{"title":"The Euler–Poisswell/Darwin equation and the asymptotic hierarchy of the Euler–Maxwell equation","authors":"Jakob Möller, Norbert J. Mauser","doi":"10.3233/asy-231864","DOIUrl":null,"url":null,"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 → ∞ 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.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":"77 4","pages":"0"},"PeriodicalIF":1.1000,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Asymptotic Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3233/asy-231864","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
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 → ∞ 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.
期刊介绍:
The journal Asymptotic Analysis fulfills a twofold function. It aims at publishing original mathematical results in the asymptotic theory of problems affected by the presence of small or large parameters on the one hand, and at giving specific indications of their possible applications to different fields of natural sciences on the other hand. Asymptotic Analysis thus provides mathematicians with a concentrated source of newly acquired information which they may need in the analysis of asymptotic problems.