Energy-transport and drift-diffusion limits of nonisentropic Euler–Poisson equations

Two relaxation limits in critical spaces for the scaled nonisentropic Euler–Poisson equations with the momentum relaxation time and energy relaxation time are considered. As the first step of this justification, the uniform (global) classical solutions to the Cauchy problem in Chemin–Lerner?s spaces with critical regularity are constructed. Furthermore, by the compactness argument, it is rigorously justified that the scaled classical solutions converge to the solutions of energy-transport equations and drift-diffusion equations, respectively, with respect to different time scales.     

The non-relativistic limit of Euler–Maxwell equations for two-fluid plasma

This paper is concerned with two-fluid time-dependent non-isentropic Euler–Maxwell equations in a torus for plasmas or semiconductors. By using the method of formal asymptotic expansions, we analyze the non-relativistic limit for periodic problems with the 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 (compressible Euler–Poisson equations) have smooth solutions. Moreover, the formal limit is rigorously justified by an iterative scheme and an analysis of asymptotic expansions up to any order.     

Classical solutions of the Vlasov–Poisson equations with external magnetic field in a half-space

We consider the first mixed problem for the Vlasov–Poisson equations with an external magnetic field in a half-space. This problem describes the evolution of the density distributions of ions and electrons in a high temperature plasma with a fixed potential of electric field on a boundary. For arbitrary potential of electric field and sufficiently large induction of external magnetic field, it is shown that the characteristics of the Vlasov equations do not reach the boundary of the halfspace. It is proved the existence and uniqueness of classical solution with the supports of charged-particle density distributions at some distance from the boundary, if initial density distributions are sufficiently small.     

Combined relaxation and non-relativistic limit of non-isentropic Euler–Maxwell equations

This paper is devoted to study the combined relaxation and non-relativistic limit of non-isentropic Euler–Maxwell equations with relaxation for semiconductors and plasmas. We prove that, as the relaxation time tends to zero and the light speed tends to infinite, periodic initial-value problem of a certain scaled non-isentropic Euler–Maxwell equations has unique smooth solution existing in the time interval where the corresponding classical driftdiffusion model has smooth solutions. It is shown that the relaxation regime plays a decisive role in the combined limit. Furthermore, the corresponding convergence rate is obtained.     

Quasineutral limit of the pressureless Euler–Poisson equation

In this short paper, we consider the quasineutral limit for the pressureless Euler–Poisson system for ions. By applying the modulated energy method, it shows that the weak solutions for the Euler–Poisson system converge weakly to the strong solutions of the compressible Euler equation as the Debye length tends to zero.     

Asymptotic behavior of solutions to Euler–Poisson equations for bipolar hydrodynamic model of semiconductors

In this paper we study the Cauchy problem for 1-D Euler–Poisson system, which represents a physically relevant hydrodynamic model but also a challenging case for a bipolar semiconductor device by considering two different pressure functions and a non-flat doping profile. Different from the previous studies (Gasser et al., 2003 [7], Huang et al., 2011 [12], Huang et al., 2012 [13]) for the case with two identical pressure functions and zero doping profile, we realize that the asymptotic profiles of this more physical model are their corresponding stationary waves (steady-state solutions) rather than the diffusion waves. Furthermore, we prove that, when the flow is fully subsonic, by means of a technical energy method with some new development, the smooth solutions of the system are unique, exist globally and time-algebraically converge to the corresponding stationary solutions. The optimal algebraic convergence rates are obtained.     

Vanishing viscosity limit for Navier–Stokes equations with general viscosity to Euler equations

In this paper, we study vanishing viscosity limit of 1-D isentropic compressible Navier–Stokes equations with general viscosity to isentropic Euler equations. Firstly, we improve estimates of the entropy flux, then we obtain that the weak solution of the isentropic Euler equations is the inviscid limit of the isentropic compressible Navier–Stokes equations with general viscosity using the compensated compactness frame recently established by G.-Q. Chen and M. Perepelitsa.     

Optimal control of a Vlasov–Poisson plasma by an external magnetic field

The aim of various technical applications (for example fusion research) is to control a plasma by magnetic fields in a desired fashion. In our model the plasma is described by the Vlasov–Poisson system that is equipped with an external magnetic field. We will prove that this model satisfies some basic properties that are necessary for calculus of variations. After that, we will analyze an optimal control problem with a tracking type cost functional with respect to the following topics: necessary conditions of first order for local optimality, derivation of an optimality system, sufficient conditions of second order for local optimality, uniqueness of the optimal control under certain conditions.     

