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arXiv · 2509.14845

Scaling-Critical Theory for the Boltzmann and Landau Equations

Ke Chen·Quoc-Hung Nguyen·Tong Yang
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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.

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Ke Chen, Quoc-Hung Nguyen, Tong Yang. 2025-09-18. Scaling-Critical Theory for the Boltzmann and Landau Equations. https://arxiv.org/abs/2509.14845

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