On the axially symmetric solutions to the spatially homogeneous Landau equation

TL;DR

This paper studies axially symmetric solutions to the spatially homogeneous Landau equation, proving existence of measure-valued solutions and instant analyticity for certain initial data, with results varying across potential types.

Contribution

It establishes the existence of axisymmetric measure-valued solutions for the Landau equation with hard potentials and shows instant analyticity if initial data is not a Dirac mass.

Findings

Existence of axisymmetric measure-valued solutions for hard potentials.

Instantaneous analyticity for non-Dirac initial data in the hard potential case.

Non-existence of solutions supported on a fixed line for soft potentials and Maxwellian molecules.

Abstract

In this paper, we consider the spatially homogeneous Landau equation, which is a variation of the Boltzmann equation in the grazing collision limit. For the Landau equation for hard potentials in the style of Desvillettes-Villani (Comm. Partial Differential Equations, 2000), we provide the proof of the existence of axisymmetric measure-valued solution for any axisymmetric $\mathcal{P}_p(\mathbb{R}^3)$ initial profile for any $p\ge 2$. Moreover, we prove that if the initial data is not a single Dirac mass, then the solution instantaneously becomes analytic for any time $t>0$ in the hard potential case. In the soft potential and the Maxwellian molecule cases, we show that there are no solutions whose support is contained in a fixed line even for any given line-concentrated data.