Euler-Poisson System Research Articles

On the Motion of a Self-Gravitating Incompressible Fluid with Free Boundary

We consider the motion of the interface separating a vacuum from an inviscid, incompressible, and irrotational fluid, subject to the self-gravitational force and neglecting surface tension, in two space dimensions. The fluid motion is described by the Euler-Poission system in moving bounded simply connected domains. A family of equilibrium solutions of the system are the perfect balls moving at constant velocity. We show that for smooth data which are small perturbations of size $\epsilon$ of these static states, measured in appropriate Sobolev spaces, the solution exists and remains of size $\epsilon$ on a time interval of length at least $c\epsilon^{-2},$ where $c$ is a constant independent of $\epsilon.$ This should be compared with the lifespan $O(\epsilon^{-1})$ provided by local well-posdness. The key ingredient of our proof is finding a nonlinear transformation which removes quadratic terms from the nonlinearity. An important difference with the related gravity water waves problem is that unlike the constant gravity for water waves, the self-gravity in the Euler-Poisson system is nonlinear. As a first step in our analysis we also show that the Taylor sign condition always holds and establish local well-posedness for this system.

Approximation of generalized Riemann solutions to compressible Euler-Poisson equations of isothermal flows in spherically symmetric space-times

In this paper, we consider the compressible Euler-Poisson system in spherically symmetric space-times. This system, which describes the conservation of mass and momentum of physical quantity with attracting gravitational potential, can be written as a $3\times 3$ mixed-system of partial differential systems or a $2\times 2$ hyperbolic system of balance laws with $global$ source. We show that, by the equation for the conservation of mass, Euler-Poisson equations can be transformed into a standard $3\times 3$ hyperbolic system of balance laws with $local$ source. The generalized approximate solutions to the Riemann problem of Euler-Poisson equations, which is the building block of generalized Glimm scheme for solving initial-boundary value problems, are provided as the superposition of Lax's type weak solutions of the associated homogeneous conservation laws and the perturbation terms solved by the linearized hyperbolic system with coefficients depending on such Lax solutions.

3-D axisymmetric subsonic flows with nonzero swirl for the compressible Euler–Poisson system

We address the structural stability of 3-D axisymmetric subsonic flows with nonzero swirl for the steady compressible Euler–Poisson system in a cylinder supplemented with non-small boundary data. A special Helmholtz decomposition of the velocity field is introduced for 3-D axisymmetric flow with a nonzero swirl (= angular momentum density) component. With the newly introduced decomposition, a quasilinear elliptic system of second order is derived from the elliptic modes in Euler–Poisson system for subsonic flows. Due to the nonzero swirl, the main difficulties lie in the solvability of a singular elliptic equation which concerns the angular component of the vorticity in its cylindrical representation, and in analysis of streamlines near the axis r=0.

Stability of periodic steady-state solutions to a non-isentropic Euler–Poisson system

We study the stability of periodic smooth solutions near non-constant steady-states for a non-isentropic Euler–Poisson system without temperature damping term. The system arises in the theory of semiconductors for which the doping profile is a given smooth function. In this stability problem, there are no special restrictions on the size of the doping profile, but only on the size of the perturbation. We prove that small perturbations of periodic steady-states are exponentially stable for large time. For this purpose, we introduce new variables and choose a non-diagonal symmetrizer of the full Euler equations to recover dissipation estimates. This also allows to make the proof of the stability result very simple and concise.

The shape of a rapidly rotating polytrope with index unity

We show that the solutions obtained in the paper `An exact solution for arbitrarily rotating gaseous polytropes with index unity' by Kong, Zhang, and Schubert represent only approximate solutions of the free-boundary Euler-Poisson system of equations describing uniformly rotating, self-gravitating polytropes with index unity. We discuss the quality of such solutions as approximations to the rigidly rotating equilibrium polytropic configurations.

