Boundary Layers and Incompressible Navier-Stokes-Fourier Limit of the Boltzmann Equation in Bounded Domain I

Ning Jiang, Nader Masmoudi

Research output: Contribution to journalArticle

Abstract

We establish the incompressible Navier-Stokes-Fourier limit for solutions to the Boltzmann equation with a general cutoff collision kernel in a bounded domain. Appropriately scaled families of DiPerna-Lions(-Mischler) renormalized solutions with Maxwell reflection boundary conditions are shown to have fluctuations that converge as the Knudsen number goes to 0. Every limit point is a weak solution to the Navier-Stokes-Fourier system with different types of boundary conditions depending on the ratio between the accommodation coefficient and the Knudsen number. The main new result of the paper is that this convergence is strong in the case of the Dirichlet boundary condition. Indeed, we prove that the acoustic waves are damped immediately; namely, they are damped in a boundary layer in time. This damping is due to the presence of viscous and kinetic boundary layers in space. As a consequence, we also justify the first correction to the infinitesimal Maxwellian that one obtains from the Chapman-Enskog expansion with Navier-Stokes scaling. This extends the work of Golse and Saint-Raymond 20, 21 and Levermore and Masmoudi 28 to the case of a bounded domain. The case of a bounded domain was considered by Masmoudi and Saint-Raymond 34 for the linear Stokes-Fourier limit and Saint-Raymond 41 for Navier-Stokes limit for hard potential kernels. Neither 34 nor 41 studied the damping of the acoustic waves. This paper extends the result of 34,41 to the nonlinear case and includes soft potential kernels. More importantly, for the Dirichlet boundary condition, this work strengthens the convergence so as to make the boundary layer visible. This answers an open problem proposed by Ukai 46.

Original languageEnglish (US)
JournalCommunications on Pure and Applied Mathematics
DOIs
StateAccepted/In press - 2016

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Incompressible Navier-Stokes
Boltzmann equation
Navier-Stokes
Boltzmann Equation
Boundary Layer
Bounded Domain
Knudsen number
Boundary layers
Boundary conditions
Acoustic Waves
kernel
Damped
Dirichlet Boundary Conditions
Damping
Chapman-Enskog Expansion
Renormalized Solutions
Limit Point
Acoustic waves
Stokes
Justify

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

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title = "Boundary Layers and Incompressible Navier-Stokes-Fourier Limit of the Boltzmann Equation in Bounded Domain I",
abstract = "We establish the incompressible Navier-Stokes-Fourier limit for solutions to the Boltzmann equation with a general cutoff collision kernel in a bounded domain. Appropriately scaled families of DiPerna-Lions(-Mischler) renormalized solutions with Maxwell reflection boundary conditions are shown to have fluctuations that converge as the Knudsen number goes to 0. Every limit point is a weak solution to the Navier-Stokes-Fourier system with different types of boundary conditions depending on the ratio between the accommodation coefficient and the Knudsen number. The main new result of the paper is that this convergence is strong in the case of the Dirichlet boundary condition. Indeed, we prove that the acoustic waves are damped immediately; namely, they are damped in a boundary layer in time. This damping is due to the presence of viscous and kinetic boundary layers in space. As a consequence, we also justify the first correction to the infinitesimal Maxwellian that one obtains from the Chapman-Enskog expansion with Navier-Stokes scaling. This extends the work of Golse and Saint-Raymond 20, 21 and Levermore and Masmoudi 28 to the case of a bounded domain. The case of a bounded domain was considered by Masmoudi and Saint-Raymond 34 for the linear Stokes-Fourier limit and Saint-Raymond 41 for Navier-Stokes limit for hard potential kernels. Neither 34 nor 41 studied the damping of the acoustic waves. This paper extends the result of 34,41 to the nonlinear case and includes soft potential kernels. More importantly, for the Dirichlet boundary condition, this work strengthens the convergence so as to make the boundary layer visible. This answers an open problem proposed by Ukai 46.",
author = "Ning Jiang and Nader Masmoudi",
year = "2016",
doi = "10.1002/cpa.21631",
language = "English (US)",
journal = "Communications on Pure and Applied Mathematics",
issn = "0010-3640",
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AU - Jiang, Ning

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N2 - We establish the incompressible Navier-Stokes-Fourier limit for solutions to the Boltzmann equation with a general cutoff collision kernel in a bounded domain. Appropriately scaled families of DiPerna-Lions(-Mischler) renormalized solutions with Maxwell reflection boundary conditions are shown to have fluctuations that converge as the Knudsen number goes to 0. Every limit point is a weak solution to the Navier-Stokes-Fourier system with different types of boundary conditions depending on the ratio between the accommodation coefficient and the Knudsen number. The main new result of the paper is that this convergence is strong in the case of the Dirichlet boundary condition. Indeed, we prove that the acoustic waves are damped immediately; namely, they are damped in a boundary layer in time. This damping is due to the presence of viscous and kinetic boundary layers in space. As a consequence, we also justify the first correction to the infinitesimal Maxwellian that one obtains from the Chapman-Enskog expansion with Navier-Stokes scaling. This extends the work of Golse and Saint-Raymond 20, 21 and Levermore and Masmoudi 28 to the case of a bounded domain. The case of a bounded domain was considered by Masmoudi and Saint-Raymond 34 for the linear Stokes-Fourier limit and Saint-Raymond 41 for Navier-Stokes limit for hard potential kernels. Neither 34 nor 41 studied the damping of the acoustic waves. This paper extends the result of 34,41 to the nonlinear case and includes soft potential kernels. More importantly, for the Dirichlet boundary condition, this work strengthens the convergence so as to make the boundary layer visible. This answers an open problem proposed by Ukai 46.

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