Long-time behavior of second order linearized Vlasov-Poisson equations near a homogeneous equilibrium

Joackim Bernier; Michel Mehrenberger

doi:10.3934/krm.2020005

Article Contents

2020, Volume 13, Issue 1: 129-168. Doi: 10.3934/krm.2020005

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Long-time behavior of second order linearized Vlasov-Poisson equations near a homogeneous equilibrium

Joackim Bernier^1, , and
Michel Mehrenberger^2,

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
^* Corresponding author: Joackim Bernier

Received: March 2019

Revised: September 2019

Published: February 2020

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 35Q83, 65Z05; Secondary: 44A10.

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Figure 1. An illustration of the geometrical constructions introduced in the proof of Lemma 3.3

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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

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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.

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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 $.

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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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