Degenerate elliptic equations for resonant wave problems

Anouk Nicolopoulos (1) , Martin Campos Pinto (2, 1) , Bruno Després (1) , Patrick Ciarlet (3)

1 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
2 CNRS - Centre National de la Recherche Scientifique
3 POEMS - Propagation des Ondes : Étude Mathématique et Simulation

The modeling of resonant waves in 2D plasma leads to the coupling of two degenerate elliptic equations with a smooth coeffcient alpha and compact terms. The coeffcient alpha changes sign. The region where alpha is positive is propagative, and the region where alpha is negative is non propagative and elliptic. The two models are coupled through the line Sigma, corresponding to alpha equal to zero. Generically, it is an ill-posed problem, and additional information must be introduced to get a satisfactory treatment at Sigma. In this work we define the solution by relying on the limit absorption principle (alpha is replaced by alpha + i0^+) in an adapted functional setting. This setting lies on the decomposition of the solution in a regular part and a singular part, which originates at Sigma, and on quasi-solutions. It leads to a new well-posed mixed variational formulation with coupling. As we design explicit quasi-solutions, numerical experiments can be carried out, which illustrate the good properties of this new tool for numerical computation.

Mots clés: Singular solutions; Weighted Sobolev; Degenerate elliptic equations; Manufactured solutions; Mixed variational formulations

Domaines: Equations aux dérivées partielles [math.AP]

Anouk Nicolopoulos, Martin Campos Pinto, Bruno Després, Patrick Ciarlet. Degenerate elliptic equations for resonant wave problems. IMA Journal of Applied Mathematics, 2020, 85 (1), pp.132-159. ⟨10.1093/imamat/hxaa001⟩. ⟨hal-02142631v2⟩