scholarly journals Local existence for the one-dimensional Vlasov–Poisson system with infinite mass

2007 ◽  
Vol 30 (5) ◽  
pp. 529-548 ◽  
Author(s):  
Stephen Pankavich
Keyword(s):  
Local Existence ◽  
Poisson System ◽  
One Dimensional ◽  
Infinite Mass ◽  
The One

scholarly journals Diffusion approximation for the one dimensional Boltzmann-Poisson system

2004 ◽  
Vol 4 (4) ◽  
pp. 1129-1142 ◽  
Author(s):  
N. Ben Abdallah ◽  
◽  
M. Lazhar Tayeb ◽  
Keyword(s):  
Diffusion Approximation ◽  
Poisson System ◽  
One Dimensional ◽  
The One

scholarly journals Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system

2011 ◽  
Vol 4 (4) ◽  
pp. 955-989 ◽  
Author(s):  
Blanca Ayuso ◽  
◽  
José A. Carrillo ◽  
Chi-Wang Shu ◽  
◽  
...  
Keyword(s):  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Methods ◽  
Galerkin Methods ◽  
Poisson System ◽  
One Dimensional ◽  
The One

scholarly journals Computer-assisted Existence Proofs for One-dimensional Schrödinger-Poisson Systems

2020 ◽  
Vol 24 (3) ◽  
pp. 373-391
Author(s):  
Jonathan Wunderlich ◽  
Michael Plum
Keyword(s):  
Approximate Solution ◽  
Ritz Method ◽  
Three Dimensional ◽  
Computer Assisted ◽  
Poisson System ◽  
Eigenvalue Bounds ◽  
One Dimensional ◽  
Existence Proofs ◽  
Norm Bound ◽  
The One

Motivated by the three-dimensional time-dependent Schrödinger-Poisson system we prove the existence of non-trivial solutions of the one-dimensional stationary Schrödinger-Poisson system using computer-assisted methods. Starting from a numerical approximate solution, we compute a bound for its defect, and a norm bound for the inverse of the linearization at the approximate solution. For the latter, eigenvalue bounds play a crucial role, especially for the eigenvalues "close to" zero. Therefor, we use the Rayleigh-Ritz method and a corollary of the Temple-Lehmann Theorem to get enclosures of the crucial eigenvalues of the linearization below the essential spectrum. With these data in hand, we can use a fixed-point argument to obtain the desired existence of a non-trivial solution "nearby" the approximate one. In addition to the pure existence result, the used methods also provide an enclosure of the exact solution.

scholarly journals A neural network closure for the Euler-Poisson system based on kinetic simulations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Léo Bois ◽  
Emmanuel Franck ◽  
Laurent Navoret ◽  
Vincent Vigon
Keyword(s):  
Neural Network ◽  
Spatial Density ◽  
Data Generation ◽  
Poisson System ◽  
Mean Velocity ◽  
One Dimensional ◽  
Poisson Equations ◽  
Wide Range ◽  
The One ◽  
Uniform Accuracy

<p style='text-indent:20px;'>This work deals with the modeling of plasmas, which are ionized gases. Thanks to machine learning, we construct a closure for the one-dimensional Euler-Poisson system valid for a wide range of collisional regimes. This closure, based on a fully convolutional neural network called V-net, takes as input the whole spatial density, mean velocity and temperature and predicts as output the whole heat flux. It is learned from data coming from kinetic simulations of the Vlasov-Poisson equations. Data generation and preprocessings are designed to ensure an almost uniform accuracy over the chosen range of Knudsen numbers (which parametrize collisional regimes). Finally, several numerical tests are carried out to assess validity and flexibility of the whole pipeline.</p>

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