The relativistic Boltzmann equation for soft potentials

The paper concerns the Cauchy problem on the relativistic Boltzmann equation for soft potentials in a periodic box. We show that the global-in-time solutions around relativistic Maxwellians exist in the weighted L∞ perturbation framework and also approach equilibrium states in large time in the weighted L2 framework at the rate of exp⁡(−λtβ) for some λ>0 and β∈(0,1). The proof is based on the nonlinear L2 energy method and nonlinear L∞ pointwise estimate with appropriate exponential weights in momentum. The results extend those on the classical Boltzmann equation by Caflisch [2,3] and Strain and Guo [31] to the relativistic version, and also improve the recent result on almost exponential time-decay by Strain [28] to the exponential rate. Moreover, we study the propagation of spatial regularity for the obtained solutions and also the large time behavior in the corresponding regular Sobolev space, provided that the spatial derivatives of initial data are bounded, not necessarily small.