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

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

1. Introduction

The Euler–Maxwell equation is an important model in plasma and semiconductor physics that models the motion of particles (e.g. electrons and ions) as the flow of a charged fluid whose density and current density act as source terms for the electromagnetic fields, described by Maxwell’s equations. The action of the electromagnetic field on the fluid is then taken into account by the Lorentz force which enters the Euler equation as a driving term.

The “non-relativistic” (or “Post-Newtonian”) limit $c \to \infty$ (c being the speed of light) of the Euler–Maxwell equation is the repulsive Euler–Poisson equation, i.e. the self-interaction of the fluid with the fields reduces to the Coulomb interaction [31,36]. This distinction is necessary since the other case, i.e. attractive interaction like gravitation, is very different also from a mathematical point of view.

We note that the Euler–Maxwell equation is in fact not consistently relativistic because it couples a nonrelativistic Galilei-invariant equation, i.e. the Euler equation, to a Lorentz invariant fully relativistic equation, i.e. Maxwell’s equations. In order to arrive at a fully relativistic system one has to consider the proper relativistic Euler equation where the unknowns are replaced by their relativistic counterparts, in particular the “gamma factor” accounting for finite speed of propagation, and then couple it to Maxwell’s equations.

However, this consistent model is rarely considered in the mathematics community – note that Guo, Ionescu and Pausader introduce this “correct” model in a J. Math. Phys. paper in 2014, but then they deal with the “nonrelativistic Euler–Maxwell” model in their substantial 2016 paper in Ann. Math.

In the semi-relativistic regime one keeps terms up to a certain order in the small parameter $\begin{matrix} (1) & ε : = 1 / c . \end{matrix}$ In the $O (ε)$ regime one obtains the Euler–Poisswell equation and in the $O (ε^{2})$ regime the Euler–Darwin equation. Here, the Poisswell equations are the $O (ε)$ approximation of Maxwell’s equations whereas the Darwin equations are the $O (ε^{2})$ approximation of Maxwell’s equations. Another important difference between the two models is that in the former the driving force is only electric since the $O (1)$ component of the magnetic field, which enters the Lorentz force, vanishes in the Poisswell approximation. In the Euler–Darwin equation the driving force is the Lorentz force like for the Euler–Maxwell equation. It should be noted that this implies that at the $O (ε)$ level (i.e. the Euler–Poisswell equation) the magnetic field decouples from the fluid dynamics, which only interacts with the electric field. The coupling of the fluid dynamics with magnetic field arises first in the $O (ε^{2})$ approximation, i.e. the Euler–Darwin model.

The Lorentz force can be added as an external force to the Euler equation in the Euler–Lorentz model which is not self-consistent since it is not consistent in ε and therefore it is a model only at $O (1)$ .

We consider the equations only in $3 d$ , i.e. 3 space dimensions, as the natural setting for magnetic fields.

We have the following diagram for the asymptotic hierarchy of the Euler–Maxwell equation in the small parameter ε. The top level represents the fully relativistic regime whereas the bottom level represents the non-relativistic regime.

Modeling a plasma consisting of ions and electrons, one employs the bipolar Euler–Maxwell equation [40]. It is a coupled system of two Euler equations, one for ions with mass $m_{i}$ and one for electrons with mass $m_{e}$ . The sources of the electromagnetic field are then the combined density and current density of ions and electrons. Usually the positive ions are much heavier and much slower than the electrons, hence the ion dynamics can be neglected when dealing with the dynamics of the electrons described by the unipolar Euler–Maxwell equation where the “static” ion background is modeled by a background density $b (x)$ as an additional source term in the Poisson equation for the electric potential $V^{ε}$ . The limit from bipolar to unipolar can be achieved as the limit $m_{i} / m_{e} \to \infty$ which is either letting the mass $m_{e}$ of the electrons go to zero (“zero-electron-mass” limit) or, equivalently, letting the mass $m_{i}$ of the ions go to infinity (“infinity-ion-mass” limit). The latter was recently worked out in full detail and rigour by Zhao in [40]. $\begin{matrix} \begin{matrix} Bipolar Euler–Maxwell \end{matrix} \overset{⟶}{m_{i} / m_{e} \to \infty} \begin{matrix} Unipolar Euler–Maxwell \end{matrix} \end{matrix}$

Remark 1.
This paper is concerned with asymptotic analysis and model hierarchies. The numerics of the Euler–Poisswell and Euler–Darwin equations are not well developed yet. For numerics of the Vlasov–Darwin and Vlasov–Poisswell equations see [2]. For numerics of the Euler–Maxwell equation see for example [9] and for the Euler–Poisson equation [7].

2. Asymptotic hierarchy of the Euler–Maxwell equation

The pressureless bipolar Euler–Maxwell equation is given by $\begin{aligned} (2a) & \partial_{t} ρ_{μ}^{ε} + \nabla \cdot (ρ_{μ}^{ε} u_{μ}^{ε}) = 0, \\ (2b) & m_{μ} \partial_{t} u_{μ}^{ε} + m_{μ} u^{ε} \cdot \nabla u^{ε} = E^{ε} + ε u_{μ}^{ε} \times B^{ε}, \\ (2c) & ε \partial_{t} E^{ε} - \nabla \times B^{ε} = ε (ρ_{i}^{ε} u_{i}^{ε} - ρ_{e}^{ε} u_{e}^{ε}), ε \partial_{t} B^{ε} + \nabla \times E^{ε} = 0, \\ (2d) & \nabla \cdot E^{ε} = ρ_{e}^{ε} - ρ_{i}^{ε}, \nabla \cdot B^{ε} = 0 . \\ (2e) & (ρ_{μ}^{ε}, u_{μ}^{ε}, E^{ε}, B^{ε}) (x, 0) = (ρ_{μ, I}^{ε}, u_{μ, I}^{ε}, E_{I}^{ε}, B_{I}^{ε}) (x), \end{aligned}$ where $μ = i, e$ for the ion and electron component, respectively. $ρ_{μ}^{ε} : R_{x}^{3} \times R_{t} \to R$ and $u_{μ}^{ε} : R_{x}^{3} \times R_{t} \to R^{3}$ are the densities and velocities of the ions and electrons. $E^{ε}, B^{ε} : R_{x}^{3} \times R_{t} \to R^{3}$ are the electric and magnetic fields. In the infinity-ion-mass limit $m_{i} \to \infty$ the bipolar Euler–Maxwell equation converges to the pressureless unipolar Euler–Maxwell equation, studied in [13,19,40], $\begin{aligned} (3a) & \partial_{t} ρ^{ε} + \nabla \cdot (ρ^{ε} u^{ε}) = 0, \\ (3b) & \partial_{t} u^{ε} + u^{ε} \cdot \nabla u^{ε} = E^{ε} + ε u^{ε} \times B^{ε}, \\ (3c) & ε \partial_{t} E^{ε} - \nabla \times B^{ε} = - ε ρ^{ε} u^{ε}, ε \partial_{t} B^{ε} + \nabla \times E^{ε} = 0, \\ (3d) & \nabla \cdot E^{ε} = ρ^{ε}, \nabla \cdot B^{ε} = 0, \\ (3e) & (ρ^{ε}, u^{ε}, E^{ε}, B^{ε}) (x, 0) = (ρ_{I}^{ε}, u_{I}^{ε}, E_{I}^{ε}, B_{I}^{ε}) (x) . \end{aligned}$ The Euler–Maxwell equation is in fact more stable than the pure Euler system without self-interaction and smooth solutions can exist globally without shock formation [19]. For more references on the analysis of both the uni- and bipolar Euler–Maxwell equation we refer to the literature in [40]. When passing to the unipolar limit from (2a)–(2e), the static ion background is represented in the limit by a background density $b (x)$ which is added as a source term in the Poisson equation, i.e. the first equation in (3d) becomes $\begin{matrix} (4) & \nabla \cdot E^{ε} = ρ^{ε} - b . \end{matrix}$ We omit this background density in the sequel without loss of generality (cp. e.g. [4]).

