On the transport operators arising from linearizing the Vlasov-Poisson or Einstein-Vlasov system about isotropic steady states

TL;DR

This paper investigates the properties of a transport operator derived from linearizing the Vlasov-Poisson and Einstein-Vlasov systems around isotropic steady states, establishing skew-adjointness and kernel characteristics crucial for stability analysis.

Contribution

It proves that the transport operator is skew-adjoint on a suitable Hilbert space and determines its kernel in the Vlasov-Poisson case, advancing stability theory.

Findings

Transport operator is skew-adjoint on a proper domain.

Kernel of the operator identified in Vlasov-Poisson case.

Results are relevant for stability analysis of steady states.

Abstract

If the Vlasov-Poisson or Einstein-Vlasov system is linearized about an isotropic steady state, a linear operator arises the properties of which are relevant in the linear as well as nonlinear stability analysis of the given steady state. We prove that when defined on a suitable Hilbert space and equipped with the proper domain of definition this transport operator $\mathcal{T}$ is skew-adjoint, i.e., $\mathcal{T}^{\ast}=-\mathcal{T}$. In the Vlasov-Poisson case we also determine the kernel of this operator.