The roe approximate riemann solver generally gives well behaved results but it does allow for expansion shocks in some cases. In this paper, we consider the compressible euler maxwell equations arising in semiconductor physics, which take the form of euler equations for the conservation laws of mass density and current density for electrons, coupled to maxwells equations for selfconsistent electromagnetic fields. Zero dielectric constant limit to the nonisentropic. Journal of hyperbolic differential equations vol 08, no 02. New explicit rungekutta methods for the incompressible. There is no electrostatic potential for the electric field in this model. Competing lagrangians for incompressible and compressible. There exist stationary solutions when the background density is a small perturbation of a positive constant state. The methods presented in this workare highorderaccurate for both velocity. Formally, we give some different limit systems according to the corresponding different scalings. The global existence of solutions near constant steady states with the vanishing electromagnetic eld is established, and also the timedecay rates of perturbed solutions in lqspace for 2 q 1are obtained.
It is shown that the small parameter problem has unique. On this page, well look at the continuity equation, which can be derived from gauss law and ampere s law. We provide sufficient conditions for a weak solution to conserve the energy. This paper deals with the cauchy problem for the compressible euler equations with timedependent damping, where the timevanishing damping in the form of. The first part is a quick introduction on maxwells equations.
Asymptotic expansions in twofluid compressible euler. Energy conservations are studied for inhomogeneous incompressible and compressible euler equations with general pressure law in a torus or a bounded domain. This paper is concerned with the convergence of the timedependent and nonisentropic eulermaxwell equations to compressible eulerpoisson equations in a torus via the nonrelativistic limit. An introduction to the incompressible euler equations. In 1746 euler modelled light as waves in a frictionless compressible uid. Pdf a new formulation of equations of compressible.
For wellprepared initial data, the convergence of the twofluid euler maxwell system to the compressible euler equations is proved in the time interval where a smooth solution of the limit problem. In this paper we consider the onefluid euler maxwell model for electrons, in 2 spatial dimensions, and prove global stability of a constant neutral background. In this paper, we will discuss asymptotic limit of nonisentropic compressible euler maxwell system arising from plasma physics. Moreover, the convergences of solutions of the former to the solutions of. The combined nonrelativistic and quasineutral limit of twofluid euler maxwell equations for plasmas is rigorously justified in this paper. In this paper we establish the uniform estimates of strong solutions with respect to the mach number and the dielectric constant to the full compressible navierstokes maxwell system in a bounded domain. Euler equation with its classical boundary condition i. Pausader for the euler maxwell electron model robust decay.
We construct an asymptotic expansion with initial layer functions and prove. For well prepared initial data the convergence of solutions is. Abstract the eulermaxwell equations as a hydrodynamic model of charge transport of semiconductors in an electromagnetic field are studied. This methodology is similar to that used by berenger for the numerical solutions of maxwell s equations. Under the assumption that the initial data are well prepared for the electric density, electric velocity, and magnetic field but not necessarily for the electric field, the convergence of the solutions of the compressible euler maxwell equations in a torus to the solutions of. We justify this singular limit rigorously in the framework of smooth solutions and obtain the nonisentropic compressible magnetohydrodynamic equations as the. Relativistic euler equations are given by the conservation of the stress energy tensor and. For well prepared initial data, the local existence of smooth solutions to the limit equations is proved by an iterative scheme.
The global existence of solutions near constant steady. In the second application in the next section, we derive the rankinehugoniot jump conditions from the weak formulation of the equations. The compressible euler equations are equations for perfect fluid. Derivation of the compressible euler equations in this section we use the divergence theorem to derive a physical interpretation of the compressible euler equations as the continuum version of newtons laws of motion. In this paper we study the combined quasineutral and nonrelativistic limit of compressible euler maxwell equations. Compressible navierstokes equations with revised maxwells.
There have been some numerical simulations 1, 2 but no mathematical studies of this model. Compressible eulermaxwell equations semantic scholar. Convergence of the eulermaxwell twofluid system to compressible euler equations. Chapter 6 maxwells equations for electromagnetic waves. The convergence for the smooth solutions to the compressible eulermaxwell equations toward the solutions to the smooth solutions to the driftdiffusion equations is proved by means of the maxwell iteration, as the relaxation time tends to zero. In this paper, we are concerned with the cauchy problem on the compressible isentropic two uids eulermaxwell equations in three dimensions. The relaxation limits of the twofluid compressible euler. A mathematical introduction to magnetohydrodynamics m16 tum. In the third application we derive the compressible euler equations from the physical principles of conservation of mass and momentum. On the euler equations of incompressible fluids peter constantin abstract. A new formulation of equations of compressible fluids.