In the equations above, Maxwell’s equations are given in field form for the electric field $E^{ε} = - \nabla V^{ε} + ε \partial_{t} A^{ε}$ and the magnetic field $B^{ε} = \nabla \times A^{ε}$ where $V^{ε}$ is the electric potential and $A^{ε}$ is the magnetic potential. In the potential formulation, one has to take into account an integration constant which corresponds to a gauge choice. For the Poisswell approximation one chooses Lorentz gauge which is $O (ε)$ $\begin{matrix} (5) & \nabla \cdot A^{ε} + ε \partial_{t} V^{ε} = 0 . \end{matrix}$ Usually the $O (1)$ Coulomb gauge $\begin{matrix} (6) & \nabla \cdot A^{ε} = 0, \end{matrix}$ is used for the Darwin approximation, although omitting the $O (ε)$ term of the relativistic Lorentz gauge is not consistent in an $O (ε^{2})$ model (cf. [2,26]).

As mentioned before, a “consistent” fully relativistic model couples the relativistic Euler equation to the relativistic Maxwell equation: the (unipolar) relativistic Euler–Maxwell equation is given by $\begin{aligned} (7a) & \partial_{t} (γ^{ε} (u^{ε}) ρ^{ε}) + \nabla \cdot (ρ^{ε} γ^{ε} (u^{ε}) u^{ε}) = 0, \\ (7b) & \partial_{t} (γ^{ε} (u^{ε}) u^{ε}) + u^{ε} \cdot \nabla (γ^{ε} (u^{ε}) u^{ε}) = E^{ε} + ε γ^{ε} (u^{ε}) u^{ε} \times B^{ε}, \\ (7c) & ε \partial_{t} E^{ε} - \nabla \times B^{ε} = - ε ρ^{ε} γ^{ε} (u^{ε}) u^{ε}, ε \partial_{t} B^{ε} + \nabla \times E^{ε} = 0, \\ (7d) & \nabla \cdot E^{ε} = ρ^{ε}, \nabla \cdot B^{ε} = 0, \\ (7e) & (ρ^{ε}, u^{ε}, E^{ε}, B^{ε} (x, 0) = (ρ_{I}^{ε}, u_{I}^{ε}, E_{I}^{ε}, B_{I}^{ε}) (x), \end{aligned}$ where the gamma factor is defined as $\begin{matrix} (8) & γ^{ε} (u^{ε}) = \frac{1}{\sqrt{1 - ε^{2} | u^{ε} |^{2}}} . \end{matrix}$ The gamma factor bounds the velocity $u^{ε}$ by $1 / ε = c$ in this scaling, consistent with the finite speed of propagation in special relativity. The gamma factor can be expanded in ε and the expansion up to order $O (ε^{2})$ is given by $\begin{matrix} (9) & γ^{ε} (u^{ε}) = 1 + \frac{1}{2} ε^{2} {| u^{ε} |}^{2} + O (ε^{4}) . \end{matrix}$

The relativistic Euler equation without coupling to electromagnetic field has been introduced by Smoller and Temple in 1993 in [33] and then used by Makino and Ukai in [25]. The relativistic Euler equation coupled to the Poisson equation for the electric potential was considered by plasma physicists in 1994 in [30] and mathematicians in [6,12].

The relativistic Euler–Maxwell equation (7a)–(7e) was introduced and studied by Guo, Ionescu and Pausader in [18]. It would be the consistent model, but in the mathematical community the focus is on the model where the non-relativistic Euler equation is coupled to the Maxwell equations, i.e. the Euler–Maxwell equation (3a)–(3e).

The (pressureless) Euler–Poisswell equation in field form is given by $\begin{aligned} (10a) & \partial_{t} ρ^{ε} + \nabla \cdot (ρ^{ε} u^{ε}) = 0, \\ (10b) & \partial_{t} u^{ε} + u^{ε} \cdot \nabla u^{ε} = E^{ε}, \\ (10c) & \nabla \cdot E^{ε} = ρ^{ε}, \nabla \times E^{ε} = 0, \\ (10d) & \nabla \cdot B^{ε} = 0, ε \partial_{t} E^{ε} - \nabla \times B^{ε} = - ε ρ^{ε} u^{ε}, \\ (10e) & (ρ^{ε}, u^{ε}) (x, 0) = (ρ_{I}^{ε}, u_{I}^{ε}) (x) . \end{aligned}$

Taking the curl of the second equation in (10d), using $\nabla \times (\nabla \times B^{ε}) = \nabla (\nabla \cdot B^{ε}) - Δ B^{ε}$ and $\nabla \times E^{ε} = 0$ we obtain $\begin{matrix} (11) & - Δ B^{ε} = \nabla \times (ε ρ^{ε} u^{ε}) . \end{matrix}$ The Euler–Poisswell equation (10a)–(10e) is the $O (ε)$ semi-relativistic approximation of the Euler–Maxwell equation and a semi-relativistic self-consistent model of a system (plasma) of charged particles with self-interaction with the electromagnetic field. The Poisswell equations (10c)–(10d) are the $O (ε)$ approximation of the Maxwell’s equations.

The pressureless Euler–Poisswell equation in potential form is given by $\begin{aligned} (12a) & \partial_{t} ρ^{ε} + \nabla \cdot (ρ^{ε} u^{ε}) = 0, \\ (12b) & \partial_{t} u^{ε} + u^{ε} \cdot \nabla u^{ε} = - \nabla V^{ε}, \\ (12c) & - Δ V^{ε} = ρ^{ε}, \\ (12d) & - Δ A^{ε} = ε ρ^{ε} u^{ε}, \\ (12e) & ε \partial_{t} V^{ε} + \nabla \cdot A^{ε} = 0, \\ (12f) & (ρ^{ε}, u^{ε}) (x, 0) = (ρ_{I}^{ε}, u_{I}^{ε}) (x) . \end{aligned}$ In the $O (ε)$ approximation the electric field and magnetic fields are given by $\begin{aligned} (13) & E^{ε} = - \nabla V^{ε}, B^{ε} = \nabla \times A^{ε} \end{aligned}$ instead of $E^{ε} = - \nabla V^{ε} + ε \partial_{t} A^{ε}$ since the $O (1)$ component of the magnetic potential vanishes, i.e $A_{0} = 0$ , cf. Section 3. Using (13), the Lorentz gauge (5) and $\nabla \times (\nabla \times A^{ε}) = \nabla (\nabla \cdot A^{ε}) - Δ A^{ε}$ , we obtain the Euler–Poisswell equation in field form (10a)–(10e).

In the Poisswell approximation the $O (1)$ part $B_{0}$ of the magnetic field vanishes since it has to satisfy (cf. Section 3) $\begin{matrix} (14) & \nabla \cdot B_{0} = \nabla \times B_{0} = 0, \end{matrix}$ i.e. $B_{0}$ is irrotational and solenoidal. However $B^{ε}$ should also decay at infinity which implies that $B_{0} \equiv 0$ (cf. Section 3). The first non-zero term in the expansion for $B^{ε}$ is then the $O (ε)$ term $B_{1}$ . In the Lorentz force $\begin{matrix} (15) & E^{ε} + ε u^{ε} \times B^{ε}, \end{matrix}$ the magnetic part scales with ε and thus the first non-zero term magnetic term is $O (ε^{2})$ . Therefore, the Lorentz force does not appear in the $O (ε)$ approximation and therefore also does not appear in the Euler–Poisswell equation. More precisely, $\begin{matrix} (16) & E^{ε} + ε u^{ε} \times B^{ε} = - \nabla V^{ε} + O (ε^{2}) . \end{matrix}$ So the Lorentz force reduces to the electric force at first order in ε.

