Figure 1.
An illustration of the geometrical constructions introduced in the proof of Lemma 3.3
-
Previous Article
Kinetic models of conservative economies with need-based transfers as welfare
- KRM Home
- This Issue
-
Next Article
Hypocoercivity of linear kinetic equations via Harris's Theorem
February
2020, 13(1): 129-168.
doi: 10.3934/krm.2020005
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.
Mathematics Subject Classification:
Primary: 35Q83, 65Z05; Secondary: 44A10.
Citation:
Joackim Bernier, Michel Mehrenberger. Long-time behavior of second order linearized Vlasov-Poisson equations near a homogeneous equilibrium. Kinetic and Related Models,
2020, 13
(1)
: 129-168.
doi: 10.3934/krm.2020005
![]()
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. |
Figure 2.
Time evolution of $ |\Re(\widehat{E}_{1,num})(t)| $ (mode 1), $ |\Re\left(\widehat{E}_{1,num}(t)- ze^{-i\omega_1t}\right)| $ (mode 1 - leading mode 1) and $ |\Re\left((z_1+t_jz_2)e^{-i\omega_b t_j}\right)| $ (Best), for coarse $ 128\times256\times0.1 $ and refined $ 2048\times4096\times0.00625 $ grids, the latter being referred as (ref) in the legend. The parameters $ [t_{\min},t_{\max}] = [17.5,35] $ and $ [\tilde{t}_{\min},\tilde{t}_{\max}] = [1.75,17.5] $ are used for the least square procedures
Figure 3.
Time evolution of
●$ |\Re(\varepsilon^{-2}\widehat{E}_{1,num})(t)| $ (simu) vs $ |z_1e^{-i\omega_{0.9}t}| $ (approx1),
●$ |\Re\left(\varepsilon^{-2}\widehat{E}_{1,num}(t)- z_1e^{-i\omega_{0.9}t}\right)| $ (simu1) vs $ |\Re\left((z_2t+z_3)e^{-i0.9\omega_1t}\right)| $ (approx2),
●$ |\Re\left(\varepsilon^{-2}\widehat{E}_{1,num}(t)- z_1e^{-i\omega_{0.9}t}-(z_2t+z_3)e^{-i0.9\omega_1t}\right)| $ (simu2) vs $ |\Re\left(z_4e^{-i(\omega_{1}+\omega_{0.1,-})t}+z_5e^{-i(\omega_{1}+\omega_{0.1,+})t}\right)| $ (approx3),
for coarse (top)$ 128\times256\times0.1 $ and refined (bottom) $ 2048\times4096\times0.00625 $ grids. The parameters for the least square procedure is $ [t_{\min},t_{\max}] = [0,30] $ for the coarse grid and $ [t_{\min},t_{\max}] = [0,35] $ for the refined grid.
●
●
●
for coarse (top)
Figure 4.
A 2D-case with Best frequency: time evolution of
●$ |\Re(\varepsilon^{-2}\widehat{\rho}_{1,1,num})(t)| $ (simu)
●$ |\Re\left(z_1e^{-i\omega_{\sqrt{2}/10}t}+(z_2t+z_3)e^{-i\frac{1}{3}\omega_{3\sqrt{2}/10}t}\right)| $ (approx)
●$ 10^{-3}|\Re\left(z_1e^{-i\omega_{\sqrt{2}/10}t}\right)| $ (main mode /1e3)
●$ 10^{-3}|\Re\left((z_2t+z_3)e^{-i\frac{1}{3}\omega_{3\sqrt{2}/10}t}\right)| $ (Best mode /1e3)
The parameters$ \lambda = 0.09 $ and $ [t_{\min},t_{\max}] = [0,60] $ are used for the least square procedure to fit (simu) by (approx) and leads to $ z_1\simeq 0.036159+0.042602i $ , $ z_2\simeq -0.0031761-0.00089598i $ and $ z_3\simeq 0.010351-0.046355i $ .
●
●
●
●
The parameters
Figure 5.
A 2D-case without Best frequency: time evolution of
●$ |\Re(\varepsilon^{-2}\widehat{\rho}_{1,1,num})(t)| $ (simu)
●$ |\Re\left(z_1e^{-i\omega_{\sqrt{2}/10}t}\right)| $ (first approx)
●$ |\Re\left(\varepsilon^{-2}\widehat{\rho}_{1,1,num}(t)-z_1e^{-i\omega_{\sqrt{2}/10}t}\right)| $ (simu - first approx)
●$ |\Re\left(z_2e^{-i(\omega_{\sqrt{5}/10,+}+\omega_{\sqrt{13}/10,-})t}+z_3e^{-i(\omega_{\sqrt{5}/10,+}+\omega_{\sqrt{13}/10,+})t}\right)| $ (approx2)
The parameters$ \lambda = 0.05 $ and $ [t_{\min},t_{\max}] = [0,240] $ are used for the least square procedure leads to $ z_1\simeq 0.052836+0.049810i $ , $ z_2\simeq -0.032921-0.0010657i $ and $ z_3\simeq -0.013703-0.0050901i $ .
