The compressible Euler system and initial layers from Boltzmann equation in soft potential case

This paper investigates the hydrodynamic limit of the Boltzmann equation under soft potential conditions, where $ -3<\gamma <0 $, when the initial data approaches a local Maxwellian in the torus. Unlike the hard potential cases in [17], the initial layer terms do not decay exponentially due to the absence of a positive lower bound on collision frequency. To overcome this difficulty, this paper establishes some new time decay rates for the initial layer terms in weighted norms through an iterative process, which are sufficient to generate estimates of the remaining term.