Long-time behavior of solutions to the non-isentropic Euler–Poisson system in $\mathbb{R}^3$

Long-time behavior of solutions to the non-isentropic Euler–Poisson system in $\mathbb{R}^3$

Boundary Layer Problem and Quasineutral Limit of Compressible Euler-Poisson System

We study the boundary layer problem and the quasineutral limit of the compressible Euler-Poisson system arising from plasma physics in a domain with boundary. The quasineutral regime is the incompressible Euler equations. Compared to the quasineutral limit of compressible Euler-Poisson equations in whole space or periodic domain, the key difficulty here is to deal with the singularity caused by the boundary layer. The proof of the result is based on a λ-weighted energy method and the matched asymptotic expansion method.

From gas dynamics with large friction to gradient flows describing diffusion theories

ABSTRACTWe study the emergence of gradient flows in Wasserstein distance as high friction limits of an abstract Euler flow generated by an energy functional. We develop a relative energy calculation that connects the Euler flow to the gradient flow in the diffusive limit regime. We apply this approach to prove convergence from the Euler–Poisson system with friction to the Keller–Segel system in the regime that the latter has smooth solutions. The same methodology is used to establish convergence from the Euler–Korteweg theory with monotone pressure laws to the Cahn–Hilliard equation.

The existence of stationary star solutions for compressible magnetohydrodynamic flows

In this paper, we are concerned with the compressible Euler-Poisson system coupled to a magnetic field in the three-dimensional space. Based on a variational method and the exact expression of the Green’s function for an elliptic equation in spherical coordinates, we prove the existence of stationary star solutions.

Absence of Shocks for One Dimensional Euler–Poisson System

It is shown that smooth solutions with small amplitude to the 1D Euler-Poisson system for electrons persist forever with no shock formation.

Quasineutral limit of the Euler-Poisson system under strong magnetic fields

The quasineutral limit of the three dimensional compressible Euler-Poisson (EP) system for ions in plasma under strong magnetic field is rigorously studied. It is proved that as the Debye length and the Larmor radius tend to zero, the solution of the compressible EP system converges strongly to the strong solution of the one-dimensional compressible Euler-equation in the external magnetic field direction. Higher order approximation and convergence rates are also given and detailed studied.

Quasi-neutral limit for the Euler–Poisson system in the presence of plasma sheaths with spherical symmetry

The purpose of this paper is to mathematically investigate the formation of a plasma sheath near the surface of a ball-shaped material immersed in a bulk plasma, and to obtain qualitative information of such a plasma sheath layer. Specifically, we study existence and the quasi-neutral limit behavior of the stationary spherical symmetric solutions for the Euler–Poisson equations in a three-dimensional annular domain. We first propose a suitable condition on the velocity at the sheath edge, referred as to Bohm criterion for the annulus, and under this condition together with the constant Dirichlet boundary conditions for the potential, we show that there exists a unique stationary spherical symmetric solution. Moreover, we study the quasi-neutral limit behavior by establishing [Formula: see text] estimate of the difference of the solutions to the Euler–Poisson equations and its quasi-neutral limiting equations, incorporated with the correctors for the boundary layers. The quasi-neutral limit analysis employing the correctors and their pointwise estimates enables us to obtain detailed asymptotic behaviors including the convergence rates in [Formula: see text] and [Formula: see text] norms as well as the thickness of the boundary layers as a consequence of the pointwise estimates.

New method for blowup of the Euler-Poisson system

In this paper, we provide a new method for establishing the blowup of C2 solutions for the pressureless Euler-Poisson system with attractive forces for RN (N ≥ 2) with ρ(0, x0) > 0 and Ω0ij(x0)=12∂iuj(0,x0)−∂jui(0,x0)=0 at some point x0 ∈ RN. By applying the generalized Hubble transformation div u(t,x0(t))=Nȧ(t)a(t) to a reduced Riccati differential inequality derived from the system, we simplify the inequality into the Emden equation ä(t)=−λa(t)N−1,a(0)=1,ȧ(0)= div u(0,x0)N. Known results on its blowup set allow us to easily obtain the blowup conditions of the Euler-Poisson system.

Time-periodic approximations of the Euler–Poisson system near Lane–Emden stars

We show a long-time validity of the time-periodic linear approximations to the gravitational Euler–Poisson system near Lane–Emden equilibria for all relevant adiabatic exponents. To prove the result, we reformulate the problem in Lagrangian coordinates and use the weighted energy estimates together with Hardy inequalities.

