Premium
Stokes‐Fourier and acoustic limits for the Boltzmann equation: Convergence proofs
Author(s) -
Golse François,
Levermore C. David
Publication year - 2002
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3011
Subject(s) - mathematics , knudsen number , boltzmann equation , mathematical analysis , fourier transform , limit (mathematics) , scaling , heat equation , collision , initial value problem , physics , mechanics , geometry , computer security , quantum mechanics , computer science
Abstract We establish a Stokes‐Fourier limit for the Boltzmann equation considered over any periodic spatial domain of dimension two or more. Appropriately scaled families of DiPerna‐Lions renormalized solutions are shown to have fluctuations that globally in time converge weakly to a unique limit governed by a solution of Stokes‐Fourier motion and heat equations provided that the fluid moments of their initial fluctuations converge to appropriate L 2 initial data of the Stokes‐Fourier equations. Both the motion and heat equations are both recovered in the limit by controlling the fluxes and the local conservation defects of the DiPerna‐Lions solutions with dissipation rate estimates. The scaling of the fluctuations with respect to Knudsen number is essentially optimal. The assumptions on the collision kernel are little more than those required for the DiPerna‐Lions theory and that the viscosity and heat conduction are finite. For the acoustic limit, these techniques also remove restrictions to bounded collision kernels and improve the scaling of the fluctuations. Both weak limits become strong when the initial fluctuations converge entropically to appropriate L 2 initial data. © 2001 John Wiley & Sons, Inc.