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

Patrick, Jr. Ciarlet, M. Kachanovska, Étienne Peillon
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Abstract

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.
研究用于模拟等离子体加热的退化非椭圆方程
在本手稿中,我们研究了异质等离子体中共振麦克斯韦方程的解。我们专注于上混合加热现象,这种现象发生在电磁波向粒子传递能量的局部区域。在二维情况下,它可以用偏微分方程 - div (α∇u) - ω2u = 0 进行数学建模,其中系数 α 是一个平滑的、符号变化的实值函数。由于符号变化的位置位于等离子体内部,因此方程是非椭圆的,而且是退化的。另一方面,利用极限吸收原理,我们可以建立一个近似于退化方程的椭圆方程组。那么,一个自然的问题是,如果存在退化方程的解,如何将其与椭圆方程的解族联系起来。为此,我们假定解的族收敛到一个极限,这个极限可分为正则部分和奇异部分,而这个极限吸收解受上文介绍的非椭圆方程支配。困难之一在于如何定义适当的规范和函数空间,以便研究非椭圆方程及其解。作为起点,我们重温了 A. Nicolopoulos、M. Campos Pinto、B. Després 和 P. Ciarlet Jr.之前关于这一主题的研究[12],他们提出了等离子体加热问题的变分公式。我们改进了他们获得的结果,特别是通过对函数空间的不同选择,建立了解的存在性和唯一性。此外,我们还提出了一系列数值测试,将尼科洛普洛斯等人的数值结果与我们的数值方法得出的结果进行比较,我们发现后者的收敛性更好。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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