This has important consequences: The magnetic field completely decouples from the fluid dynamics, i.e. (10a)–(10b) is not coupled to B and (12a)–(12b) is not coupled to (12d), the Poisson equation for the magnetic potential $A^{ε}$ . The first time the magnetic field acts nontrivially on the fluid is in the $O (ε^{2})$ approximation.

The correct $O (ε^{2})$ equation, which is the order where the full Lorentz force appears for the first time, is the pressureless Euler–Darwin equation in field form, $\begin{aligned} (17a) & \partial_{t} ρ^{ε} + \nabla \cdot (ρ^{ε} u^{ε}) = 0, \\ (17b) & \partial_{t} u^{ε} + u^{ε} \cdot \nabla u^{ε} = E^{ε} + ε u^{ε} \times B^{ε}, \\ (17c) & \nabla \cdot E_{L}^{ε} = ρ^{ε}, \nabla \times E_{L}^{ε} = 0, \nabla \cdot E_{T}^{ε} = 0, \nabla \cdot B^{ε} = 0, \\ (17d) & ε \partial_{t} B^{ε} + \nabla \times E_{T}^{ε} = 0, ε \partial_{t} E_{L}^{ε} - \nabla \times B^{ε} = - ε ρ^{ε} u^{ε}, \\ (17e) & E^{ε} = E_{T}^{ε} + E_{L}^{ε}, \\ (17f) & (ρ^{ε}, u^{ε}) (x, 0) = (ρ_{I}^{ε}, u_{I}^{ε}) (x) . \end{aligned}$ Here $E_{T}^{ε}$ stands for the transversal or solenoidal part and $E_{L}^{ε}$ stands for the longitudinal or irrotational part. In the definition of the electric field we have $\begin{aligned} (18) & E_{L}^{ε} = - \nabla V^{ε}, E_{T}^{ε} = ε \partial_{t} A^{ε} . \end{aligned}$ By taking the curl of the first equation (Faraday’s law) and the time derivative of the second equation (Ampère’s law) in (17d) we obtain an elliptic equation for $E_{T}^{ε}$ : $\begin{matrix} (19) & - Δ E_{T}^{ε} = - ε^{2} \partial_{t} {(ρ^{ε} u^{ε})}_{T}, \end{matrix}$ where $\begin{matrix} (20) & \partial_{t} {(ρ^{ε} u^{ε})}_{T} = \partial_{t} (ρ^{ε} u^{ε}) + \partial_{t}^{2} E_{L}^{ε} \end{matrix}$ with the subscript ${(\cdot)}_{T}$ denotes the transversal or solenoidal, i.e. divergence free part, cf. [24]. In view of the boundary condition ${lim}_{| x | \to \infty} E^{ε} = 0$ this decomposition is unique.

Choosing Coulomb gauge (21e) (cf. [2,24]) the Euler–Darwin equation in potential form is given by $\begin{aligned} (21a) & \partial_{t} ρ^{ε} + \nabla \cdot (ρ^{ε} u^{ε}) = 0, \\ (21b) & \partial_{t} u^{ε} + u^{ε} \cdot \nabla u^{ε} = - \nabla V^{ε} + ε \partial_{t} A^{ε} + ε (u^{ε} \times (\nabla \times A^{ε})), \\ (21c) & - Δ V^{ε} = ρ^{ε}, \\ (21d) & - Δ A^{ε} = ε {((ρ^{ε} u^{ε}))}_{T}, \\ (21e) & \nabla \cdot A^{ε} = 0, \\ (21f) & (ρ^{ε}, u^{ε}) (x, 0) = (ρ_{I}^{ε}, u_{I}^{ε}) (x) . \end{aligned}$

We make the important observation that the Euler–Darwin model includes a non-trivial interaction of the fluid with the magnetic field via the full Lorentz force while maintaining the elliptic nature of the equations for the potentials (21c)–(21d).

Remark 2.
Starting from the relativistic Euler–Maxwell equation (7a)–(7e) and using the $O (ε^{2})$ expansion of the gamma factor, i.e. (9), one obtains the relativistic Euler–Darwin equation, which would be a more consistent model than (17a)–(17f) and which we include for completeness. The relativistic Euler–Darwin equation in field form is given by $\begin{aligned} (22a) & (1 + \frac{1}{2} ε^{2} {| u^{ε} |}^{2}) (\partial_{t} ρ^{ε} + \nabla \cdot (ρ^{ε} u^{ε})) + ε^{2} (ρ^{ε} u^{ε}) (\partial_{t} u^{ε} + u^{ε} \cdot \nabla u^{ε}) = 0, \\ (22b) & (1 + \frac{1}{2} ε^{2} {| u^{ε} |}^{2}) (\partial_{t} u^{ε} + u^{ε} \cdot \nabla u^{ε}) + ε^{2} u^{ε} \cdot ((u^{ε} \cdot \nabla u^{ε}) u^{ε}) = E^{ε} + ε u^{ε} \times B^{ε}, \\ (22c) & \nabla \cdot E_{L}^{ε} = ρ^{ε}, \nabla \times E_{L}^{ε} = 0, \nabla \cdot E_{T}^{ε} = 0, \nabla \cdot B^{ε} = 0, \\ (22d) & ε \partial_{t} B^{ε} + \nabla \times E_{T}^{ε} = 0, ε \partial_{t} E_{L}^{ε} - \nabla \times B^{ε} = - ε ρ^{ε} u^{ε}, \\ (22e) & E^{ε} = E_{T}^{ε} + E_{L}^{ε}, \\ (22f) & (ρ^{ε}, u^{ε}) (x, 0) = (ρ_{I}^{ε}, u_{I}^{ε}) (x) . \end{aligned}$ Note that the Lorentz force and the second equation in (22d) do not have a factor $1 + (1 / 2) ε^{2} | u^{ε} |^{2}$ because of the ε in front of the magnetic force and the current, which makes these terms $O (ε^{3})$ and thus they should be dropped in a $O (ε^{2})$ approximation.

The non-relativistic limit $c \to \infty$ of the Euler–Maxwell equation (3a)–(3e) to the Euler–Poisson equation was shown in [31,36]. The (repulsive) pressureless Euler–Poisson equation is given by $\begin{aligned} (23a) & \partial_{t} ρ + \nabla \cdot (ρ u) = 0, \\ (23b) & \partial_{t} (u) + u \cdot \nabla u = - \nabla V, \\ (23c) & (ρ, u) (x, 0) = (ρ_{I}, u_{I}) (x), \end{aligned}$ where $\begin{aligned} (24) & - Δ V & = ρ . \end{aligned}$ The Euler–Poisson equation also arises as the semiclassical limit $ℏ \to 0$ of vanishing Planck constant using WKB analysis of the Schrödinger–Poisson equation (cf. Zhang [38,39] and Alazard and Carles [1]). This is a more general behavior for Schrödinger-type equations, which converge to Euler-type equations by WKB analysis (on the other hand, Wigner analysis leads to Vlasov-type equations) and which is related to the Madelung transform and the hydrodynamic formulation of the Schrödinger equation [5]. Grenier [15] proved the semiclassical limit of the cubic NLS and recently, Gui and Zhang [16] proved the semiclassical limit of the Gross–Pitaevskii equation.

