Zero dielectric constant limit to the non-isentropic compressible Euler-Maxwell system

We investigate the zero dielectric constant limit to the non-isentropic compressible Euler-Maxwell system.We justify this singular limit rigorously in the framework of smooth solutions and obtain the nonisentropic compressible magnetohydrodynamic equations as the dielectric constant tends to zero.     

Asymptotic limits of compressible Euler-Maxwell system in plasma physics

In this talk we will discuss asymptotic limit of compressible Euler-Maxwell system in plasma physics. Some recent results about the convergence of compressible Euler-Maxwell system to the incompressible Euler equations or incompressible e-MHD equations will be given via the quasi-neutral regime. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The diffusive relaxation limit of non-isentropic Euler-Maxwell equations for plasmas

This paper concerns the non-isentropic Euler-Maxwell equations for plasmas with short momentum relaxation time. With the help of the Maxwell-type iteration, it is obtained that, as the relaxation time tends to zero, periodic initial-value problem of certain scaled non-isentropic Euler-Maxwell equations has unique smooth solutions existing in the time interval where the corresponding classical drift-diffusion model has smooth solutions. Meanwhile, we justify a formal derivation of the corresponding drift-diffusion model from the non-isentropic Euler-Maxwell equations.     

Stability of stationary solutions for the non-isentropic Euler-Maxwell system in the whole space

In this paper we discuss the asymptotic stability of stationary solutions for the non-isentropic Euler-Maxwell system in R³. It is known in the authors’ previous works [17, 18, 19] that the Euler-Maxwell system verifies the decay property of the regularity-loss type. In this paper we first prove the existence and uniqueness of a small stationary solution. Then we show that the non-stationary problemhas a global solution in a neighborhood of the stationary solution under smallness condition on the initial perturbation. Moreover, we show the asymptotic convergence of the solution toward the stationary solution as time tends to infinity. The crucial point of the proof is to derive a priori estimates by using the energy method.     

Convergence of the nonisentropic Euler-Poisson equations to incompressible type Euler equations

In this paper, we investigate a multidimensional nonisentropic hydrodynamic (Euler-Poisson) model for semiconductors. We study the convergence of the nonisentropic Euler-Poisson equation to the incompressible nonisentropic Euler type equation via the quasi-neutral limit. The local existence of smooth solutions to the limit equations is proved by an iterative scheme. The method of asymptotic expansion and energy methods are used to rigorously justify the convergence of the limit.     

Large-time behavior for fluid and kinetic plasmas with collisions

The motion of collisional plasmas can be governed either by the Euler-Maxwell system with damping at the fluid level or by the Vlasov-Maxwell-Boltzmann system at the kinetic level. In the note, we present some recent results in [8] and [7] for the study of the non-trivial large-time behavior of solutions to the Cauchy problem on the related models in perturbation framework.     

Quasi-neutral limit of the full bipolar Euler-Poisson system

The quasi-neutral limit of the multi-dimensional non-isentropic bipolar Euler-Poisson system is considered in the present paper. It is shown that for well-prepared initial data the smooth solution of the nonisentropic bipolar Euler-Poisson system converges strongly to the compressible non-isentropic Euler equations as the Debye length goes to zero.     

Continuous conjugation of special nonisentropic one-dimensional gas motions

The exact partially invariant solution of equations of motion of a compressible fluid describing the collapse of particles to a point and an instantaneous source from the point in a one-dimensional nonisentropic motion is cut off by the characteristics and glued into a continuous solution of a one-dimensional submodel in a finite domain. The possibility of a continuous periodic nonisentropic motion of a compressible fluid in a bounded domain under the action of a piston is shown.     

Diffusion relaxation limit of a nonisentropic hydrodynamic model for semiconductors

In this paper, we discussed a general multidimensional nonisentropic hydrodynamical model for semiconductors with small momentum relaxation time. The model is self‐consistent in the sense that the electric field, which forms a forcing term in the momentum equation, is determined by the coupled Poisson equation. With the help of the Maxwell‐type iteration, we prove that, as the relaxation time tends to zero, periodic initial‐value problem of certain scaled multidimensional nonisentropic hydrodynamic model has a unique smooth solution existing in the time interval where the corresponding classical drift‐diffusion model has smooth solutions. Meanwhile, we justify a formal derivation of the drift‐diffusion models from the nonisentropic hydrodynamic models. Copyright © 2007 John Wiley & Sons, Ltd.     

Global existence and asymptotic behavior of solutions to the nonisentropic bipolar hydrodynamic models

In this paper, we consider a one-dimensional bipolar nonisentropic hydrodynamical model from semiconductor devices. This system takes the nonisentropic Euler-Poisson form with electric field and frictional damping added to the momentum equations. First, we prove global existence of smooth solutions to the Cauchy problem. Next, we also discuss the asymptotic behavior of the smooth solutions. We find that in large time, the densities of electron and hole tend to the same nonlinear diffusive wave, the momentums tend to the Darcy's law, and the temperatures tend to the ambient device temperature. Finally, we can obtain the algebraic decay rate of the densities to the same nonlinear diffusive wave, the momentums to the Darcy's law and the temperatures to the ambient device temperature, and the exponential decay of their difference and the electric field to zero. We can show our results by precise energy methods.     

