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Hydrodynamic limits with shock waves of the Boltzmann equation
Author(s) -
Yu ShiHsien
Publication year - 2005
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.20027
Subject(s) - boltzmann equation , knudsen number , lattice boltzmann methods , mathematics , boltzmann constant , bhatnagar–gross–krook operator , direct simulation monte carlo , shock wave , mathematical analysis , euler's formula , shock (circulatory) , euler equations , boltzmann relation , statistical physics , mechanics , physics , thermodynamics , medicine , statistics , dynamic monte carlo method , monte carlo method
We show that piecewise smooth solutions with shocks of the Euler equations in gas dynamics can be obtained as the zero Knudsen number limit of solutions of the Boltzmann equation for hard sphere collision model. The construction of the Boltzmann solutions is done in two steps. First we introduce a generalized Hilbert expansion with shock layer correction to construct approximations to the solutions of the Boltzmann equations with small Knudsen numbers. We then apply the recently developed macro‐micro decomposition and energy method for Boltzmann shock layers to construct the exact Boltzmann solutions through the stability analysis. © 2004 Wiley Periodicals, Inc.