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On the transport operators arising from linearizing the Vlasov-Poisson or Einstein-Vlasov system about isotropic steady states
Fakultät für Mathematik, Physik und Informatik, Universität Bayreuth, D-95440 Bayreuth, Germany |
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.
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. |
| [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. |
| [3] |
J. Binney and S. Tremaine, Galactic Dynamics, Princeton University Press, Princeton, 1987.
|
| [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. |
| [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. |
| [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. |
| [7] |
M. Hadžić, Z. Lin and G. Rein, Stability and instability of self-gravitating relativistic matter distributions, preprint, arXiv: 1810.00809. |
| [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. |
| [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. |
| [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. |
| [11] |
G. Rein, The Vlasov-Einstein System with Surface Symmetry, Habilitationsschrift, Universität München, 1995. |
| [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. |
| [13] |
W. Rudin, Functional Analysis, McGraw-Hill, New York, 1973. |
| [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. |
| [15] |
C. Straub, Stability of the King model—A coercivity-based approach, Master thesis, Universität Bayreuth, 2019. |
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. |
| [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. |
| [3] |
J. Binney and S. Tremaine, Galactic Dynamics, Princeton University Press, Princeton, 1987.
|
| [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. |
| [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. |
| [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. |
| [7] |
M. Hadžić, Z. Lin and G. Rein, Stability and instability of self-gravitating relativistic matter distributions, preprint, arXiv: 1810.00809. |
| [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. |
| [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. |
| [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. |
| [11] |
G. Rein, The Vlasov-Einstein System with Surface Symmetry, Habilitationsschrift, Universität München, 1995. |
| [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. |
| [13] |
W. Rudin, Functional Analysis, McGraw-Hill, New York, 1973. |
| [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. |
| [15] |
C. Straub, Stability of the King model—A coercivity-based approach, Master thesis, Universität Bayreuth, 2019. |
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