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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 & Related Models,
2020, 13
(1)
: 129-168.
doi: 10.3934/krm.2020005
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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. Google Scholar |
| [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. Google Scholar |
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. Google Scholar |
| [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. Google Scholar |
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
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