Euler-Poisson Equations Research Articles

Hyers–Ulam Stability of Order k for Euler Equation and Euler–Poisson Equation in the Calculus of Variations

In this paper, we define and study Hyers–Ulam stability of order 1 for Euler’s equation and Hyers–Ulam stability of order m−1 for the Euler–Poisson equation in the calculus of variations in two special cases, when these equations have the form y″(x)=f(x) and y(m)(x)=f(x), respectively. We prove some estimations for Jyx−Jy0x, where y is an approximate solution and y0 is an exact solution of the corresponding Euler and Euler-Poisson equations, respectively. We also give two examples.

Linearized Wave-Damping Structure of Vlasov--Poisson in $\mathbb{R}^3$

In this paper we study the linearized Vlasov-Poisson equation for localized disturbances of an infinite, homogeneous Maxwellian background distribution in $\mathbb R^3_x \times \mathbb R^3_v$. In contrast with the confined case $\mathbb T^d _x \times \mathbb R_v ^d$, or the unconfined case $\mathbb R^d_x \times \mathbb R^d_v$ with screening, the dynamics of the disturbance are not scattering towards free transport as $t \to \pm \infty$: we show that the electric field decomposes into a very weakly-damped Klein-Gordon-type evolution for long waves and a Landau-damped evolution. The Klein-Gordon-type waves solve, to leading order, the compressible Euler-Poisson equations linearized about a constant density state, despite the fact that our model is collisionless, i.e. there is no trend to local or global thermalization of the distribution function in strong topologies. We prove dispersive estimates on the Klein-Gordon part of the dynamics. The Landau damping part of the electric field decays faster than free transport at low frequencies and damps as in the confined case at high frequencies; in fact, it decays at the same rate as in the screened case. As such, neither contribution to the electric field behaves as in the vacuum case.

Global solutions of the Cauchy problem to Euler–Poisson equations of two-carrier types

In this paper, we introduce the flux approximation coupled with the classical viscosity method to study the global entropy solutions to the Cauchy problem of the inhomogeneous Euler–Poisson equations of two-carrier types in one dimension with arbitrarily large initial data, and arbitrary adiabatic exponent γ>1.

Energy conserving and well-balanced discontinuous Galerkin methods for the Euler–Poisson equations in spherical symmetry

ABSTRACT This paper presents high-order Runge–Kutta (RK) discontinuous Galerkin methods for the Euler–Poisson equations in spherical symmetry. The scheme can preserve a general polytropic equilibrium state and achieve total energy conservation up to machine precision with carefully designed spatial and temporal discretizations. To achieve the well-balanced property, the numerical solutions are decomposed into equilibrium and fluctuation components that are treated differently in the source term approximation. One non-trivial challenge encountered in the procedure is the complexity of the equilibrium state, which is governed by the Lane–Emden equation. For total energy conservation, we present second- and third-order RK time discretization, where different source term approximations are introduced in each stage of the RK method to ensure the conservation of total energy. A carefully designed slope limiter for spherical symmetry is also introduced to eliminate oscillations near discontinuities while maintaining the well-balanced and total-energy-conserving properties. Extensive numerical examples – including a toy model of stellar core collapse with a phenomenological equation of state that results in core bounce and shock formation – are provided to demonstrate the desired properties of the proposed methods, including the well-balanced property, high-order accuracy, shock-capturing capability, and total energy conservation.

Free boundary value problem for damped Euler equations and related models with vacuum

This paper is concerned with the local well-posedness for the free boundary value problem of smooth solutions to the cylindrical symmetric Euler equations with damping and related models, including the compressible Euler equations and the Euler-Poisson equations. The free boundary is moving in the radial direction with the radial velocity, which will affect the angular velocity but does not affect the axial velocity. However, the compressible Euler equations or Euler-Poisson equations with damping become a degenerate system at the moving boundary. By setting a suitable weighted Sobolev space and using Hardy's inequality, we successfully overcome the singularity at the center point and the vacuum occurring on the moving boundary, and obtain the well-posedness of local smooth solutions. We also summarize the recent related results on the free boundary value problem for the Euler equations with damping, compressible Euler equations and Euler-Poisson equations.

