The Euler–Poisswell/Darwin equation and the asymptotic hierarchy of the Euler–Maxwell equation

Article type: Research Article

Authors: Möller, Jakob^{; *} | Mauser, Norbert J.

Affiliations: Research Platform MMM “Mathematics-Magnetism-Materials” c/o Fak. Mathematik, Univ. Wien, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria

Correspondence: [*] Corresponding author. E-mail: jakob.moeller@univie.ac.at.

Abstract: In this paper we introduce the (unipolar) pressureless Euler–Poisswell equation for electrons as the O(1/c) semi-relativistic approximation and the (unipolar) pressureless Euler–Darwin equation as the O(1/c2) semi-relativistic approximation of the (unipolar) pressureless Euler–Maxwell equation. In the “infinity-ion-mass” limit, the unipolar Euler–Maxwell equation arises from the bipolar Euler–Maxwell equation, describing a coupled system for a plasma of electrons and ions. The non-relativistic limit c→∞ of the Euler–Maxwell equation is the repulsive Euler–Poisson equation with electric force. We propose that the Euler–Poisswell equation, where the Euler equation with electric force is coupled to the magnetostatic O(1/c) approximation of Maxwell’s equations, is the correct semi-relativistic O(1/c) approximation of the Euler–Maxwell equation. In the Euler–Poisswell equation the fluid dynamics are decoupled from the magnetic field since the Lorentz force reduces to the electric force. The first non-trivial interaction with the magnetic field is at the O(1/c2) level in the Euler–Darwin equation. This is a consistent and self-consistent model of order O(1/c2) and includes the full Lorentz force, which is relativistic at O(1/c2). The Euler–Poisswell equation also arises as the semiclassical limit of the quantum Pauli–Poisswell equation, which is the O(1/c) approximation of the relativistic Dirac–Maxwell equation. We also present a local wellposedness theory for the Euler–Poisswell equation. The Euler–Maxwell system couples the non-relativistic Euler equation and the relativistic Maxwell equations and thus it is not fully consistent in 1/c. The consistent fully relativistic model is the relativistic Euler–Maxwell equation where Maxwell’s equations are coupled to the relativistic Euler equation – a model that is neglected in the mathematics community. We also present the relativistic Euler–Darwin equation resulting as a O(1/c2) approximation of this model.

Keywords: Quantum physics, mathematical modeling, Euler equation, non-relativistic limit, semi-relativistic approximation

DOI: 10.3233/ASY-231864

Journal: Asymptotic Analysis, vol. 135, no. 3-4, pp. 525-543, 2023