Yue-Jun Peng

Convergence of Compressible Euler–Maxwell Equations to Incompressible Euler Equations

In this paper we study the combined quasineutral and non-relativistic limit of compressible Euler–Maxwell equations. 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 ε-weighted Liapunov-type functional. One main ingredient of establishing uniformly a priori estimates with respect to ε is to use the curl–div decomposition of the gradient.

A HIERARCHY OF HYDRODYNAMIC MODELS FOR PLASMAS ZERO-RELAXATION-TIME LIMITS

We present a model hierarchy of hydrodynamic and quasihydrodynamic equations for plasmas consisting of electrons and ions, and give a rigorous proof of the zero-relaxation-time limits in the hydrodynamic equations. described by the Euler equations coupled with a linear or nonlinear Poisson equation. The proof is based on the high energy estimates for the Euler equations together with compactness arguments.

Rigorous Derivation of Incompressible e-MHD Equations from Compressible Euler–Maxwell Equations

We derive incompressible e-MHD equations from compressible Euler–Maxwell equations via the quasi-neutral regime. 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 the incompressible e-MHD equations is justified rigorously by studies on a weighted energy. One of the main ingredients for establishing uniform a priori estimates is to use the curl-div decomposition of the gradient and the wave-type equations of the Maxwell equations.

Quasi-neutral limit of the non-isentropic Euler-Poisson system

This paper is concerned with multi-dimensional non-isentropic Euler–Poisson equations for plasmas or semiconductors. By using the method of formal asymptotic expansions, we analyse the quasi-neutral limit for Cauchy problems with prepared initial data. It is shown that the small-parameter problems have unique solutions existing in the finite time interval where the corresponding limit problems have smooth solutions. Moreover, the formal limit is justified.

Relaxation Limit and Global Existence of Smooth Solutions of Compressible Euler–Maxwell Equations

We consider smooth periodic solutions for the Euler–Maxwell equations, which are a symmetrizable hyperbolic system of balance laws. We proved that as the relaxation time tends to zero, the Euler–Maxwell system converges to the drift-diffusion equations at least locally in time. The global existence of smooth solutions is established near a constant state with an asymptotic stability property.

The mixed initial-boundary value problem for reducible quasilinear hyperbolic systems with linearly degenerate characteristics

In this paper, we prove that the C0 boundedness of solution implies the global existence and uniqueness of C1 solution to the mixed initial-boundary value problem for linearly degenerate, reducible quasilinear hyperbolic systems with nonlinear boundary conditions and we show by an example that the C0 norm of solution may blow up in a finite time. This gives the mechanism of the formation of singularities caused by the interaction of boundary conditions with nonlinear hyperbolic waves. The same result is still valid for the quasilinear hyperbolic system of rich type.

FINITE VOLUME APPROXIMATION FOR DEGENERATE DRIFT-DIFFUSION SYSTEM IN SEVERAL SPACE DIMENSIONS

This paper is devoted to a finite volume discretization for multidimensional nonlinear drift-diffusion system. Such system arises in semiconductors modelling and is composed of two degenerate parabolic equations and an elliptic one. We prove the convergence of the finite volume scheme and then the existence of solutions to the problem. Several numerical tests show the efficiency of the method.

Boundary layers and quasi-neutral limit in steady state Euler–Poisson equations for potential flows

We study the quasi-neutral limit in the steady state Euler–Poisson system for potential flows. Boundary layers occur when the boundary conditions are not in equilibrium. We perform a formal asymptotic expansion of solutions and derive the boundary layer equations. Under the subsonic condition on the boundary and the smallness assumption on the data, the existence, uniqueness and exponential decay of the boundary layer profiles are proved by applying the centre manifold theorem to a dynamical system. We also give a rigorous justification of the asymptotic expansion up to first order in one space dimension.

Explicit solutions for 2 × 2 linearly degenerate systems

Abstract We provide explicit formulae for the weak solutions of the Cauchy problems associated with two kinds of 2 × 2 systems of conservation laws, namely, the linearly degenerate system and the Keyfitz-Kranzer model. The method used allows us to show easily the existence and uniqueness of weak solutions for the Cauchy problem, and impose precisely the hypotheses on the initial data for the well-posedness of the problem.

A hierarchy of hydrodynamic models for plasmas. Zero-electron-mass limits in the drift-diffusion equations

Abstract A model hierarchy of hydrodynamic and quasi-hydrodynamic equations for plasmas consisting of electrons and ions is presented. The various model equations are obtained from the transient Euler–Poisson system for electrons and ions in the zero-electron-mass limit and/or in the zero-relaxation-time limit. A rigorous proof of the zero-electron-mass limit in the quasi-hydrodynamic equations is given. This model consists of two parabolic equations for the electrons and ions and the Poisson equation for the electric potential, subject to initial and mixed boundary conditions. The remaining asymptotic limits will be proved in forthcoming publications. Furthermore, the existence of solutions to the limit problem which can be of degenerate type is proved without the assumptions needed for the zero-electron-mass limit (essentially, positivity of the particle densities). Finally, the uniqueness of solutions to the limit problem is studied.