Issue |
ESAIM: M2AN
Volume 58, Number 5, September-October 2024
|
|
---|---|---|
Page(s) | 1785 - 1821 | |
DOI | https://doi.org/10.1051/m2an/2024053 | |
Published online | 10 October 2024 |
Study of a degenerate non-elliptic equation to model plasma heating
POEMS, CNRS, Inria, ENSTA Paris, Institut Polytechnique de Paris, 828 Boulevard des Maréchaux, 91120 Palaiseau, France
* Corresponding author: maryna.kachanovska@inria.fr
Received:
10
November
2023
Accepted:
26
June
2024
In this manuscript, we study solutions to resonant Maxwell’s equations in heterogeneous plasmas. We concentrate on the phenomenon of upper-hybrid heating, which occurs in a localized region where electromagnetic waves transfer energy to the particles. In the 2D case, it can be modelled mathematically by the partial differential equation − div(α∇u) − w2u = 0, where the coefficient α is a smooth, sign-changing, real-valued function. Since the locus of the sign change is located within the plasma, the equation is non-elliptic, and degenerate. On the other hand, using the limiting absorption principle, one can build a family of elliptic equations that approximate the degenerate equation. Then, a natural question is to relate the solution of the degenerate equation, if it exists, to the family of solutions of the elliptic equations. For that, we assume that the family of solutions converges to a limit, which can be split into a regular part and a singular part, and that this limiting absorption solution is governed by the non-elliptic equation introduced above. One of the difficulties lies in the definition of appropriate norms and function spaces in order to be able to study the non-elliptic equation and its solutions. As a starting point, we revisit a prior work [Nicolopoulos et al., IMA J. Appl. Math. 85 (2020) 132–159] on this topic by Nicolopoulos et al., who proposed a variational formulation for the plasma heating problem. We improve the results they obtained, in particular by establishing existence and uniqueness of the solution, by making a different choice of function spaces. Also, we propose a series of numerical tests, comparing the numerical results of Nicolopoulos et al. to those obtained with our numerical method, for which we observe better convergence.
Mathematics Subject Classification: 78M10 / 78M30 / 65N30 / 35A21 / 35G99 / 78A40
Key words: Degenerate partial differential equations / singular solutions / mixed variational formulations / limiting absorption principle / upper-hybrid plasma resonance
© The authors. Published by EDP Sciences, SMAI 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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