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Hypocoercivity of linear kinetic equations via Harris's Theorem
Long-time behavior of second order linearized Vlasov-Poisson equations near a homogeneous equilibrium
1. | Univ Rennes, INRIA, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France |
2. | Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France |
The asymptotic behavior of the solutions of the second order linearized Vlasov-Poisson system around homogeneous equilibria is derived. It provides a fine description of some nonlinear and multidimensional phenomena such as the existence of Best frequencies. Numerical results for the $ 1D\times1D $ and $ 2D\times2D $ Vlasov-Poisson system illustrate the effectiveness of this approach.
References:
[1] |
W. Arendt, C. J. K. Batty and M. Hieber, Vector-valued Laplace Transforms and Cauchy Problems, 2nd edition, Birkhäuser/Springer Basel AG, Basel, 2011.
doi: 10.1007/978-3-0348-0087-7. |
[2] |
M. Badsi and M. Herda,
Modelling and simulating a multispecies plasma, ESAIM: ProcS, 53 (2016), 22-37.
doi: 10.1051/proc/201653002. |
[3] |
J. Barré and Y. Y. Yamaguchi, On algebraic damping close to inhomogeneous Vlasov equilibria in multi-dimensional spaces, J. Phys. A, 46 (2013), 225501, 19 pp.
doi: 10.1088/1751-8113/46/22/225501. |
[4] |
Y. Barsamian, J. Bernier, S. Hirstoaga and M. Mehrenberger,
Verification of 2D 2D and Two-Species Vlasov-Poisson Solvers, ESAIM: ProcS, 63 (2018), 78-108.
doi: 10.1051/proc/201863078. |
[5] |
J. Bedrossian, N. Masmoudi and C. Mouhot, Landau damping: Paraproducts and Gevrey regularity, Ann. PDE, 2 (2016), Art. 4, 71pp.
doi: 10.1007/s40818-016-0008-2. |
[6] |
F. Casas, N. Crouseilles, E. Faou and M. Mehrenberger,
High-order Hamiltonian splitting for the Vlasov-Poisson equations, Numer. Math., 135 (2017), 769-801.
doi: 10.1007/s00211-016-0816-z. |
[7] |
P. Degond,
Spectral theory of the linearized Vlasov-Poisson equation, Trans. Amer. Math. Soc., 294 (1986), 435-453.
doi: 10.1090/S0002-9947-1986-0825714-8. |
[8] |
J. Denavit,
First and second order landau damping in maxwellian plasmas, Physics of Fluids, 8 (1965), 471-478.
doi: 10.1063/1.1761247. |
[9] |
R. Horsin, Comportement en Temps Long D'équations de Type Vlasov: Études Mathématiques et Numériques, Ph.D thesis, Université Rennes 1, 2017. |
[10] |
L. Landau,
On the vibrations of the electronic plasma, J. Phys. (USSR), 10 (1946), 25-34.
doi: 10.1016/B978-0-08-010586-4.50066-3. |
[11] |
C. Mouhot and C. Villani,
On Landau damping, Acta Math., 207 (2011), 29-201.
doi: 10.1007/s11511-011-0068-9. |
[12] |
F. Nicola and L. Rodino, Global Pseudo-differential Calculus on Euclidean Spaces, Birkhäuser Verlag, Basel, 2010.
doi: 10.1007/978-3-7643-8512-5. |
[13] |
Z. Sedláček and L. Nocera,
Second-order oscillations of a Vlasov-Poisson plasma in Fourier-transformed velocity space, Journal of Plasma Physics, 48 (1992), 367-389.
doi: 10.1017/S0022377800016639. |
[14] |
M. M. Shoucri and R. R. Gagné, A Multistep Technique for the Numerical Solution of a Two-Dimensional Vlasov Equation, Journal of Computational Physics, 23 (1977), 243–262.
doi: 10.1016/0021-9991(77)90093-6. |
[15] |
E. Sonnendrücker, Numerical Methods for the Vlasov-Maxwell Equations, book in preparation, version of february, 2015. |
show all references
References:
[1] |
W. Arendt, C. J. K. Batty and M. Hieber, Vector-valued Laplace Transforms and Cauchy Problems, 2nd edition, Birkhäuser/Springer Basel AG, Basel, 2011.
doi: 10.1007/978-3-0348-0087-7. |
[2] |
M. Badsi and M. Herda,
Modelling and simulating a multispecies plasma, ESAIM: ProcS, 53 (2016), 22-37.
doi: 10.1051/proc/201653002. |
[3] |
J. Barré and Y. Y. Yamaguchi, On algebraic damping close to inhomogeneous Vlasov equilibria in multi-dimensional spaces, J. Phys. A, 46 (2013), 225501, 19 pp.
doi: 10.1088/1751-8113/46/22/225501. |
[4] |
Y. Barsamian, J. Bernier, S. Hirstoaga and M. Mehrenberger,
Verification of 2D 2D and Two-Species Vlasov-Poisson Solvers, ESAIM: ProcS, 63 (2018), 78-108.
doi: 10.1051/proc/201863078. |
[5] |
J. Bedrossian, N. Masmoudi and C. Mouhot, Landau damping: Paraproducts and Gevrey regularity, Ann. PDE, 2 (2016), Art. 4, 71pp.
doi: 10.1007/s40818-016-0008-2. |
[6] |
F. Casas, N. Crouseilles, E. Faou and M. Mehrenberger,
High-order Hamiltonian splitting for the Vlasov-Poisson equations, Numer. Math., 135 (2017), 769-801.
doi: 10.1007/s00211-016-0816-z. |
[7] |
P. Degond,
Spectral theory of the linearized Vlasov-Poisson equation, Trans. Amer. Math. Soc., 294 (1986), 435-453.
doi: 10.1090/S0002-9947-1986-0825714-8. |
[8] |
J. Denavit,
First and second order landau damping in maxwellian plasmas, Physics of Fluids, 8 (1965), 471-478.
doi: 10.1063/1.1761247. |
[9] |
R. Horsin, Comportement en Temps Long D'équations de Type Vlasov: Études Mathématiques et Numériques, Ph.D thesis, Université Rennes 1, 2017. |
[10] |
L. Landau,
On the vibrations of the electronic plasma, J. Phys. (USSR), 10 (1946), 25-34.
doi: 10.1016/B978-0-08-010586-4.50066-3. |
[11] |
C. Mouhot and C. Villani,
On Landau damping, Acta Math., 207 (2011), 29-201.
doi: 10.1007/s11511-011-0068-9. |
[12] |
F. Nicola and L. Rodino, Global Pseudo-differential Calculus on Euclidean Spaces, Birkhäuser Verlag, Basel, 2010.
doi: 10.1007/978-3-7643-8512-5. |
[13] |
Z. Sedláček and L. Nocera,
Second-order oscillations of a Vlasov-Poisson plasma in Fourier-transformed velocity space, Journal of Plasma Physics, 48 (1992), 367-389.
doi: 10.1017/S0022377800016639. |
[14] |
M. M. Shoucri and R. R. Gagné, A Multistep Technique for the Numerical Solution of a Two-Dimensional Vlasov Equation, Journal of Computational Physics, 23 (1977), 243–262.
doi: 10.1016/0021-9991(77)90093-6. |
[15] |
E. Sonnendrücker, Numerical Methods for the Vlasov-Maxwell Equations, book in preparation, version of february, 2015. |



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