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October  2020, 13(5): 933-949. doi: 10.3934/krm.2020032

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

Received  December 2019 Revised  April 2020 Published  August 2020

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.

Keywords: Vlasov-Poisson system, Einstein-Vlasov system, linearization, skew-adjointness, transport operator.
Mathematics Subject Classification: Primary: 85A05, 47F05; Secondary: 35B35, 35Q75.
Citation: Gerhard Rein, Christopher Straub. On the transport operators arising from linearizing the Vlasov-Poisson or Einstein-Vlasov system about isotropic steady states. Kinetic & Related Models, 2020, 13 (5) : 933-949. doi: 10.3934/krm.2020032
References:
[1]

H. Andréasson, The Einstein-Vlasov system/kinetic theory, Living Rev. Relativ., 14 (2011), Available from: https://doi.org/10.12942/lrr-2011-4. Google Scholar

[2]

J. Batt, W. Faltenbacher and E. Horst, Stationary spherically symmetric models in stellar dynamics, Arch. Rational Mech. Anal., 93 (1986), 159-183.  doi: 10.1007/BF00279958.  Google Scholar

[3] J. Binney and S. Tremaine, Galactic Dynamics, Princeton University Press, Princeton, 1987.   Google Scholar
[4]

M. Bostan, Transport equations with disparate advection fields. Application to the gyrokinetic models in plasma physics, J. Differential Equations, 249 (2010), 1620-1663.  doi: 10.1016/j.jde.2010.07.010.  Google Scholar

[5]

Y. Guo and Z. Lin, Unstable and stable galaxy models, Commun. Math. Phys., 279 (2008), 789–813. doi: 10.1007/s00220-008-0439-z.  Google Scholar

[6]

Y. Guo and G. Rein, A non-variational approach to nonlinear stability in stellar dynamics applied to the King model, Commun. Math. Phys., 271 (2007), 489-509.  doi: 10.1007/s00220-007-0212-8.  Google Scholar

[7]

M. Hadžić, Z. Lin and G. Rein, Stability and instability of self-gravitating relativistic matter distributions, preprint, arXiv: 1810.00809. Google Scholar

[8]

J. Ipser and K. S. Thorne, Relativistic, spherically symmetric star clusters I. Stability theory for radial perturbations, Astrophys. J., 154 (1968), 251-270.  doi: 10.1086/149755.  Google Scholar

[9]

M. Lemou, F. Mehats and P. Raphaël, A new variational approach to the stability of gravitational systems, Commun. Math. Phys., 302 (2011), 161-224.  doi: 10.1007/s00220-010-1182-9.  Google Scholar

[10]

T. Ramming and G. Rein, Spherically symmetric equilibria for self-gravitating kinetic or fluid models in the non-relativistic and relativistic case—A simple proof for finite extension, SIAM Journal on Mathematical Analysis, 45 (2013), 900-914.  doi: 10.1137/120896712.  Google Scholar

[11]

G. Rein, The Vlasov-Einstein System with Surface Symmetry, Habilitationsschrift, Universität München, 1995. Google Scholar

[12]

G. Rein, Collisionless kinetic equations from astrophysics—The Vlasov-Poisson system, in Handbook of Differential Equations, Evolutionary Equations (eds. C. M. Dafermos and E. Feireisl), 3, Elsevier, 2007,383–476. doi: 10.1016/S1874-5717(07)80008-9.  Google Scholar

[13]

W. Rudin, Functional Analysis, McGraw-Hill, New York, 1973.  Google Scholar

[14]

J. Schaeffer, A class of counterexamples to Jeans' theorem for the Vlasov-Einstein system, Commun. Math. Phys., 204 (1999), 313-327.  doi: 10.1007/s002200050647.  Google Scholar

[15]

C. Straub, Stability of the King model—A coercivity-based approach, Master thesis, Universität Bayreuth, 2019. Google Scholar

show all references

References:
[1]

H. Andréasson, The Einstein-Vlasov system/kinetic theory, Living Rev. Relativ., 14 (2011), Available from: https://doi.org/10.12942/lrr-2011-4. Google Scholar

[2]

J. Batt, W. Faltenbacher and E. Horst, Stationary spherically symmetric models in stellar dynamics, Arch. Rational Mech. Anal., 93 (1986), 159-183.  doi: 10.1007/BF00279958.  Google Scholar

[3] J. Binney and S. Tremaine, Galactic Dynamics, Princeton University Press, Princeton, 1987.   Google Scholar
[4]

M. Bostan, Transport equations with disparate advection fields. Application to the gyrokinetic models in plasma physics, J. Differential Equations, 249 (2010), 1620-1663.  doi: 10.1016/j.jde.2010.07.010.  Google Scholar

[5]

Y. Guo and Z. Lin, Unstable and stable galaxy models, Commun. Math. Phys., 279 (2008), 789–813. doi: 10.1007/s00220-008-0439-z.  Google Scholar

[6]

Y. Guo and G. Rein, A non-variational approach to nonlinear stability in stellar dynamics applied to the King model, Commun. Math. Phys., 271 (2007), 489-509.  doi: 10.1007/s00220-007-0212-8.  Google Scholar

[7]

M. Hadžić, Z. Lin and G. Rein, Stability and instability of self-gravitating relativistic matter distributions, preprint, arXiv: 1810.00809. Google Scholar

[8]

J. Ipser and K. S. Thorne, Relativistic, spherically symmetric star clusters I. Stability theory for radial perturbations, Astrophys. J., 154 (1968), 251-270.  doi: 10.1086/149755.  Google Scholar

[9]

M. Lemou, F. Mehats and P. Raphaël, A new variational approach to the stability of gravitational systems, Commun. Math. Phys., 302 (2011), 161-224.  doi: 10.1007/s00220-010-1182-9.  Google Scholar

[10]

T. Ramming and G. Rein, Spherically symmetric equilibria for self-gravitating kinetic or fluid models in the non-relativistic and relativistic case—A simple proof for finite extension, SIAM Journal on Mathematical Analysis, 45 (2013), 900-914.  doi: 10.1137/120896712.  Google Scholar

[11]

G. Rein, The Vlasov-Einstein System with Surface Symmetry, Habilitationsschrift, Universität München, 1995. Google Scholar

[12]

G. Rein, Collisionless kinetic equations from astrophysics—The Vlasov-Poisson system, in Handbook of Differential Equations, Evolutionary Equations (eds. C. M. Dafermos and E. Feireisl), 3, Elsevier, 2007,383–476. doi: 10.1016/S1874-5717(07)80008-9.  Google Scholar

[13]

W. Rudin, Functional Analysis, McGraw-Hill, New York, 1973.  Google Scholar

[14]

J. Schaeffer, A class of counterexamples to Jeans' theorem for the Vlasov-Einstein system, Commun. Math. Phys., 204 (1999), 313-327.  doi: 10.1007/s002200050647.  Google Scholar

[15]

C. Straub, Stability of the King model—A coercivity-based approach, Master thesis, Universität Bayreuth, 2019. Google Scholar

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