Degenerate elliptic equations for resonant wave problems
Anouk Nicolopoulos, Martin Campos Pinto, Bruno Després, Patrick Ciarlet
Abstract
Abstract The modelling of resonant waves in 2D plasma leads to the coupling of two degenerate elliptic equations with a smooth coefficient $\alpha $ and compact terms. The coefficient $\alpha $ changes sign. The region where $\{\alpha>0\}$ is propagative, and the region where $\{\alpha <0\}$ is non propagative and elliptic. The two models are coupled through the line $\varSigma =\{\alpha =0\}$. Generically, it is an ill-posed problem and additional information must be introduced to get a satisfactory treatment at $\varSigma $. In this work, we define the solution by relying on the limiting absorption principle ($\alpha $ is replaced by $\alpha +i0^+$) in an adapted functional setting. This setting lies on the decomposition of the solution in a regular and a singular part, which originates at $\varSigma $, 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.
BibTeX
@article{Nicolopoulos_2020,
title={Degenerate elliptic equations for resonant wave problems},
volume={85},
ISSN={1464-3634},
url={http://dx.doi.org/10.1093/imamat/hxaa001},
DOI={10.1093/imamat/hxaa001},
number={1},
journal={IMA Journal of Applied Mathematics},
publisher={Oxford University Press (OUP)},
author={Nicolopoulos, Anouk and Campos Pinto, Martin and Després, Bruno and Ciarlet, Patrick},
year={2020},
month=feb,
pages={132–159} }