The semi-relativistic Pauli–Poisswell equation converges to the Euler–Poisswell equation (10a)–(10e) in the semiclassical limit via the WKB method which is shown in [35]. For more details see Section 5.

The convergence of the Dirac–Maxwell to the Euler–Maxwell equation (3a)–(3e) is an open question.

Recently, Golse and Paul [14] showed that the bosonic N-particle Schrödinger dynamics with Coulomb interaction converge to the Euler–Poisson equation in the combined semiclassical and mean field limit $ℏ + 1 / N \to 0$ if the first marginal of the initial data has monokinetic Wigner measure which is equivalent to the WKB formulation.

Depending on the sign in front of the term $ρ \nabla V$ on the LHS of the second equation in (23a) the Euler–Poisson equation is called repulsive (with a negative sign) or attractive (with a positive sign). In other words, the Euler–Poisson equation is repulsive if the sign on the LHS of the second equation in (23a) and the sign on the RHS of the Poisson equation (24) are the same and attractive if they are opposite.

In the repulsive case the Euler–Poisson equation is a non-relativistic self-consistent model (for the self-interaction with the electric field, coupled via the Poisson equation (24)) for a system (plasma) of charged particles of charge of the same sign (e.g. electrons). This is because the Coulomb interaction is repulsive for charges of the same sign.

In the attractive case the Euler–Poisson equation is a self-consistent model for non-relativistic gravitational interaction, e.g. gaseous stars, which is attractive.

The analytical properties of the Euler–Poisson equation depend heavily on the sign. The repulsive case is less delicate than the attractive case: The dispersive nature of the electric field may prevent the formation of singularities [20]. In $3 d$ Guo and Pausader [17,20] proved the global existence of small smooth irrotational flows. Recently, Hadžić and Jang constructed global solutions for both the gravitational and electrostatic Euler–Poisson equation in [21].

However for Euler–Poisson it is also possible to have blow-up even for the repulsive case, e.g. [37] or even non-existence, e.g. [32] for the $1 d$ case. Further references can also be found in [11].

In the Euler–Maxwell and Euler–Darwin equations there is no distinction between repulsive or attractive interaction as for the Euler–Poisson equation (which describes either the attractive gravitational or the repulsive electrostatic self-interaction depending on the sign) since the force described by Maxwell’s equations is given by the Lorentz force $E^{ε} + ε u^{ε} \times B^{ε}$ (in the non-relativistic Euler–Poisson equation and in the semi-relativistic Euler–Poisswell equation it reduces to the electrostatic force).

The Euler–Lorentz equation is a $O (1)$ model with given electromagnetic fields E, B. Here, the Lorentz force acts as the driving force in the Euler equation. It was studied by Degond et al. in [3,8] and is given by $\begin{aligned} (25a) & \partial_{t} ρ + \nabla \cdot (ρ u) = 0, \\ (25b) & \partial_{t} (u) + u \cdot \nabla u = E + u \times B, \\ (25c) & (ρ, u) (x, 0) = (ρ_{I}, u_{I}) (x) . \end{aligned}$ Note that this model is not self-consistent and not consistent in the small parameter ε since it contains both $O (1)$ and $O (ε)$ and $O (ε^{2})$ terms (the Lorentz force is $O (ε^{2})$ as discussed in equations (15)–(16)).
3. Formal asymptotic expansion in $ε = 1 / c$

In this section we consider the asymptotic expansion of the Euler–Maxwell equation (3a)–(3e). In accordance with most mathematical work, we do not deal with the “consistent” relativistic Euler–Maxwell equation (7a)–(7e).

Let us suppose that the initial data of the unipolar Euler–Maxwell equation satisfy $\begin{matrix} (26) & (ρ_{I}^{ε} (x), u_{I}^{ε} (x), E_{I}^{ε} (x), B_{I}^{ε} (x)) = \sum_{j = 0}^{m} ε^{j} (ρ_{I, j} (x), u_{I, j} (x), E_{I, j} (x), B_{I, j} (x)) + O (ε^{m + 1}) . \end{matrix}$ We take the ansatz for the solutions of the Euler–Maxwell equation $\begin{matrix} (27) & (ρ^{ε} (x), u^{ε} (x), E^{ε} (x), B^{ε} (x)) = \sum_{j ⩾ 0} ε^{j} (ρ_{j} (x), u_{j} (x), E_{j} (x), B_{j} (x)) . \end{matrix}$ In order to find the $O (ε)$ asymptotic expansion which yields the Euler–Poisswell equation we make the ansatz $\begin{aligned} (28) & \begin{aligned} (ρ^{ε} (x), u^{ε} (x), E^{ε} (x), B^{ε} (x)) & = (ρ_{0} (x), u_{0} (x), E_{0} (x), B_{0} (x)) \\ + ε (ρ_{1} (x), u_{1} (x), E_{1} (x), B_{1} (x)) \\ + O (ε^{2}) . \end{aligned} \end{aligned}$ We also make an ansatz for the potentials: $\begin{aligned} (29) & A^{ε} = A_{0} + ε A_{1} + O (ε^{2}), V^{ε} = V_{0} + ε V_{1} + O (ε^{2}) . \end{aligned}$ The leading order equation is given by the Euler–Poisson equation, $\begin{aligned} (30a) & \partial_{t} ρ_{0} + \nabla \cdot (ρ_{0} u_{0}) = 0, \\ (30b) & \partial_{t} u_{0} + u_{0} \cdot \nabla u_{0} = E_{0}, \\ (30c) & \nabla \times B_{0} = 0, \nabla \times E_{0} = 0, \\ (30d) & \nabla \cdot E_{0} = ρ_{0}, \nabla \cdot B_{0} = 0, \\ (30e) & (ρ_{0}, u_{0}) (x, 0) = (ρ_{I, 0}, u_{I, 0}) (x) . \end{aligned}$ For the Poisswell equations we choose the Lorentz gauge which reads $\begin{matrix} (31) & ε \partial_{t} V^{ε} + \nabla \cdot A^{ε} = 0 \end{matrix}$ and becomes $\begin{matrix} (32) & \nabla \cdot A_{0} = 0 \end{matrix}$ at $O (1)$ .

Now $\nabla \times E_{0} = 0$ together with $\nabla \cdot E_{0} = ρ_{0}$ implies that $E^{ε} = E_{0}$ is the gradient of a potential $V^{ε}$ which obeys $\begin{matrix} (33) & - Δ V^{ε} = ρ^{ε} . \end{matrix}$

The two equations $\nabla \cdot B_{0} = 0$ and $\nabla \times B_{0} = 0$ in the zeroth order equation yield that $B_{0}$ is both solenoidal and irrotational. Together with the requirement that ${lim}_{| x | \to \infty} B_{0} (x) = 0$ this implies that $B_{0} \equiv 0$ . Indeed, the curl condition implies that $B_{0}$ is the gradient of a harmonic function (by the divergence condition). However, by the maximum principle, such a function must vanish everywhere. By (32), the same holds for the magnetic potential at $O (1)$ , i.e. $A_{0} = 0$ .

