The Euler Limit of the Relativistic Boltzmann Equation

In this work we prove the existence and uniqueness theorems of the solutions to the relativistic Boltzmann equation for analytic initial fluctuations on a time interval independent of the Knudsen number ǫ > 0. As ǫ → 0, we prove that the solution of the relativistic Boltzmann equation tends to the local relativistic Maxwellian, whose fluid-dynamical parameters solve the relativistic Euler equations and the convergence rate is also obtained. Due to this convergence rate, the Hilbert expansion is verified in the short time interval for the relativistic Boltzmann equation. We also consider the physically important initial layer problem. As a by-product, an existence theorem for the relativistic Euler equations without the assumption of the non-vacuum fluid states is obtained. AMS subject classifications: 35Q99, 82A47