Scaling limits of nonisentropic eulermaxwell equations. In this paper, we study the convergence of timedependent euler maxwell equations to incompressible type euler equations in a torus via the combined quasineutral and nonrelativistic limit. In fact, the full bipolar eulermaxwell system is more complicated, containing 16 equations. In this paper, we consider the compressible eulermaxwell equations arising in semiconductor physics, which take the form of euler. In this paper, we consider the low mach number limit of the full compressible mhd equations in a 3d bounded domain with dirichlet boundary condition for velocity field, neumann boundary condition for temperature and perfectly conducting boundary condition for magnetic field. Decay estimates of solutions to the bipolar nonisentropic. In 1861, maxwell derived two of his equations of electromagnetism by modelling a magnetic line of force. Asymptotic stability of stationary solutions to the. The wavenumber k is actually a vector, as is the velocity v. By using the method of formal asymptotic expansions, we analyze the nonrelativistic limit for periodic problems with the prepared initial data. Jerome abstract the hydrodynamic maxwell equations are studied, as a compressible model of charge transport induced by an electromagnetic eld in semiconductors. Perfect fluids have no heat conduction and no viscosity, so in the comoving frame the stress energy tensor is. Research open access scaling limits of nonisentropic.
Global classical solutions to the compressible eulermaxwell. Dec 10, 2009 this paper is concerned with the convergence of the timedependent and nonisentropic eulermaxwell equations to compressible eulerpoisson equations in a torus via the nonrelativistic limit. The euler maxwell system regarded as a hydrodynamic model for plasma physics describes the dynamics of compressible electrons in a constant, charged, non. The global wellposedness is proved as well as the compatibility with the classical compressible navierstokes system in the sense that, for vanishing relaxation parameters, the. For each limit with wellprepared initial data, we show the existence and uniqueness of an asymptotic expansion up to any order. In this paper, we consider the compressible euler maxwell equations arising in semiconductor physics, which take the form of euler equations for the conservation laws of mass density and current. Low mach number limit of strong solutions for 3d full. The absorbinglayer equations are simply obtained by splitting the governing equations in the coordinate directions and introducing absorption coefficients in each split equation. Convergence of the nonisentropic eulermaxwell equations to. A convergent finite element method for the compressible. In this paper, we consider the compressible eulermaxwell equations arising in semiconductor physics, which take the form of euler equations for the conservation laws of mass density and current. And nally, in the fourth application, we introduce the heat equation and derive the maximum principle for solutions of laplaces equation, motivating this by the condition that solutions of the heat equation decay in time to solutions of. Convergence of the eulermaxwell twofluid system to.
Maxwells equations vector theorems constitutive relationships outline 1 maxwell equations, units, and vectors units and conventions maxwells equations vector theorems constitutive relationships 2 basic theory generalized current derivation of poyntings theorem 3 the frequency domain phasors and maxwells equations complex power. Euler maxwell system will be more interesting and challenging works. This system can be reformulated in a form analogous to that of electromagnetism governed by maxwell s equations with source terms. Under suitable restriction on the size of the initial departure from the constant state, we establish theorems which give a complete description of the maximal development. There have been studies of the eulerpoisson equations by many authors, including ourselves see 8, 6, 12 and the references therein. Rigorous derivation of incompressible emhd equations. A basic model for describing plasma dynamics is given by the euler maxwell system, in which compressible ion and electron fluids interact with their own selfconsistent electromagnetic field. They present important open physical and mathematical problems. The local existence of smooth solutions to both systems is proved by using energy estimates for first order symmetrizable hyperbolic systems. Compressibility of ionic liquids and molten salts can be expressed as a sum of the contribution of the ionic lattice and of the holes. We also map them to solutions of eulers equations, obtained. Incompressible type euler as scaling limit of compressible.
