Uniform Stability and the Propagation of Regularity for the Relativistic Boltzmann Equation

Abstract

We present a priori uniform $L^q$-stability estimates of classical solutions to the relativistic Boltzmann equation. Our uniform stability analysis does not require any smallness in the amplitudes of solutions, but we need exponentially decaying far-field conditions on density functions in phase space. The uniform stability analysis follows directly from the integrable temporal decay of the collision frequencies along the particle trajectory. Our uniform stability results improve on the previous results [S. Y. Ha, E. Jeong, and R. M. Strain, Comm. Pure Appl. Anal., 12 (2013), pp. 1141--1161] from a uniform $L^1$-stability analysis of the relativistic Boltzmann equation, which relied crucially on the nonlinear functional approach and smallness in the amplitude of the distribution function.