Time-asymptotic expansion and fluid approximation of the 1D Boltzmann equation with hard potentials
We establish a quantitative large-time comparison between the macroscopic Boltzmann solution and the full nonlinear compressible Navier-Stokes solution in one space dimension. For Maxwellian molecules and hard potentials with angular cutoff, we consider small, spatially compactly supported perturbations of a global Maxwellian with polynomial velocity decay, without assuming spatial differentiability. Nonlinear diffusion waves persist at leading order, making comparison through linearized equations alone insufficient. We construct a coupled system for the leading nonlinear fluid waves and derive cancellation identities from the exact kinetic spectral representation to control interactions between distinct characteristic families. Combined with a particle-fluid decomposition, these yield pointwise estimates separating the leading waves, their interactions, and a faster-decaying remainder.