Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators

TL;DR

This paper establishes a precise mathematical relationship between the linearized non-cutoff Boltzmann and Landau operators, showing the Boltzmann operator as a fractional power of the Landau operator, leading to explicit coercive estimates.

Contribution

It proves that the non-cutoff Boltzmann operator equals a fractional power of the Landau operator with explicit spectral properties, clarifying their connection.

Findings

Boltzmann operator is a fractional power of the Landau operator.

Derived explicit sharp coercive estimates for the Boltzmann operator.

Established the exact equivalence for Maxwellian molecules.

Abstract

In many works, the linearized non-cutoff Boltzmann operator is considered to behave essentially as a fractional Laplacian. In the present work, we prove that the linearized non-cutoff Boltzmann operator with Maxwellian molecules is exactly equal to a fractional power of the linearized Landau operator which is the sum of the harmonic oscillator and the spherical Laplacian. This result allows to display explicit sharp coercive estimates satisfied by the linearized non-cutoff Boltzmann operator for both Maxwellian and non-Maxwellian molecules.