The convergence of the compressible eulermaxwell equations to the incompressible euler equations is justified as. Abstract the euler maxwell equations as a hydrodynamic model of charge transport of semiconductors in an electromagnetic field are studied. Although there have been many results mentioned above that referred to the compressible euler maxwell system, studies of the full bipolar case are few and far between. Global and blowup solutions for compressible euler. The aim of this paper is to study multidimensional euler maxwell equations for plasmas with short momentum relaxation time. For well prepared initial data the convergences of solutions of compressible euler maxwell equations to the solutions of incompressible euler equations are justified rigorously by an analysis of asymptotic expansions and a careful use of. Convergence of a singular eulermaxwell approximation of. Dense oscillations for the compressible euler equations in this chapter it is proved that the compressible euler equations have a cascade of resonant nonlinear interactions that can create waves moving in a dense set of directions from three incoming waves. Introduction to maxwells equations sources of electromagnetic fields differential form of maxwells equation stokes and gauss law to derive integral form of maxwells equation some clarifications on all four equations timevarying fields wave equation example.
This work is concerned with the twofluid euler maxwell equations for plasmas with small parameters. Zero dielectric constant limit to the nonisentropic compressible euler maxwell system. Eulers compressible fluid obeys the wave equa tion to first. In this paper, we are concerned with the compressible euler maxwell equations with a nonconstant background density e. Siam journal on mathematical analysis siam society for. Electromagnetism lecture 8 maxwells equations continuity equation displacement current modi cation to amp eres law maxwells equations in vacuo solution of maxwells equations introduction to electromagnetic waves 1. In this paper, we consider the compressible eulermaxwell equations arising in semiconductor physics, which take the form of euler equations for the conservation laws of mass density and current density for electrons, coupled to maxwells equations for selfconsistent electromagnetic fields.
Furthermore, some recent results about the convergence of nonisentropic compressible euler maxwell system to the compressible euler poisson equations will be. By energy estimation and the curldiv decomposition of the gradient, we rigorously justify a singular approximation of the incompressible euler equations via a quasineutral regime. We consider the classical compressible euler s equations in three space dimensions with an arbitrary equation of state, and whose initial data corresponds to a constant state outside a sphere. In this paper, we are concerned with the cauchy problem on the compressible isentropic twouids euler maxwell equations in three dimensions. Stability of nonconstant equilibrium solutions for twofluid non. The cauchy problem on the compressible twofluids eulermaxwell equations renjun duan, qingqing liu, and changjiang zhu abstract. Global classical solutions to the compressible euler. This eulermaxwell system in the isentropic case assumes the following form 1, 2, 16. The relaxation limits of the twofluid compressible euler maxwell equations. This system can be reformulated in a form analogous to that of electromagnetism governed by maxwells equations with source terms. The eulermaxwell equations are much more complicated than the eulerpoisson equations, not only because of maxwells equations, but also because of the complicated coupling of the lorentz force.
For this new model, we show that for some special large initial data, the life span of any c 1 solution must be finite. In this paperwe consider the relaxation limits of the twofluid euler maxwell systems with initial layer. Pdf a compressible ideal fluid is governed by euler s equation of motion and equations of continuity, entropy and vorticity. The global existence of solutions near constant steady states with. The effect of temperature diffusion in nonisentropic eulermaxwell equations has been re. We study, by means of asymptotic expansions, the zerorelaxation limit, the nonrelativistic limit and the combined nonrelativistic and quasineutral limit. A suitable geometric generalization of the wave equation 1. Jeromey, dehua wang z abstract the euler maxwell equations as a hydrodynamic model of charge transport of semiconductors in an electromagnetic eld are studied. The eulermaxwell equations as a hydrodynamic model of charge transport of semiconductors in an electromagnetic field are studied.
The cauchy problem of the compressible euler equations with damping in multidimensions is considered when the initial perturbation in h 3norm is small. The continuity equation is approximated by discontinuous galerkin method. For linear materials the polarization p and magnetization m is given by m h e m o e f hf and the d and b fields are related to e and h by isthemagneticsusceptibilityof materialand. A compressible ideal fluid is governed by euler s equation of motion and equations of continuity, entropy and vorticity. Global existence for the euler maxwell equation benoit pausader joint work with y. The approximate solutions are shown convergent by weak convergence methods. In this discussion, vectors are denoted by boldfaced underscored lowercase letters, e. The eulermaxwell equations are much more complicated than the eulerpoisson equations, not only because of maxwell s equations, but also because of the complicated coupling of the lorentz force.
This paper studies the euler maxwell system which is a model of a collisionless plasma. Residual distribution schemes for maxwells equations. Compressible euler maxwell equations guiqiang chen, joseph w. An adaptive shockcapturing hdg method for compressible flows. An introduction to the incompressible euler equations john k. The cauchy problem for compressible hydrodynamic maxwell systems. Compressible navierstokes equations with hyperbolic heat. Convergence of compressible eulermaxwell equations to.