Optimal convergence rate to nonlinear diffusion waves for bipolar Euler–Poisson equations with critical overdamping

This paper is concerned with the large time behaviors of smooth solutions to the Cauchy problem of the one dimensional bipolar Euler-Poisson equations with the time dependent critical overdamping. We show that in this critical overdamping case the bipolar Euler-Poisson system admits a unique global smooth solution that asymptotically converges to the nonlinear diffusion wave. In particular, the optimal convergence rate in logarithmic form is derived when the initial perturbations are L 2 sense by using the technical time-weighted energy method.

Power-law solutions of the f(G) gravity model with electromagnetic and scalar field

In this article, the Gauss-Bonnet model of gravity is investigated, the action of which includes the Gauss-Bonnet invariant, the Maxwell term  and the scalar field. The model is considered in a flat, isotropic and homogeneous Friedman-Robertson-Walker universe in four dimensions at the late stages of the Universe evolution. The Minkowski metric is used. For the researched model, a system of equations of motion and a solution with a scale factor with a power and modified power dependence on time are found. To find a solution, the method of variation was used. Similar solutions are obtained using the Euler-Poisson equation and the method of lowering the order of the derivative, followed by the usage of the Euler-Lagrange equation. The values of , energy density and isotropic pressure are obtained, the graphs of which correspond to modern cosmological data. In the course of the research, it turned out that for a scale factor with a power-law dependence on time within the framework of the researched model, the parameter of the equation of state is equal to the value obtained when solving a model with a modified power-law scale factor. The deceleration parameter is negative, which confirms the realism of the proposed model within the framework of an accelerated expanding universe. When considering the energy conditions, it turned out that the conditions are fulfilled in both models, but the SEC is not fulfilled. A comparison of the values of the energy conditions of the two models is shown.

Rigorous proof of the slightly nonlinear Jeans instability in the expanding Newtonian universe

Due to the nonlinearity of the Euler{Poisson equations, it is possible that the nonlinear Jeans instability may lead to a faster density growing rate than the rate in the standard theory of linearized Jeans instability, which motivates us to study the nonlinear Jeans instability. The aim of this article is to develop a method proving the Jeans instability for slightly nonlinear Euler-Poisson equations in the expanding Newtonian universe. The standard proofs of the Jeans instability rely on the Fourier analysis. However, it is difficult to generalize Fourier method to a nonlinear setting, and thus there is no result in the nonlinear analysis of Jeans instability. We firstly develop a non-Fourier-based method to reprove the linearized Jeans instability in the expanding Newtonian universe. Secondly, we generalize this idea to a slightly nonlinear case. This method relies on the Cauchy problem of the Fuchsian system due to the recent developments of this system in mathematics. The fully nonlinear Jeans instability for the Euler-Poisson and Einstein-Euler equations are in progress.

