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

* Corresponding author: Joackim Bernier

Received  March 2019 Revised  September 2019 Published  December 2019

Fund Project: This work was granted access to the HPC resources of Aix-Marseille Université financed by the project Equip@Meso (ANR-10-EQPX-29-01) of the program "Investissements d'Avenir"? supervised by the Agence Nationale de la Recherche. This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014-2018 and 2019-2020 under grant agreement No 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission.

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.

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. BarsamianJ. BernierS. 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. CasasN. CrouseillesE. 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. BarsamianJ. BernierS. 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. CasasN. CrouseillesE. 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 1.  An illustration of the geometrical constructions introduced in the proof of Lemma 3.3
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.
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 $.
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 $.
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