Kinetic equations: Fluid dynamical limits and viscous heating
by Claude Bardos   C. David Levermore   Seiji Ukai   Tong Yang  

Vol. 3 No. 1 (2008) P.1~P.49

ABSTRACT

In the long-time scale, we consider the fluid dynamical limits for the kinetic equations when the fluctuation is decomposed into even and odd parts with respect to the microscopic velocity with different scalings. It is shown that when the background state is an absolute Maxwellian, the limit fluid dynamical equations are the incompressible Navier-Stokes equations with viscous heating. This is different from the case when the even and odd parts of the fluctuation have the same scaling where the standard incompressible Navier-Stokes equations without viscous heating are obtained. On the other hand, when the background is a local Maxwellian, it is shown that the above even-odd decomposition leads to a non-classical fluid dynamical system without viscous heating which has been used to describe the ghost effect in the kinetic theory. In addition, the above even-odd decomposition is justified rigorously for the Boltzmann equation for the former case when the background is an absolute Maxwellian.

KEYWORDS
Kinetic equation, Boltzmann equation, fluid dynamical limits, Navier-Stokes limit, viscous heating, Ghost-effect, macro-micro decomposition, even-odd decomposition

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 34E05, 34E10, 35B40, 35Q30, 76D03, 76P05, 82B40, 84C40

MILESTONES

Received: 2007-08-01
Revised :
Accepted:

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