Local Existence and Finite-Time Singularity Formation in the Vlasov-Poisson-Isotropic Landau System

TL;DR

This paper establishes local existence and explores conditions under which finite-time singularities can form in the Vlasov-Poisson-Isotropic Landau system, highlighting the role of initial energy and regularity assumptions.

Contribution

It provides the first local existence theory for the system and identifies a mechanism for finite-time singularity formation under specific energy and regularity conditions.

Findings

Local existence of solutions under small initial data

Finite-time singularity can occur when gravitational energy exceeds kinetic energy

Singularity formation is demonstrated via bounds on the second spatial moment

Abstract

The isotropic Landau (Coulomb) operator was introduced in kinetic theory by Krieger and Strain (Comm. Partial Differential Equations, 2012). In this work, we study the spatially inhomogeneous Vlasov--Poisson--isotropic Landau system. We first establish a local--in--time existence theory for the Cauchy problem: for initial data satisfying a suitable smallness condition in an appropriate norm, there exists a non--negative solution on a time interval $[0,T]$, where the lifespan $T$ depends on the size of the initial data. Beyond the local theory, we investigate a mechanism that may lead to the breakdown of global existence. We show that finite--time singularity formation can occur in the gravitationally attractive case, provided that the weak solution satisfies certain a priori regularity and decay assumptions, the initial gravitational field energy exceeds the kinetic energy, and the resulting energy gap dominates the diffusive effect of the collision operator. As a consequence, if the solution is further assumed to belong to a suitable measure space up to the maximal existence time, it collapses to a single point in physical space at that time. The proof of finite--time singularity formation is based on deriving an upper bound for the second spatial moment, which becomes negative in finite time.