Low Mach limit to one-dimensional nonisentropic planar compressible magnetohydrodynamic equations

This paper is concerned with a one-dimensional nonisentropic compressible planar magnetohydrodynamic flow with general initial data, whose behaviors at far fields x→±∞ are different. The low Mach limit for the system is rigorously justified. The limit relies on the uniform estimates including weighted time derivatives and an extended convergence lemma.     

GLOBAL EXISTENCE AND EXPONENTIAL STABILITY OF SMOOTH SOLUTIONS TO A MULTIDIMENSIONAL NONISENTROPIC EULER-POISSON EQUATIONS

The global existence and large time behavior of smooth solutions to multi-dimensional nonisentropic Euler-Poisson equations are established.     

Global existence and asymptotic behavior for a multidimensional nonisentropic hydrodynamic semiconductor model with the heat source

In this paper, a multidimensional nonisentropic hydrodynamic model for semiconductors with the nonconstant lattice temperature is studied. The model is self-consistent in the sense that the electric field, which forms a forcing term in the momentum equation, is determined by the coupled Poisson equation. Global existence to the Cauchy problem for the multidimensional nonisentropic hydrodynamic semiconductor model with the small perturbed initial data is established, and the asymptotic behavior of these smooth solutions is investigated, namely, that the solutions converge to the general steady-state solution exponentially fast as t→+∞ is obtained. Moreover, the existence and uniqueness of the stationary solutions are investigated.     

Rigorous derivation of incompressible type Euler equations from non-isentropic Euler-Maxwell equations

In this paper, we investigate the convergence of the time-dependent and non-isentropic Euler-Maxwell equations to incompressible Euler equations in a torus via the combined quasi-neutral and non-relativistic limit. For well prepared initial data, the convergences of solutions of the former to the solutions of the latter are justified rigorously by an analysis of asymptotic expansions and energy method.     

Global existence and decay of solution for the nonisentropic Euler–Maxwell system with a nonconstant background density

In this paper, the Cauchy problem for the nonisentropic Euler–Maxwell system with a nonconstant background density is studied. The global existence of classical solution is constructed in three space dimensions provided the initial perturbation is sufficiently small. The proof is mainly based on classical energy estimate and the techniques of symmetrizer. And the time decay of the solution is also established by combining the decay estimate of the Green’s function with some time-weighted estimate.     

The zero-electron-mass limit for a stationary nonisentropic hydrodynamic semiconductor model

In this paper, we study a general multidimensional nonisentropic hydrodynamical model for semiconductors. The model is self-consistent in the sense that the electric field, which forms a forcing term in the momentum equation, is determined by the coupled Poisson equation. For steady state, subsonic and potential flows, we discuss the zero-electron-mass limit of system by using the method of asymptotic expansions. We show the existence and uniqueness of profiles, and justify the asymptotic expansions up to any order.     

Shock Formation and Vorticity Creation for 3d Euler

We analyze the shock formation process for the 3D nonisentropic Euler equations with the ideal gas law, in which sound waves interact with entropy waves to produce vorticity. Building on our theory for isentropic flows in [3, 4], we give a constructive proof of shock formation from smooth initial data. Specifically, we prove that there exist smooth solutions to the nonisentropic Euler equations which form a generic stable shock with explicitly computable blowup time, location, and direction. This is achieved by establishing the asymptotic stability of a generic shock profile in modulated self-similar variables, controlling the interaction of wave families via: (i) pointwise bounds along Lagrangian trajectories, (ii) geometric vorticity structure, and (iii) high-order energy estimates in Sobolev spaces. © 2022 Wiley Periodicals LLC.     

A NOTE ON THE EXISTENCE AND NONEXISTENCE OF GLOBALLY BOUNDED CLASSICAL SOLUTIONS FOR NONISENTROPIC GAS DYNAMICS

Existence of globally bounded classical solution for nonisentropic gas dynamics system has long been studied, especially in the case of polytropic gas. In [4], Liu claimed that sufficient condition has been established. However, the authors find that the argument he used is not true in general. In this article, the authors give a counter example of his argument. Hence, his claim is not valid. The authors believe that it is difficult to impose general conditions on the initial data to obtain globally bounded classical solution.     

Convergence of Compressible Euler-Maxwell Equations to Compressible Euler-Poisson Equations

In this paper,the convergence of time-dependent Euler-Maxwell equations to compressible Euler-Poisson equations in a torus via the non-relativistic limit is studied. The local existence of smooth solutions to both systems is proved by using energy esti- mates for first order symmetrizable hyperbolic systems.For well prepared initial data the convergence of solutions is rigorously justified by an analysis of asymptotic expansions up to any order.The authors perform also an initial layer analysis for general initial data and prove the convergence of asymptotic expansions up to first order.     

The Riemann problem for thermoelastic materials with phase change

We consider the Riemann problem for a system of conservation laws related to a phase transition problem. The system is nonisentropic and we treat the case where the latent heat is not zero. We study the cases where the initial data are given in the same phase and in the different phases. The role of the entropy condition is studied as well as the kinetic relation and the entropy rate admissibility criterion. We confine our attention to the case where the speeds of phase boundaries are close to zero. This is one interesting case in physics. We discuss the number of phase boundaries consistent with the above criteria and the uniqueness and nonuniqueness issue of the solution to the Riemann problem.