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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 $O (1 / c^{2})$ 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 $c \to \infty$ 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 $O (1 / c^{2})$ level in the Euler–Darwin equation. This is a consistent and self-consistent model of order $O (1 / c^{2})$ and includes the full Lorentz force, which is relativistic at $O (1 / c^{2})$ .

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 $O (1 / c^{2})$ approximation of this model.

Keywords

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

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

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