A Poisson map from kinetic theory to hydrodynamics with non-constant entropy

Kinetic theory describes a dilute monatomic gas using a distribution function f(q,p,t), the expected phase-space density of particles at a given position q with a given momentum p. The distribution function evolves according to the collisionless Boltzmann equation in the high Knudsen number limit. Fluid dynamics provides an alternative description of the gas using macroscopic hydrodynamic variables that are functions of position and time only. The mass, momentum and entropy densities of the gas evolve according to the compressible Euler equations in the limit of vanishing viscosity and thermal diffusivity. Both systems can be formulated as noncanonical Hamiltonian systems. Each configuration space is an infinite-dimensional Poisson manifold, and the dynamics is the flow generated by a Hamiltonian functional via a Poisson bracket. We construct a map J1 from the space of distribution functions to the space of hydrodynamic variables that respects the Poisson brackets on the two spaces. This map is therefore a Poisson map. It maps the p-integral of the Boltzmann entropy flogf to the hydrodynamic entropy density. This map belongs to a family of Poisson maps to spaces that include generalised entropy densities as additional hydrodynamic variables. The whole family can be generated from the Taylor expansion of a further Poisson map that depends on a formal parameter. If the kinetic-theory Hamiltonian factors through the Poisson map J1, an exact reduction of kinetic theory to fluid dynamics is possible. However, this is not the case. Nonetheless, by ignoring the contribution to the Hamiltonian from the entropy of the distribution function relative to its local Maxwellian, a distribution function defined by the p-moments ∫dnp(1,p,|p|2)f, we construct an approximate Hamiltonian that factors through the map. The resulting reduced Hamiltonian, which depends on the hydrodynamic variables only, generates the compressible Euler equations. We can thus derive the compressible Euler equations as a Hamiltonian approximation to kinetic theory. We also give an analogous Hamiltonian derivation of the compressible Euler–Poisson equations with non-constant entropy, starting from the Vlasov–Poisson equation.

Voltage Stabilisation of a Drive System Including a Power Transformer and Asynchronous and Synchronous Motors of Susceptible Motion Transmission

A mathematical model is developed of the master circuit of an electric driver system including a power transformer and susceptible motion transmission of asynchronous and synchronous drives. Electric motors drive water pumps by means of motion transmission that comprises two elastic couplings of lumped mechanical parameters and a long shaft of distributed mechanical parameters. Differential equations for oscillatory processes for the long shaft and the elastic couplings are different. The shaft is described with partial derivative Euler–Poisson equations, which, combined with the boundary conditions, form mixed problems from the mathematical point of view. The elastic couplings, on the other hand, are described with the ordinary second type Lagrange equations. Based on the theory of electromagnetic field, the partial differential equations describe the skin effects across the rotor age bars. Vertical pumps are presented by means of a loading torque waveform as a function of the input shaft’s angular velocity. The complex mathematical model serves to analyse electromechanical transient processes across the integrated drive system. Starting from there, conditions of stabilisation of the drive system voltage are determined. Electromechanical state equations are presented in the normal Cauchy form and integrated using the fourth-order Runge–Kutta method. Results of computer simulations are shown with graphics, which are interpreted and described.

Newtonian limit for the relativistic Euler-Poisson equations with vacuum

This paper is concerned with the Newtonian limit of the relativistic Euler-Poisson equation. Under conditions of the free boundary and vacuum, we prove the existence and uniqueness of local smooth solutions, which converge to the solutions of the classical Euler-Poisson equation at the rate of c−2, where c is the speed of light. From the mathematical standpoint, we successfully overcome the strong nonlinearity caused by the Lorentz factor, the vacuum occurring on the moving boundary and the singularity at the center point by applying the weighted Sobolev space, respectively.

Sub-exponential convergence to steady-states for 1-D Euler–Poisson equations with time-dependent damping

Sub-exponential convergence to steady-states for 1-D Euler–Poisson equations with time-dependent damping

Necessary conditions for blow-up solutions to the restricted Euler–Poisson equations

In this work, we study the behavior of blow-up solutions to the multidimensional restricted Euler--Poisson equations which are the localized version of the full Euler--Poisson system. We provide necessary conditions for the existence of finite-time blow-up solutions in terms of the initial data, and describe the asymptotic behavior of the solutions near blow up times. We also identify a rich set of the initial data which yields global bounded solutions.

On the quasineutral limit for the compressible Euler-Poisson equations

<p style='text-indent:20px;'>In this paper, we consider the quasineutral limit of compressible Euler-Poisson equations based on the concept of dissipative measure-valued solutions. In the case of well-prepared initial data under periodic boundary condictions, we prove that dissipative measure-valued solutions of the compressible Euler-Poisson equations converge to the smooth solution of the incompressible Euler system when the Debye length tends to zero.</p>

Large-time behavior of solutions to bipolar Euler-Poisson equations with time-dependent damping in the half space