The $O (ε)$ equation is given by, $\begin{aligned} (34a) & \partial_{t} ρ_{1} + \nabla \cdot (ρ_{0} u_{1}) + \nabla \cdot (ρ_{1} u_{0}) = 0, \\ (34b) & \partial_{t} u_{1} + u_{0} \cdot \nabla u_{1} + u_{1} \cdot \nabla u_{0} = E_{1}, \\ (34c) & \partial_{t} E_{0} - \nabla \times B_{1} = ρ_{0} u_{0}, \nabla \times E_{1} = 0, \\ (34d) & \nabla \cdot E_{1} = ρ_{1}, \nabla \cdot B_{1} = 0, \\ (34e) & (ρ_{1}, u_{1}) (x, 0) = (ρ_{I, 1}, u_{I, 1}) (x) . \end{aligned}$ At $O (ε)$ the Lorentz gauge becomes $\begin{matrix} (35) & \partial_{t} V_{0} + \nabla \cdot A_{1} = 0 . \end{matrix}$ Equation (35) and the first equation in (34c) yield that $A^{ε} = ε A_{1} + O (ε^{2})$ satisfies $\begin{matrix} (36) & - Δ A^{ε} = ε (ρ^{ε} u^{ε}) . \end{matrix}$ These observations yield the Euler–Poisswell equation in the field form (10a)–(10e) and in the potential form (12a)–(12f).

We now turn to the $O (ε^{2})$ equation in order to formally derive the Euler–Darwin equation. We make the additional assumption $\begin{matrix} (37) & ρ_{j} = 0, j ⩾ 1 . \end{matrix}$ Or equivalently, $\begin{matrix} (38) & ρ^{ε} = ρ_{0} . \end{matrix}$ The ansatz becomes $\begin{aligned} (39) & \begin{aligned} (ρ^{ε} (x), u^{ε} (x), E^{ε} (x), B^{ε} (x)) & = (ρ_{0} (x), u_{0} (x), E_{0} (x), B_{0} (x)) \\ + ε (0, u_{1} (x), E_{1} (x), B_{1} (x)) \\ + ε^{2} (0, u_{2} (x), E_{2} (x), B_{2} (x)) \\ + O (ε^{3}) . \end{aligned} \end{aligned}$ This assumption is necessary in order to arrive at the form of the Darwin approximation (17c)–(17d), see for instance the analysis of Degond and Raviart in [10]. For the potentials we make the ansatz: $\begin{aligned} (40) & A^{ε} = A_{0} + ε A_{1} + ε^{2} A_{2} + O (ε^{3}), V^{ε} = V_{0} + ε V_{1} + ε^{2} V_{2} + O (ε^{3}) . \end{aligned}$ Then the $O (ε^{2})$ equation is given by $\begin{aligned} (41a) & \nabla \cdot (ρ_{0} u_{2}) = 0, \\ (41b) & \partial_{t} u_{2} + u_{0} \cdot \nabla u_{2} + u_{2} \cdot \nabla u_{0} + u_{1} \cdot \nabla u_{1} = E_{2} + (u_{0} \times B_{1}), \\ (41c) & \partial_{t} E_{1} - \nabla \times B_{2} = ρ_{0} u_{1} + ρ_{1} u_{0}, \partial_{t} B_{1} + \nabla \times E_{2} = 0, \\ (41d) & \nabla \cdot E_{2} = 0, \nabla \cdot B_{2} = 0, \\ (41e) & (ρ_{2}, u_{2}) (x, 0) = (ρ_{I, 2}, u_{I, 0}) (x) . \end{aligned}$ Here, due to the fact that $ρ_{1} = 0$ we have that $\nabla \cdot E_{1} = 0$ and $\nabla \times E_{1} = 0$ together with ${lim}_{| x | \to \infty} E_{1} (x) = 0$ this implies that $E_{1} \equiv 0$ . The electric field is now given by $E^{ε} = E_{0} + ε^{2} E_{2}$ where $\nabla \times E_{0} = 0$ and $\nabla \cdot E_{2} = 0$ . Therefore we can define the irrotational part $E_{L}^{ε} : = E_{0}$ and the solenoidal part $E_{T}^{ε} : = ε^{2} E_{2}$ and we recover (17a)–(17f).

The potential form of the Darwin approximation is then obtained by considering Maxwell’s equations for the potentials in Coulomb gauge $\begin{array}{r} (Δ - ε^{2} \partial_{t}^{2}) A^{ε} = - ε (ρ^{ε} u^{ε}) + ε \partial_{t} \nabla V^{ε}, Δ V^{ε} = - ρ^{ε} . \end{array}$ Since $\nabla \times (\nabla V^{ε}) = 0$ , the second term in the wave equation for $A^{ε}$ is rotation-free. Since any vector field can be decomposed into a transversal, divergence-free and a longitudinal, irrotational part we can write $\begin{array}{r} (Δ - ε^{2} \partial_{t}^{2}) A^{ε} = - ε {(ρ^{ε} u^{ε})}_{T}, Δ V^{ε} = - ρ^{ε} . \end{array}$ Using the ansatz for the asymptotic expansion and plugging it into Maxwell’s equations, and using the Coulomb gauge and the field equations: $\begin{aligned} B_{0} = 0, E_{1} = 0, \\ V_{1} = 0, A_{0} = 0, \end{aligned}$ and $\begin{array}{r} Δ V_{0} = - ρ^{ε}, Δ A_{1} = - (ρ^{ε} u^{ε}) + \partial_{t} \nabla V_{0}, \end{array}$ as well as $\begin{matrix} div A_{1} = 0 . \end{matrix}$ Again, $(ρ^{ε} u^{ε}) - \partial_{t} \nabla V_{0}$ is the divergence-free part of $ρ^{ε} u^{ε}$ , so $\begin{array}{r} - Δ V^{ε} = ρ^{ε}, - Δ A^{ε} = ε {(ρ^{ε} u^{ε})}_{T} . \end{array}$ So we obtain a magnetostatic approximation of Maxwell’s equations with a divergence free current as source term.

Remark 3.
Note that for the derivation of the Euler–Poisswell equation we need not assume (37)–(38). It is necessary for the Euler–Darwin equation because otherwise the term $E_{2}$ would not be solenoidal since the first equation in (41d) would become $\nabla \cdot E_{2} = ρ_{2}$ .

4. Local/global wellposedness of the Euler–Poisswell equation

The Euler–Poisswell equation (12a)–(12f) has local solutions in $H^{s} (R^{3}) \times H^{s - 1} (R^{3})$ which obey a blow-up alternative.

Theorem 1 ([35]).

Let $s > 7 / 2$ and let $\begin{matrix} {‖ u_{I}^{ε} ‖}_{H^{s}} + {‖ ρ_{I}^{ε} ‖}_{H^{s}} ⩽ Q . \end{matrix}$ There exists a unique local solution $(ρ, u) \in C ([0, T^{0}), H^{s - 1} (R^{3})) \times C ([0, T^{0}), H^{s} (R^{3}))$ to the Euler–Poisswell equation ( 12a )–( 12f ) with initial data $(ρ_{I}^{ε}, u_{I}^{ε}) \in H^{s} \times H^{s - 1}$ . Here $T^{0} > 0$ is the maximal (possibly infinite) time of existence.

The time of existence is either global, or finite time blow up occurs. More precisely, if $T^{0} < \infty$ , then the solution $(ρ^{ε}, u^{ε})$ of ( 12a )–( 12f ) blows up at T such that $\begin{matrix} lim_{t ↑ T^{0}} ({‖ ρ^{ε} (t) ‖}_{L^{\infty}} + {‖ u^{ε} (t) ‖}_{W^{1, \infty}}) = \infty . \end{matrix}$

The proof consists of a standard fixed point argument using a priori estimates where the Poisson equations for A and V are taken into account. The regularity $s > 7 / 2$ is necessary due to Sobolev’s embedding. In fact, this result follows from a more general local wellposedness result for the Pauli–Poisswell–WKB equation which can be found in [35].