Convergence of the quantum Navier–Stokes–Poisson equations to the incompressible Euler equations for general initial data

In this paper, we are concerned with the rigorous proof of the convergence of the quantum Navier–Stokes-Poisson system to the incompressible Euler equations via the combined quasi-neutral, vanishing damping coefficient and inviscid limits in the three-dimensional torus for general initial data. Furthermore, the convergence rates are obtained.     

Low Mach number limit of the full compressible Navier–Stokes–Maxwell system

This paper deals with the low Mach number limit of the full compressible Navier–Stokes–Maxwell system. It is justified rigorously that, for the well-prepared initial data, the solutions of the full compressible Navier–Stokes–Maxwell system converge to that of the incompressible Navier–Stokes–Maxwell system as the Mach number tends to zero.     

Improved blowup results for the Euler and Euler–Poisson equations with repulsive forces

Some results are presented on the formation of singularities in the solutions of the radially-symmetric N-dimensional Euler or Euler–Poisson equations with repulsive forces. Based on the integration method of M.W. Yuen, we generalize the blowup results with constant compact radius R   of solutions to the case with general compact radius R(t)$R(t)$ and to the case with no compact support restriction.     

Small amplitude limit of solitary waves for the Euler–Poisson system

The one-dimensional Euler–Poisson system arises in the study of phenomena of plasma such as plasma solitons, plasma sheaths, and double layers. When the system is rescaled by the Gardner–Morikawa transformation, the rescaled system is known to be formally approximated by the Korteweg–de Vries (KdV) equation. In light of this, we show existence of solitary wave solutions of the Euler–Poisson system in the stretched moving frame given by the transformation, and prove that they converge to the solitary wave solution of the associated KdV equation as the small amplitude parameter tends to zero. Our results assert that the formal expansion for the rescaled system is mathematically valid in the presence of solitary waves and justify Sagdeev's formal approximation for the solitary wave solutions of the pressureless Euler–Poisson system. Our work extends to the isothermal case.     

Chern–Simons limit of the standing wave solutions for the Schrödinger equations coupled with a neutral scalar field

We show the existence of standing wave solutions to the Schrödinger equation coupled with a neutral scalar field. We also verify the Chern–Simons limit for these solutions. More precisely we prove that solutions to Eqs. (1.3)–(1.4) converge to the unique positive radially symmetric solution of the nonlinear Schrödinger equation (1.6) as the coupling constant q goes to infinity.     

The Convergence of the Navier–Stokes–Poisson System to the Incompressible Euler Equations

ABSTRACT

The combining quasineutral and inviscid limit of the Navier–Stokes–Poisson system in the torus 𝕋 ^d , d ≥ 1 is studied. The convergence of the Navier–Stokes–Poisson system to the incompressible Euler equations is proven for the global weak solution and for the case of general initial data.     

Convergence of the Euler–Maxwell two-fluid system to compressible Euler equations

The combined non-relativistic and quasi-neutral limit of two-fluid Euler–Maxwell equations for plasmas is rigorously justified in this paper. For well-prepared initial data, the convergence of the two-fluid Euler–Maxwell system to the compressible Euler equations is proved in the time interval where a smooth solution of the limit problem exists.     

Solutions of two-dimensional Schrödinger-type equations in a magnetic field

We use the method of dressing by a linear operator of general form to construct new solutions of Schrödinger-type two-dimensional equations in a magnetic field. In the case of a nonunit metric, we integrate the class of solutions that admit a variable separation before dressing. In particular, we show that the ratio of the coefficients of the differential operators in the unit metric case satisfies the Hopf equation. We establish a relation between the solutions of the two-dimensional eikonal equation with the unit right-hand side and solutions of the Hopf equation.     

Invariance of functionals and related Euler–Lagrange equations

We establish a connection between symmetries of functionals and symmetries of the corresponding Euler–Lagrange equations. A similar problem is investigated for equations with quasi-B _u -potential operators.     

Quasi-Neutral Limit of the Euler–Poisson and Euler–Monge–Ampère Systems

ABSTRACT

This paper studies the pressureless Euler–Poisson system and its fully nonlinear counterpart, the Euler–Monge–Ampère system, where the fully nonlinear Monge–Ampère equation substitutes for the linear Poisson equation. While the first is a model of plasma physics, the second is derived as a geometric approximation to the Euler incompressible equations. Using energy estimates, convergence of both systems to the Euler incompressible equations is proved.