Hypoelliptic and spectral estimates for the linearized Landau operator

TL;DR

This paper establishes hypoelliptic estimates, spectral localization, and resolvent bounds for the linearized Landau operator with hard potentials and Maxwellian molecules, using advanced pseudo-differential calculus techniques.

Contribution

It provides new spectral and hypoelliptic estimates for the linearized Landau operator, advancing understanding of its mathematical properties near Maxwellian equilibrium.

Findings

Spectral localization of the linear Landau operator.

Hypoelliptic regularity estimates for solutions.

Bounds on the resolvent operator in relevant function spaces.

Abstract

We are interested in the inhomogeneous Landau equation which describes the evolution of a particle density f = f (t, x, v) representing at time t $\ge$ 0, the density of particles at position x $\in$ R 3 and velocity v $\in$ R 3. The study is motivated by the linearization of the Landau equation near Maxwellian distribution. In this article, we establish hypoelliptic estimates, a localization of the spectrum and estimates of the resolvent of the linear Landau operator with hard potentials and Maxwellian molecules. The proof is based on a multiplier method and requires fine pseudo-differential calculus tools.