The Initial Value Problem of the Boltzmann Equation and Its Fluid Dynamical Limit at the Level of Compressible Euler Equation

Abstract

This chapter discusses the initial value problem of Boltzmann equation and its fluid dynamical limit at the level of compressible Euler equation. The initial value problem was investigated locally in time by Grad and globally in time by Ukai, Nishida and Imai, and then Shizuta. If the initial deviation from the absolute Maxwellian distribution is small with ɛ fixed, then the solution is proved to exist uniquely in the large in time and to converge to the absolute Maxwellian as time goes to infinity. The spectral theory for the linear Boltzmann equation gives the decay estimates on solutions of the linear Boltzmann equation; an arguement of Grad gives solutions to the full Boltzmann equation as a nonlinear perturbation from the linear equation. The initial-boundary value problems in the bounded domains are solved globally in time by Guiraud for the boundary condition of random reflection and by Shizuta and Asano for the specular boundary condition.