Scaling-Critical Theory for the Boltzmann and Landau Equations

Abstract

For sufficiently small initial perturbations in a localized, weighted, anisotropic Riesz-potential norm, we prove global well-posedness near a Maxwellian in the whole space. This critical phase-space norm captures the Boltzmann--Landau scaling, the velocity-dependent anisotropy, the hypoelliptic transport structure, and the nonnegativity constraint. The proof combines frozen-operator estimates, a critical fixed-point argument, and weighted hypocoercive energy estimates. We also develop a short-time pointwise Green-function theory for variable-coefficient kinetic equations with a nonnegative H"older background. We first construct the small-jump Green function by freezing coefficients along kinetic characteristics and then recover the full kernel through a convergent parametrix expansion. The resulting bounds capture the fractional Kolmogorov geometry near characteristics and rapid decay away from them, providing the analytic foundation for the scaling-critical theory.

This version contains substantial revisions. In particular, we have added a short-time pointwise Green-function theory for variable-coefficient kinetic equations and revised the title to reflect this new contribution. Comments are welcome