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

Gerhard Rein; Christopher Straub

doi:10.3934/krm.2020032

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2020, Volume 13, Issue 5: 933-949. Doi: 10.3934/krm.2020032

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

Gerhard Rein^, and
Christopher Straub

Fakultät für Mathematik, Physik und Informatik, Universität Bayreuth, D-95440 Bayreuth, Germany

^* Corresponding author: Gerhard Rein
^* Corresponding author: Gerhard Rein

Received: December 2019

Revised: April 2020

Published: August 2020

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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.

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Mathematics Subject Classification: Primary: 85A05, 47F05; Secondary: 35B35, 35Q75.

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