●
●
●
●
The parameters
| [1] |
Meixia Xiao, Xianwen Zhang. On global solutions to the Vlasov-Poisson system with radiation damping. Kinetic and Related Models, 2018, 11 (5) : 1183-1209. doi: 10.3934/krm.2018046 |
| [2] |
Miroslav Grmela, Michal Pavelka. Landau damping in the multiscale Vlasov theory. Kinetic and Related Models, 2018, 11 (3) : 521-545. doi: 10.3934/krm.2018023 |
| [3] |
Katherine Zhiyuan Zhang. Focusing solutions of the Vlasov-Poisson system. Kinetic and Related Models, 2019, 12 (6) : 1313-1327. doi: 10.3934/krm.2019051 |
| [4] |
Blanca Ayuso, José A. Carrillo, Chi-Wang Shu. Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system. Kinetic and Related Models, 2011, 4 (4) : 955-989. doi: 10.3934/krm.2011.4.955 |
| [5] |
Silvia Caprino, Guido Cavallaro, Carlo Marchioro. Time evolution of a Vlasov-Poisson plasma with magnetic confinement. Kinetic and Related Models, 2012, 5 (4) : 729-742. doi: 10.3934/krm.2012.5.729 |
| [6] |
Jack Schaeffer. Global existence for the Vlasov-Poisson system with steady spatial asymptotic behavior. Kinetic and Related Models, 2012, 5 (1) : 129-153. doi: 10.3934/krm.2012.5.129 |
| [7] |
Gang Li, Xianwen Zhang. A Vlasov-Poisson plasma of infinite mass with a point charge. Kinetic and Related Models, 2018, 11 (2) : 303-336. doi: 10.3934/krm.2018015 |
| [8] |
Gianluca Crippa, Silvia Ligabue, Chiara Saffirio. Lagrangian solutions to the Vlasov-Poisson system with a point charge. Kinetic and Related Models, 2018, 11 (6) : 1277-1299. doi: 10.3934/krm.2018050 |
| [9] |
Zili Chen, Xiuting Li, Xianwen Zhang. The two dimensional Vlasov-Poisson system with steady spatial asymptotics. Kinetic and Related Models, 2017, 10 (4) : 977-1009. doi: 10.3934/krm.2017039 |
| [10] |
Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361 |
| [11] |
Yulia O. Belyaeva, Björn Gebhard, Alexander L. Skubachevskii. A general way to confined stationary Vlasov-Poisson plasma configurations. Kinetic and Related Models, 2021, 14 (2) : 257-282. doi: 10.3934/krm.2021004 |
| [12] |
Jack Schaeffer. On time decay for the spherically symmetric Vlasov-Poisson system. Kinetic and Related Models, 2022, 15 (4) : 721-727. doi: 10.3934/krm.2021021 |
| [13] |
Gerhard Rein, Christopher Straub. On the transport operators arising from linearizing the Vlasov-Poisson or Einstein-Vlasov system about isotropic steady states. Kinetic and Related Models, 2020, 13 (5) : 933-949. doi: 10.3934/krm.2020032 |
| [14] |
Xianglong Duan. Sharp decay estimates for the Vlasov-Poisson and Vlasov-Yukawa systems with small data. Kinetic and Related Models, 2022, 15 (1) : 119-146. doi: 10.3934/krm.2021049 |
| [15] |
Yemin Chen. Smoothness of classical solutions to the Vlasov-Poisson-Landau system. Kinetic and Related Models, 2008, 1 (3) : 369-386. doi: 10.3934/krm.2008.1.369 |
| [16] |
Silvia Caprino, Guido Cavallaro, Carlo Marchioro. A Vlasov-Poisson plasma with unbounded mass and velocities confined in a cylinder by a magnetic mirror. Kinetic and Related Models, 2016, 9 (4) : 657-686. doi: 10.3934/krm.2016011 |
| [17] |
Hyung Ju Hwang, Jaewoo Jung, Juan J. L. Velázquez. On global existence of classical solutions for the Vlasov-Poisson system in convex bounded domains. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 723-737. doi: 10.3934/dcds.2013.33.723 |
| [18] |
Francis Filbet, Roland Duclous, Bruno Dubroca. Analysis of a high order finite volume scheme for the 1D Vlasov-Poisson system. Discrete and Continuous Dynamical Systems - S, 2012, 5 (2) : 283-305. doi: 10.3934/dcdss.2012.5.283 |
| [19] |
Dongming Wei. 1D Vlasov-Poisson equations with electron sheet initial data. Kinetic and Related Models, 2010, 3 (4) : 729-754. doi: 10.3934/krm.2010.3.729 |
| [20] |
Mihaï Bostan. Asymptotic behavior for the Vlasov-Poisson equations with strong uniform magnetic field and general initial conditions. Kinetic and Related Models, 2020, 13 (3) : 531-548. doi: 10.3934/krm.2020018 |
2021 Impact Factor: 1.398
Tools
Metrics
Other articles
by authors
[Back to Top]