Well-posedness of the Boltzmann and Landau Equations in Critical Spaces

Abstract

This paper investigates the well-posedness of the inhomogeneous Boltzmann and Landau equations in critical function spaces, a fundamental open problem in kinetic theory. We develop a new analytical framework to establish local well-posedness near a global Maxwellian for both equations, under the assumption that the initial perturbation is small in a critical norm. A major contribution lies in the introduction of a novel anisotropic norm adapted to the intrinsic scaling invariance of the equations, which provides precise control over the high-frequency behavior of solutions. By leveraging the regularizing effect and a decomposition of the linearized collision operator, we further extend the local solution globally in time and establish pointwise decay estimates. Our work not only resolves a fundamental issue in the theory of kinetic equations in critical spaces, but also provides a new approach applicable to a broader class of kinetic models.