Solutions to the cold plasma model at resonances

Després, Bruno; Imbert-Gérard, Lise-Marie; Lafitte, Olivier

doi:10.5802/jep.41

Cited by 2 publications

(6 citation statements)

References 16 publications

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“…We make use of the Hardy inequality to prove the correctness of the functional setting. An important asset of the dissipative formulation (Problem 3) is a direct measure of the limit resonant heating defined in [13,12], see Remark 12.…”

Section: Resultsmentioning

confidence: 99%

“…This will be illustrated with the example of the Budden problem where singular solutions are encountered at x = 0, where the extra diagonal part of the 2 × 2 block tensor dominates the corresponding diagonal part [13,12]. Since such a problem may be ill-posed in standard functional spaces like L 2 (Ω) 3 or H(curl, Ω) = F ∈ L 2 (Ω) 3 , ∇ × F ∈ L 2 (Ω) 3 , it is usual to regularize it in the context of the limit absorption principle.…”

Section: General Strategymentioning

confidence: 99%

“…In this section we detail the construction of such manufactured solutions for the 1D case. In the setting of the regularized Budden problem (17), we thus consider the non homogeneous system Here we have considered the symmetrized system, i.e., (9), since our analysis will mostly address weak constraints of the form (12) which are simpler in their expression. We note however that in 1D, the manufactured solutions designed for the original system (5) only differ by a couple of sign changes, see Section 4.…”

Section: Manufactured Solutions For the 1d Casementioning

confidence: 99%

See 2 more Smart Citations

Constructive Formulations of Resonant Maxwell's Equations

Pinto¹,

Després²

2017

SIAM J. Math. Anal.

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In this article we propose new constructive weak formulations for resonant time-harmonic wave equations with singular solutions. Our approach follows the limiting absorption principle and combines standard weak formulations of PDEs with properties of elementary special functions adapted to the singularity of the solutions, called manufactured solutions. We show the well-posedness of several formulations obtained by these means for the limit problem in dimension one, and propose a generalization in dimension two.

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Section: Resultsmentioning

confidence: 99%

Section: General Strategymentioning

confidence: 99%

Section: Manufactured Solutions For the 1d Casementioning

confidence: 99%

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Constructive Formulations of Resonant Maxwell's Equations

Pinto¹,

Després²

2017

SIAM J. Math. Anal.

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“…Up to our knowledge, the limiting absorption principle can be justified in 1D, as well as for particular values of α(x, y) in slab geometries, cf. [7,6]. Let us provide an illuminating example whose goal is two-fold.…”

Section: Mathematical Settingmentioning

confidence: 99%

“…where N e (x) is the particle density (assumed to be a continuous function) and C α,ω , C δ,ω and C β,ω are three real-valued frequency-dependent constants which do not vary in space. We consider an upper hybrid resonance in the plasma, see [15, and recent works [11,7,4,5,8,6,12,14,13], which is characterized by the fact that α = 0 on some curve inside the region. Like in the above cited works, we will be particularly interested in the cases when the density N e (x) is s.t.…”

Section: Introductionmentioning

confidence: 99%

Study of a degenerate non-elliptic equation to model plasma heating

Ciarlet,

Kachanovska,

Peillon

2024

ESAIM: M2AN

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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) − ω2u = 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 [12] on this topic by A. Nicolopoulos, M. Campos Pinto, B. Després and P. Ciarlet Jr., 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.

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Solutions to the cold plasma model at resonances

Cited by 2 publications

References 16 publications

Constructive Formulations of Resonant Maxwell's Equations

Constructive Formulations of Resonant Maxwell's Equations

Study of a degenerate non-elliptic equation to model plasma heating

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