Remark 4.
The local wellposedness result above should hold analogously for the Euler–Darwin equation, since in [35] the $O (ε^{2})$ terms are partially kept in the Poisswell model (cf. the discussion at the end of Section 5) and the proof should be easily adapted for the Euler–Darwin equation. This will be discussed in future work.

5. Euler–Poisswell as the semiclassical limit of Pauli–Poisswell

The self-consistent Pauli–Poisswell equation arises as the semi-relativistic approximation at first order $O (1 / c)$ of the fully relativistic Dirac–Maxwell equation for a 4-spinor (cf. [26]) and is the correct semi-relativistic $O (1 / c)$ self-consistent model for charged spin- $1 / 2$ particles with self-interaction with the electromagnetic field where the electromagnetic potentials $V^{ℏ, ε}$ and $A^{ℏ, ε}$ , which now depend on the semiclassical parameter ℏ (Planck constant), are given by the magnetostatic (i.e. slow in the intermediate regime with respect to c) description via $1 + 3$ Poisson equations with density and current density as source terms.

The Pauli–Poisswell equation for a 2-spinor $ψ^{ℏ, ε} (x, t) \in {(L^{2} (R^{3} \times R, C))}^{2}$ and the electromagnetic potentials $A^{ℏ, ε} : R^{3} \to R^{3}$ and $V^{ℏ, ε} : R^{3} \to R$ (cf. [26]) $\begin{array}{c} (42) & i ℏ \partial_{t} ψ^{ℏ, ε} = - \frac{1}{2} {(ℏ \nabla + i ε A^{ℏ, ε})}^{2} ψ^{ℏ, ε} + V^{ℏ, ε} ψ^{ℏ, ε} - \frac{1}{2} ℏ ε (σ \cdot B^{ℏ, ε}) ψ^{ℏ, ε}, \\ (43) & - Δ V^{ℏ, ε} = ρ^{ℏ, ε} : = {| ψ^{ℏ, ε} |}^{2}, \\ (44) & - Δ A^{ℏ, ε} = ε J^{ℏ, ε}, \end{array}$ where the Pauli current density is given by $\begin{matrix} (45) & J^{ℏ, ε} (ψ^{ℏ, ε}, A^{ℏ, ε}) = ℑ (\overline{ψ^{ℏ, ε}} (ℏ \nabla + i ε A^{ℏ, ε}) ψ^{ℏ, ε}) - ℏ \nabla \times (\overline{ψ^{ℏ, ε}} σ ψ^{ℏ, ε}), \end{matrix}$ with initial data $\begin{matrix} (46) & ψ^{ℏ, ε} (x, 0) = ψ_{I}^{ℏ, ε} (x) \in {(L^{2} (R^{3}))}^{2} . \end{matrix}$ The Stern–Gerlach term is $σ \cdot B^{ℏ, ε} : = \sum_{k = 1}^{3} σ_{k} B_{k}^{ℏ, ε}$ where the $σ_{k}$ are the Pauli matrices $\begin{aligned} (47) & σ_{1} = (\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}), σ_{2} = (\begin{array}{cc} 0 & - i \\ i & 0 \end{array}), σ_{3} = (\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array}) . \end{aligned}$ The expressions $\overline{ψ^{ℏ, ε}} \nabla ψ^{ℏ, ε}$ and $\overline{ψ^{ℏ, ε}} σ ψ^{ℏ, ε}$ are to be understood as the vectors with components $\begin{array}{r} (\overline{ψ_{1}^{ℏ, ε}}, \overline{ψ_{2}^{ℏ, ε}}) \partial_{k} (\begin{array}{c} ψ_{1}^{ℏ, ε} \\ ψ_{2}^{ℏ, ε} \end{array}), (\overline{ψ_{1}^{ℏ, ε}}, \overline{ψ_{2}^{ℏ, ε}}) σ_{k} (\begin{array}{c} ψ_{1}^{ℏ, ε} \\ ψ_{2}^{ℏ, ε} \end{array}) . \end{array}$ The WKB ansatz consists of assuming that the initial data $ψ_{I}^{ℏ, ε}$ of the Pauli–Poisswell equation are of the form $\begin{matrix} (48) & ψ_{I}^{ℏ, ε} (x) = a_{I}^{ℏ, ε} (x) e^{\frac{i}{ℏ} S_{I}^{ℏ, ε} (x)}, \end{matrix}$ where $a_{I}^{ℏ, ε}$ is the initial amplitudes of the 2-spinor. The initial phase $S_{I}^{ℏ, ε}$ , which we choose to be the same for both components, is real-valued. This “multicomponent WKB ansatz” is also usually chosen by physicists (see e.g. [23,34]) and is used in order to avoid oscillatory cross terms. The gradient of $S_{I}^{ℏ, ε}$ is the initial velocity $u^{ℏ_{I}} : = \nabla S_{I}^{ℏ, ε}$ . One then expects that at least for short times the solution $ψ^{ℏ, ε}$ will be of the form $\begin{matrix} (49) & ψ^{ℏ, ε} (x, t) = a^{ℏ, ε} (x, t) e^{\frac{i}{ℏ} S^{ℏ, ε} (x, t)}, \end{matrix}$ where $a^{ℏ, ε}$ is the amplitude and $S^{ℏ, ε}$ is the phase. The velocity $u^{ℏ, ε}$ is defined as the gradient of the phase, $u^{ℏ, ε} : = \nabla S^{ℏ, ε}$ .

Substituting $ψ^{ℏ, ε}$ given by (49) into (42) yields the following system: $\begin{matrix} (50) & \begin{aligned} \partial_{t} a^{ℏ, ε} + ε A^{ℏ, ε} \cdot \nabla a^{ℏ, ε} + \nabla a^{ℏ, ε} \cdot \nabla S^{ℏ, ε} + \frac{1}{2} a^{ℏ, ε} (Δ S^{ℏ, ε} + \nabla \cdot ε A^{ℏ, ε}) \\ = \frac{i ℏ}{2} Δ a^{ℏ, ε} + \frac{i ε}{2} (σ \cdot B^{ℏ, ε}) a^{ℏ, ε}, \\ \partial_{t} S^{ℏ, ε} + \frac{1}{2} {| \nabla S^{ℏ, ε} |}^{2} + ε A^{ℏ, ε} \cdot \nabla S^{ℏ, ε} + (\frac{| ε A^{ℏ, ε} |^{2}}{2} + V^{ℏ, ε}) = 0, \\ (a^{ℏ, ε}, S^{ℏ, ε}) (x, 0) = (a_{I}^{ℏ, ε}, S_{I}^{ℏ, ε}) (x) . \end{aligned} \end{matrix}$ Let now $u^{ℏ, 0} = \nabla S^{ℏ, 0}$ and $u^{ℏ, ε} = \nabla S^{ℏ, ε}$ be the initial velocity and velocity, respectively. By differentiating the second equation in (50) we obtain the Pauli–Poisswell–WKB equation $\begin{aligned} (51) & \begin{aligned} \partial_{t} a^{ℏ, ε} + (u^{ℏ, ε} + ε A^{ℏ, ε}) \cdot \nabla a^{ℏ, ε} + \frac{1}{2} a^{ℏ, ε} \nabla \cdot (u^{ℏ, ε} + ε A^{ℏ, ε}) = \frac{i ℏ}{2} Δ a^{ℏ, ε} + \frac{i ε}{2} (σ \cdot B^{ℏ, ε}) a^{ℏ, ε}, \\ \partial_{t} u^{ℏ, ε} + (u^{ℏ, ε} + ε A^{ℏ, ε}) \cdot \nabla u^{ℏ, ε} + u^{ℏ, ε} \cdot \nabla ε A^{ℏ, ε} + \nabla (\frac{| ε A^{ℏ, ε} |^{2}}{2} + V^{ℏ, ε}) = 0, \\ (a^{ℏ, ε}, u^{ℏ, ε}) (x, 0) = (a_{I}^{ℏ, ε}, u_{I}^{ℏ, ε}) (x), \end{aligned} \end{aligned}$ where $\begin{array}{c} (52) & - Δ V^{ℏ, ε} = ρ^{ℏ, ε}, \\ (53) & - Δ A^{ℏ, ε} = ε (ℏ w^{ℏ, ε} - ℏ v^{ℏ, ε} + ρ^{ℏ, ε} u^{ℏ, ε} + ε ρ^{ℏ, ε} A^{ℏ, ε}) \\ (54) & = ε J^{ℏ, ε} \end{array}$ with $\begin{array}{c} (55) & ρ^{ℏ, ε} : = {| a^{ℏ, ε} |}^{2}, \\ (56) & v^{ℏ, ε} : = \frac{1}{2} \nabla \times {(\overline{a^{ℏ, ε}} σ_{1} a^{ℏ, ε}, \overline{a^{ℏ, ε}} σ_{2} a^{ℏ, ε}, \overline{a^{ℏ, ε}} σ_{3} a^{ℏ, ε})}^{T}, \\ (57) & w^{ℏ, ε} : = \frac{i}{2} \sum_{j = 1}^{2} (\overline{a_{j}^{ℏ, ε}} \nabla a_{j}^{ℏ, ε} - a_{j}^{ℏ, ε} \nabla \overline{a_{j}^{ℏ, ε}}) . \end{array}$

