Incompressible Magnetohydrodynamics Equations with Free Surface Boundary
• Phase transitions and macroscopic limits in a BGK model of body-attitude coordination
• The large amplitude solution of the Boltzmann equation with soft potential
• Asymptotic Analysis of Unsteady Neutron Transport Equation
• Small Knudsen rate of convergence to rarefaction wave for the Landau equation
• Boundary Layer of Boltzmann Equation in 2D Convex Domains
• Justification of Diffusion limit for the Boltzmann Equation with a non-trivial Profile
• Stationary quantum BGK model for bosons and fermions in a bounded interval
• Heat transfer problem for the Boltzmann equation in a channel with diffusive boundary condition
• Diffusive limit for a Boltzmann-like equation with non-conserved momentum
• Stationary BGK models for chemically reacting gas in a slab
• Strong diffusive limit of the Boltzmann equation with Maxwell boundary condition
• Regularity of Boltzmann equation with external fields in convex domains of diffuse reflection
• Compressible Navier-Stokes approximation for the Boltzmann equation in bounded domains
• The Boltzmann equation with large-amplitude initial data and specular reflection boundary condition
• A note on two species collisional plasma in bounded domains
• On smooth solutions to the thermostated Boltzmann equation with deformation
• Solutions to a moving boundary problem on the Boltzmann equation
• Stationary Flows of the ES-BGK model with the correct Prandtl number
• The Boltzmann equation with time-periodic boundary temperature
• Boltzmann-Grad limit of a hard sphere system in a box with diffusive boundary conditions
• Boundary conditions and polymeric drag reduction for the Navier-Stokes equations
• Rarefied gas dynamics with external fields under specular reflection boundary condition
• Hydrodynamic limit of the Boltzmann equation to the planar rarefaction wave in three dimensional space
• The Boltzmann equation with an external force on the torus: Incompressible Navier-Stokes-Fourier hydrodynamical limit
• Incompressible hydrodynamic approximation with viscous heating to the Boltzmann equation