Global weak entropy solutions to the Euler–Poisson system with spherical symmetry

In this paper, we study the initial boundary value problem for the isentropic Euler–Poisson system in an exterior domain with spherical symmetry. The initial data is supposed to be bounded and satisfy other suitable assumptions. Using a fractional step Godunov scheme, we construct the approximate solutions and prove the uniform [Formula: see text] estimates for the approximate solutions. Then the compensated compactness argument implies the convergence of the solutions. The weak entropy solution also satisfies the initial value and boundary value in the sense of trace.

Blow-up conditions for two dimensional modified Euler–Poisson equations

The multi-dimensional Euler–Poisson system describes the dynamic behavior of many important physical flows, yet as a hyperbolic system its solution can blow-up for some initial configurations. This article strives to advance our understanding on the critical threshold phenomena through the study of a two-dimensional modified Euler–Poisson system with a modified Riesz transform where the singularity at the origin is removed. We identify upper-thresholds for finite time blow-up of solutions for the modified Euler–Poisson equations with attractive/repulsive forcing.

Pointwise estimates of solutions for the multi-dimensional bipolar Euler–Poisson system

In the paper, we consider a multi-dimensional bipolar hydrodynamic model from semiconductor devices and plasmas. This system takes the form of Euler–Poisson with electric field and frictional damping added to the momentum equations. By making a new analysis on Green’s functions for the Euler system with damping and the Euler–Poisson system with damping, we obtain the pointwise estimates of the solution for the multi-dimensions bipolar Euler–Poisson system. As a by-product, we extend decay rates of the densities \({\rho_i(i=1,2)}\) in the usual L2-norm to the Lp-norm with \({p\geq1}\) and the time-decay rates of the momentums mi(i = 1,2) in the L2-norm to the Lp-norm with p > 1 and all of the decay rates here are optimal.

Stability of non-constant steady state solutions for non-isentropic Euler-Poisson system in semiconductors

This work is concerned with the non-isentropic Euler-Poisson system in semiconductors. We investigate, by means of the techniques of symmetrizer and an induction argument on the order of the mixed time-space derivatives of solutions in energy estimates, the periodic problem in a three-dimensional torus. Under the assumption that the initial data is close to a non-constant steady state solutions, we prove that the smooth solutions of this problem converge to a steady state with exponential decay rates as the time goes to the infinity. This phenomenon on the charge transport shows the essential relation between the isentropic and the non-isentropic compressible Euler-Poisson system.

Existence and uniqueness of two dimensional Euler-Poisson system and WKB approximation to the nonlinear Schrödinger-Poisson system

In this paper, we study a dispersive Euler-Poisson system in two dimensional Euclidean space. Our aim is to show unique existence and the zero-dispersion limit of the time-local weak solution. Since one may not use dispersive structure in the zero-dispersion limit, when reducing the regularity, lack of critical embedding H1⊊L∞ becomes a bottleneck. We hence employ an estimate on the best constant of the Gagliardo-Nirenberg inequality. By this argument, a reasonable convergence rate for the zero-dispersion limit is deduced with a slight loss. We also consider the semiclassical limit problem of the Schrödinger-Poisson system in two dimensions.

Relaxation limit of the one-dimensional bipolar Euler–Poisson system in the bound domain

In this paper, we present a one-dimensional bipolar hydrodynamic model from semiconductor devices and plasmas, which takes the form of bipolar isothermal Euler–Poisson with electric field and frictional damping added to the momentum equations. From proper scaling, when the relaxation time in the bipolar Euler–Poisson system tends to zero, we can obtain the bipolar drift-diffusion equation. First, we show that the solutions to the initial boundary value problems of the bipolar Euler–Poisson system and the corresponding drift-diffusion equation converge to their stationary solutions as time tends to infinity, respectively. Then, it is shown that the solution for the bipolar Euler–Poisson equation converges to that of the corresponding bipolar drift-diffusion equations as the relaxation time tends to zero with the initial layer.