This paper is concerned with the large-time behavior of solutions to an initial boundary value problem for the one-dimensional bipolar Euler-Poisson equations with time-dependent damping effects Ji(1+t)λ(i=1,2) for −1<λ<1. We first show the decay rates of the corresponding asymptotic profiles, the so-called nonlinear diffusion waves, then by means of the time-weighted energy method, we prove that the smooth solutions to the initial-boundary value problem exist uniquely and globally, and time-asymptotically converge to the nonlinear diffusion waves, provided that the initial perturbation around the diffusion wave is small enough. The convergence rates are in the forms that O(t−34(1+λ)) for −1<λ<35 and O(tλ−32) for 35<λ<1, respectively, where λ=35 is the critical point, and the convergence rate at the critical point is O(t65lnt). The results are different from those of the Cauchy problem in Li et al. (2019) [20].

Blow up for the solutions of the pressureless Euler–Poisson equations with time‐dependent damping

The Euler–Poisson equations can be used to describe the important physical phenomena in many areas, such as semiconductor modeling and plasma physics. In this paper, we show the singularity formation mechanism for the solutions of the pressureless Euler–Poisson equations with time‐dependent damping for the attractive forces in and the repulsive forces in . We obtain the blow up of the derivative of the velocity under the appropriate assumptions.

Asymptotic stability of a stationary solution for the bipolar full Euler–Poisson equation in a bounded domain

In this paper, we present a bipolar full hydrodynamic model from semiconductor devices, which takes the form of bipolar full Euler–Poisson with electric field and relaxation terms added to the momentum equations and energy equations. We firstly prove the existence of the stationary solutions under proper boundary value conditions. Next, we present the global existence and the asymptotic behavior of smooth solutions to the initial boundary value problem for the one-dimensional bipolar full Euler–Poisson system in a bounded domain. The result is shown by an elementary energy method. Compared with the corresponding initial data case, we find that the asymptotic state of the solutions is the stationary solution.

Variational rotating solutions to non-isentropic Euler-Poisson equations with prescribed total mass

This paper proves the existence of variational rotating solutions to the compressible non-isentropic Euler-Poisson equations with prescribed total mass. This extends the result of the isentropic case (see Auchmuty and Beals (1971)) to the non-isentropic case. Compared with the previous result of variational rotating solutions in the non-isentropic case (see Wu (2015)), to keep the constraint of prescribed finite total mass, we establish a new variational structure of the non-isentropic Euler-Poisson equations.

Large-time behavior of solutions to the time-dependent damped bipolar Euler-Poisson system

This paper concerns with the Cauchy problem of the 1-D bipolar hydrodynamic model for semiconductors, a system of Euler-Poisson equations with time-dependent damping effects and for . Here, we consider a more physical case that allows the two pressure functions can be different and the doping profile can be non-zero. Different from the previous study [Li HT, Li JY, Mei. M, et al. Asymptotic behavior of solutions to bipolar Euler-Poisson equations with time-dependent damping. J Math Anal Appl. 2019;437:1081-1121] which considered two identical pressure functions and zero doping profile, the asymptotic profiles of the solutions to this model are constant states rather than the nonlinear diffusion waves. When the initial perturbation around the constant states are sufficiently small in the sense of , by means of the time-weighted energy method, we prove the global existence and uniqueness of the smooth solutions to the Cauchy problem, and obtain the optimal convergence rates of the solutions toward the constant states.

Effect of the Gravity and Magnetic Field to Find Regular Precessions of a Satellite-gyrostat with Principal Axes on a Circular Orbit

We consider the motion of a magnetized satellite-gyrostat in a circular orbit due to the combined influence of uniform gravity and magnetic fields. Based on the Lagrangian equations, the necessary conditions for the existence of regular precessions are determined in which the axis of precession is perpendicular to the orbital plane. All possible regular precessions and permanent rotations are determined and classified. We show the usage of Lagrange equations taking Eulerian angles as generalized coordinates for determining the regular precessions is more effective and accurate than utilization of Euler-Poisson equations.