Remark 5.
Note that in the WKB ansatz we choose a complex-valued amplitude as opposed to the Madelung transform where one uses a real-valued amplitude. This additional degree of freedom allows us to avoid singular quantum pressure in the equation for the velocity $u^{ℏ, ε}$ . Instead one obtains a skew-symmetric term in the transport equation (due to the term $(i / ℏ) / 2 Δ a^{ℏ, ε}$ ). For a detailed discussion of WKB vs. Madelung transform and its connection to the hydrodynamic formulation of nonlinear Schrödinger equations we refer to the survey by Carles, Danchin and Saut in [5].

Formally passing to the limit $ℏ \to 0$ in system (51) we recover the following form of Euler–Poisswell equation, $\begin{array}{c} (58a) & \partial_{t} a^{ε} + (u^{ε} + ε A^{ε}) \cdot \nabla a^{ε} + \frac{1}{2} a^{ε} \nabla \cdot (u^{ε} + ε A^{ε}) = \frac{i ε}{2} (σ \cdot B^{ε}) a^{ε}, \\ (58b) & \partial_{t} u^{ε} + (u^{ε} + ε A^{ε}) \cdot \nabla u^{ε} + u^{ε} \cdot \nabla ε A^{ε} + \nabla (\frac{| ε A^{ε} |^{2}}{2} + V^{ε}) = 0, \\ (58c) & (a^{ε}, u^{ε}) (x, 0) = (a_{I}^{ε}, u_{I}^{ε}) (x), \end{array}$ where $\begin{array}{c} (59) & - Δ V^{ε} = ρ^{ε}, \\ (60) & - Δ A^{ε} = ε (ρ^{ε} u^{ε} + ε ρ^{ε} A^{ε}), \end{array}$ and $\begin{matrix} ρ^{ε} = {| a^{ε} |}^{2} . \end{matrix}$ Multiplying the first equation in (58a) by $\overline{a^{ε}}$ and taking the real part yields $\begin{aligned} (61a) & \partial_{t} ρ^{ε} + \nabla \cdot (ρ (u^{ε} + ε A^{ε})) = 0, \\ (61b) & \partial_{t} u + (u^{ε} + ε A^{ε}) \cdot \nabla u^{ε} + u^{ε} \cdot \nabla ε A^{ε} + \nabla (\frac{| ε A^{ε} |^{2}}{2} + V^{ε}) = 0, \\ (61c) & (ρ^{ε}, u^{ε}) (x, 0) = (ρ_{I}^{ε}, u_{I}^{ε}) (x), \end{aligned}$ and $\begin{array}{c} (62) & - Δ V^{ε} = ρ^{ε}, \\ (63) & - Δ A^{ε} = ε (ρ^{ε} u^{ε} + ε ρ^{ε} A^{ε}), \end{array}$ since $ℜ (i \overline{a^{ε}} (σ \cdot B^{ε}) a^{ε}) = 0$ .

The form (51)–(57) is used in [35], however in order to arrive at (12a)–(12f) we have to transform (50) to arrive at the correct equation. We switch to Einstein notation and differentiate the second equation in (50): $\begin{matrix} \partial_{t} \partial_{i} S^{ℏ, ε} + \partial_{i} \partial_{k} S^{ℏ, ε} \partial_{k} S^{ℏ, ε} + ε \partial_{i} A_{k}^{ℏ, ε} \partial_{k} S^{ℏ, ε} + ε A_{k}^{ℏ, ε} \partial_{i} \partial_{k} S^{ℏ, ε} + ε^{2} \partial_{i} A_{k}^{ℏ, ε} A_{k}^{ℏ, ε} + \partial_{i} V^{ℏ, ε} = 0 . \end{matrix}$ Setting $v_{i}^{ℏ, ε} : = \partial_{i} S^{ℏ, ε}$ . We obtain $\begin{matrix} \partial_{t} v_{i}^{ℏ, ε} + \partial_{i} v_{k}^{ℏ, ε} (v_{k}^{ℏ, ε} + ε A_{k}^{ℏ, ε}) + ε \partial_{i} A_{k}^{ℏ, ε} (v_{k}^{ℏ, ε} + ε A_{k}^{ℏ, ε}) + \partial_{i} V^{ℏ, ε} = 0 . \end{matrix}$ Now set $u_{i}^{ℏ, ε} = v_{i}^{ℏ, ε} + ε A_{k}^{ℏ, ε}$ which yields $\begin{matrix} \partial_{t} u_{i}^{ℏ, ε} - ε \partial_{t} A^{ℏ, ε} + \partial_{i} u_{k}^{ℏ, ε} u_{k}^{ℏ, ε} + \partial_{i} V^{ℏ, ε} = 0 . \end{matrix}$ Switching back from Einstein notation and using the definition of electric force $E^{ℏ, ε} = - \nabla V^{ℏ, ε} + ε \partial_{t} A^{ℏ, ε}$ we obtain $\begin{matrix} (64) & \partial_{t} u^{ℏ, ε} + u^{ℏ, ε} \cdot \nabla u^{ℏ, ε} = E^{ℏ, ε} . \end{matrix}$ The first equation becomes $\begin{matrix} (65) & \partial_{t} a^{ℏ, ε} + u^{ℏ, ε} \cdot \nabla a^{ℏ, ε} + \frac{1}{2} a^{ℏ, ε} \nabla \cdot u^{ℏ, ε} = \frac{i ℏ}{2} Δ a^{ℏ, ε} + \frac{i ε}{2} (σ \cdot B^{ℏ, ε}) a^{ℏ, ε} . \end{matrix}$ Then passing to the limit $ℏ \to 0$ yields the equation $\begin{array}{c} (66a) & \partial_{t} a^{ε} + u^{ε} \cdot \nabla a^{ε} + \frac{1}{2} a^{ε} \nabla \cdot u^{ε} = \frac{i ε}{2} (σ \cdot B^{ε}) a^{ε}, \\ (66b) & \partial_{t} u^{ε} + u^{ε} \cdot \nabla u^{ε} = - \nabla V^{ε} + ε \partial_{t} A^{ε}, \\ (66c) & (a^{ε}, u^{ε}) (x, 0) = (a_{I}^{ε}, u_{I}^{ε}) (x), \end{array}$ where $\begin{aligned} (67) & - Δ V^{ε} & = ρ^{ε}, \\ (68) & - Δ A^{ε} & = ε ρ^{ε} u^{ε} . \end{aligned}$ Multiplying the first equation in (66a) by $\overline{a^{ε}}$ and taking the real part yields $\begin{matrix} (69) & \partial_{t} ρ^{ε} + \nabla \cdot (ρ^{ε} u^{ε}) = 0, \end{matrix}$ since $ℜ (i \overline{a^{ε}} (σ \cdot B^{ε}) a^{ε}) = 0$ . This yields the Euler–Poisswell equation in potential form (12a)–(12f).