Compressible eulermaxwell equations northwestern mathematics. An adaptive leastsquares method for the compressible euler equations. Convergence of the nonisentropic eulermaxwell equations. A blowup criterion for the full compressible navierstokes equations huang, xiangdi, methods and applications of analysis, 2009. The assumptions of his theorem on the initial data were that there is an annular region. There have been studies of the eulerpoisson equations by many authors, including ourselves. Convergence of the full compressible navierstokesmaxwell. Consider a compressible inviscid uid such as the air in the idealization it is continuous and has no viscosity or thermal conductivity.
The continuity equation if we do some simple mathematical tricks to maxwell s equations, we can derive some new equations. Based on these uniform estimates, we obtain the convergence of the full compressible navierstokes maxwell system to the incompressible magnetohydrodynamic equations for wellprepared data. Convergence of the full compressible navierstokes maxwell system to the incompressible magnetohydrodynamic equations in a bounded domain. The irrotational motion of a compressible inviscid uid. Global classical solutions to the compressible eulermaxwell equations article pdf available in siam journal on mathematical analysis 436 september 2011 with 71 reads how we measure reads. Coupled with maxwells equations, they can be used to model and study magnetohydrodynamics. The interaction obeys maxwells equations and its strength is characterized by a ne structure constant. In this paper, we are concerned with the cauchy problem on the compressible isentropic twofluids euler maxwell equations in three dimensions. Large time behavior of solution for the full compressible. In this paper, we are concerned with the compressible eulermaxwell equations with a nonconstant background density e. Blowup of solutions for compressible navierstokes equations.
We are concerned with the cauchy problem on the compressible isentropic twofluids euler maxwell equations in three dimensions. A multiplegrid scheme for solving the euler equations aiaa. The cauchy problem for compressible hydrodynamicmaxwell. For example, in the compressible euler equations the pressure is a thermodynamic variable, and it is a function of any other pair of thermodynamic variables, such. Based on the renormalized scheme, we derive the stability of the.
A compressible ideal fluid is governed by eulers equation of motion and. Pdf the cauchy problem on the compressible twofluids. We investigate the zero dielectric constant limit to the nonisentropic compressible eulermaxwell system. Sideris considered the compressible euler equations in the case of a classical ideal gas with adiabatic index. Note that maxwells equations predict a unique velocity for the electromagnetic waves, which is just c, the speed of light. We employ quadratic finite elements to approximate the velocity and nedelec edge elements to approximate the magnetic induction. There exist stationary solutions when the background density is a small perturbation of a positive. In this paper, the convergence of timedependent eulermaxwell equations to compressible euler poisson equations in a torus via the nonrelativistic limit is studied. Pdf global classical solutions to the compressible euler. The study of compressible eulermaxwell equations began in 2000, chen, jerome and wang 1 prove the existence of global weak solutions of the simplified euler maxwell equations by using the. In this paper, we consider the compressible euler maxwell equations arising in semiconductor physics, which take the form of euler equations for the. Blowup solutions for small irrotational initial data for the pure compressible euler equations john, sideris.
Where istheelectricsusceptibilityof material, 1 1 m e. A new formulation of equations of compressible fluids by. Remarks on the blowup of solutions for the 3d euler equations kim, namkwon, differential and integral equations, 2001. We consider blowup of classical solutions to compressible navierstokes equations with revised maxwells law which can be regarded as a relaxation to the classical newtonian flow. The global approximate solutions to the initialboundary value problem are constructed by the fractional godunov scheme. Maxwells equations inside matter maxwells equations are modified for polarized and magnetized materials.
The navierstokes equations, in their full and simplified forms, help with the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, and many other things. We investigate the system of compressible navierstokes equations with hyperbolic heat conduction, i. See relations between specific heats for a derivation. Thus, for maxwells equations to be correct in all reference frames we are led to einsteins theory of special relativity. Jan 16, 2020 this paper is devoted to the study of finite element method for the isentropic compressible magnetohydrodynamics system. It is shown that singularity will be developed in finite time for the original without symmetry multidimensional compressible euler equations for generalized chaplygin gas if the initial momentum is sufficiently strong and an initial averaged quantity is nonnegative. May 21, 2016 we investigate the compressible navierstokes equations where the constitutive law for the stress tensor given by maxwells law is revised to a system of relaxation equations for two parts of the tensor. The cauchy problem on the compressible twofluids euler. We derive incompressible emhd equations from compressible euler maxwell equations via the quasineutral regime. The relaxationtime limit in the compressible euler.
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