# Hydrodynamic Limits of the Boltzmann Equation with Contact Discontinuities

Project: Research

## Description

In the gas and fluid dynamics, many famous equations of motion have been derived by focusing on different aspects of gas and fluid in different physical scales. In the macroscopic scales where gas and fluid are regarded as continua, the motion is described by the macroscopic quantities. And the Euler equations and Navier-Stokes equations are the most famous equations proposed so far in fluid dynamics.The extreme contrary is the microscopic scale where the gas and fluid are treated as a many-body system of microscopic particles (atom/molecule). Thus, the motion of the system is governed by the coupled Newton equations.Although the Newton equation is the first principle of the classical mechanics, it is not of practical use when the number of the equations is enormous (Avogadro number»6×1023). On the other hand, the macroscopic (fluid dynamical) quantities are related to the statistical average of quantities depending on the microscopic state. Thus, the kinetic theory that gives the mesoscopic description of the gas and fluid is a key theory that links the microscopic and macroscopic scales. The most classical and fundamental kinetic equation is the Boltzmann equation derived in 1872.The first derivation of fluid dynamics from the kinetic equations can be traced back to Maxwell and Boltzmann. Their early derivation rests on arguments as how the various terms in a kinetic equation balance each other. These balance arguments seem arbitrary to some extent. Hence, Hilbert proposed a systematic expansion in 1912, and Enskog and Chapman independently proposed another expansion in 1916/17.Either the Hilbert expansion or Chapman-Enskog expansion yields the compressible Euler equations in the leading order, and the compressible Navier-Stokes equations, Burnett equations in the subsequent orders. To justify these formal approximations in rigorous mathematics, that is, hydrodynamic limits, has been proven to be difficult, in part because many basic well-posedness and regularity questions are still mostly open for these fluid equations.Since there are three basic wave patterns for the Euler equations, that is, shock, rarefaction wave and contact discontinuity, in this project, we will investigate the hydrodynamic limits of the Boltzmann equation with contact discontinuities in several settings explained later.

## Detail(s)

Project number 9041547 GRF Finished 1/08/10 → 6/03/14