Note that the term $ε \partial_{t} A^{ε}$ has to be dropped in the $O (ε)$ approximation since it is of second order in ε. This shows that the $O (ε)$ semi-relativistic approximation at the quantum level is different than at the classical level: In the Pauli–Poisswell equation (42)–(46) there is a non-trivial interaction between the magnetic field and the particle, whereas the asymptotic analysis of the Euler–Maxwell equation shows that at $O (ε)$ , the fluid dynamics and the magnetic field decouple.

It is interesting that there is an apparent “gap” between Euler–Poisswell equation arising from the semiclassical limit of the $O (ε)$ Pauli–Poisswell equation and the Euler–Poisswell equation as the semi-relativistic approximation of the Euler–Maxwell equation. This could be traced back to the fact that the $O (ε)$ Pauli equation still contains a $O (ε^{2})$ term in the magnetic Laplacian which is due to the term $ε^{2} | A^{ℏ, ε} |^{2}$ in the magnetic Laplacian $\begin{matrix} \frac{1}{2} {(ℏ \nabla + i ε A^{ℏ, ε})}^{2} = \frac{ℏ^{2}}{2} Δ + i ε ℏ A^{ℏ, ε} \cdot \nabla + \frac{ε ℏ}{2} (\nabla \cdot A^{ℏ, ε}) - \frac{ε^{2}}{2} {| A^{ℏ, ε} |}^{2} . \end{matrix}$
Remark 6.
Note that there are two different uses of the name “Darwin”: On the one hand it denotes the Darwin approximation, where Maxwell’s equations are approximated at $O (ε^{2})$ in the fashion described in Section 3: the electric field is split into a solenoidal and a irrotational component and the solenoidal part is dropped from the time derivative $\partial_{t} E^{ε}$ in Maxwell’s equations. The resulting “Darwin equations” are then coupled to the $O (ε^{2})$ Euler equations, yielding the Euler–Darwin equation (17a)–(17f).

On the other hand, in relativistic quantum physics, the Darwin term is an additional $O (ε^{2})$ term in the second order approximation of the Dirac equation, which has to be included in the Pauli equation in addition to the Stern–Gerlach term, cf. [22,27,28]. In the $O (ε^{2})$ Pauli equation one also obtains another term, representing the Zitterbewegung. The $O (ε^{2})$ Pauli equation with Darwin term and Zitterbewegung should then consistently be coupled to the Darwin equations, yielding the Pauli–Darwin equation, see e.g. [29].

The semiclassical limit of the Pauli–Poisswell equation to the Euler–Poisswell equation can be formalized in a Theorem which is proven in [35]: Theorem 2 ([35]).

Let $s > 7 / 2$ and let $T^{0}$ and $T^{ℏ, ε}$ be the maximal times of existence of the Euler–Poisswell equation ( 58a )–( 60 ) and the Pauli–Poisswell–WKB equation ( 51 )–( 54 ). Assume that $(a_{I}^{ℏ, ε}, u_{I}^{ℏ, ε})$ converges to $(a^{ε}, u^{ε})$ in $H^{s - 2} \times H^{s - 3} (R^{3})$ as $ℏ \to 0$ . Suppose that $T < T^{0}$ is a lower bound of $T^{ℏ, ε}$ . Then we have the following semiclassical limit $\begin{matrix} (70) & {‖ (a^{ℏ, ε} - a^{ε}, u^{ℏ, ε} - u^{ε}) ‖}_{L^{\infty} ([0, T], H^{s - 2} \times H^{s - 3})} \underset{ℏ \to 0}{⟶} 0 . \end{matrix}$ For the density $ρ^{ℏ, ε}$ and the current density $J^{ℏ, ε}$ it holds that $\begin{array}{c} ρ^{ℏ, ε} = {| a^{ℏ, ε} |}^{2} \underset{ℏ \to 0}{⟶} ρ^{ε} = {| a^{ε} |}^{2} in L^{\infty} ([0, T], H^{s - 3} (R^{3})), \\ J^{ℏ, ε} = ℏ w^{ℏ, ε} - ℏ v^{ℏ, ε} + ρ^{ℏ, ε} (u^{ℏ, ε} + ε A^{ℏ, ε}) \underset{ℏ \to 0}{⟶} J^{ε} = ρ^{ε} (u^{ε} + ε A^{ε}) in L^{\infty} ([0, T], H^{s - 3} (R^{3})) . \end{array}$

6. Conclusion

The Euler–Poisswell equation is the consistent $O (ε)$ and the Euler–Darwin equation is the consistent $O (ε^{2})$ approximation of the (unipolar) Euler–Maxwell fluid model for charged particles with self-interaction with the electromagnetic field.

It is important to note that in the Euler–Poisswell equation the interaction between magnetic field and fluid is trivial, since the Lorentz force reduces to the electric force in the $O (ε)$ approximation.

In order to take the Lorentz force into account one has to consider $O (ε^{2})$ terms which yields the Euler–Darwin equation, where the Darwin equations describe the electromagnetic self-interaction as the $O (ε^{2})$ semi-relativistic approximation of Maxwell’s equations.

The Euler–Poisswell equation can also be introduced as the semiclassical limit by WKB of the semi-relativistic self-consistent Pauli–Poisswell equation as the consistent $O (ε)$ approximation of the relativistic Dirac–Maxwell equation. However, in the semiclassical limit the $O (ε^{2})$ term $ε \partial_{t} A^{ε}$ remains in the Euler–Poisswell equation, yielding a non-trivial interaction between magnetic field and fluid dynamics.

The Lorentz force can also be added to the Euler–Poisson equation at the $O (1)$ level, where the electromagnetic fields are externally given. This is the Euler–Lorentz model considered in [3,8] which mixes terms of $O (1)$ , $O (ε)$ and $O (ε^{2})$ .

We also explain that the widely used Euler–Maxwell equation is not fully consistent since it couples the Galilei-invariant non-relativistic Euler equation to the Lorentz-invariant relativistic Maxwell equations. The fully consistent relativistic model is the relativistic Euler–Maxwell equation.

Footnotes

Acknowledgements

We acknowledge support by the Austrian Science Fund (FWF) via grants SFB F65 & W1245 and by the Vienna Science & Technology Fund (WWTF) MA16-066 “SEQUEX”.

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

Abstract

Keywords

1. Introduction

Theorem 1 ([35]).

6. Conclusion

Footnotes

Acknowledgements

References