Euler-Poisson System Research Articles

Stability of planar diffusion wave for three‐dimensional bipolar Euler–Poisson system with time‐dependent over‐damping

This paper is concerned with the asymptotic behavior of the solution to the Cauchy problem for the three‐dimensional bipolar Euler–Poisson system with time‐dependent over‐damping. We prove that the Cauchy problem admits a unique global‐in‐time smooth solution that tends to the planar diffusion wave as time goes to infinity provided that the initial perturbation is sufficiently small. Moreover, we also obtain the convergence rate of the solution toward the planar diffusion wave. The proof is completed by the time‐weighted energy method.

Quantified asymptotic analysis for the relativistic quantum mechanical system with electromagnetic fields

We study the asymptotic analysis of a relativistic quantum mechanical system interacting with self-consistent electromagnetic fields, with a specific focus on the Maxwell–Klein–Gordon (MKG) system. Firstly, we show that the MKG system is globally well-posed under the Coulomb gauge condition, even in the presence of self-interacting potential. Furthermore, to derive the asymptotic analysis, we consider the non-relativistic and semi-classical regime simultaneously, introducing a single scaling parameter that serves to parameterize both the speed of light and the Planck constant. As the scaling parameter approaches zero, we obtain rigorous and quantified estimates on the asymptotic convergence of the electrostatic MKG system toward the classical Euler–Poisson system. Our analytical framework relies on the modulated energy estimate.

Hunter Self-Similar Implosion Profiles for the Gravitational Euler–Poisson System

Our result is a construction of infinitely many radial self-similar implosion profiles for the gravitational Euler–Poisson system. The problem can be expressed as solving a system of non-autonomous non-linear ODEs. The first rigorous existence result for a non-trivial solution to these ODEs is due to Guo et al. (Commun Math Phys 386(3):1551–1601, 2021), in which they construct a solution found numerically by Larson (Mon Not R Astron Soc 145(3):271–295, 1969) and Penston (Mon Not R Astron Soc 144(4):425–448, 1969) independently. The solutions we construct belong to a different regime and correspond to a strict subset of the family of profiles discovered numerically by Hunter (Astrophys J 218:834, 1977). Our proof adapts a technique developed by Collot et al. (Mem Am Math Soc 260(1255):v+97, 2019), in which they study blowup for a family of energy-supercritical focusing semilinear heat equations. In our case, the quasilinearity presents complications, most severely near the sonic point where the system degenerates.

Nonexistence of multi-dimensional solitary waves for the Euler–Poisson system

We study the nonexistence of multi-dimensional solitary waves for the Euler–Poisson system governing ion dynamics. It is well-known that the one-dimensional Euler–Poisson system has solitary waves that travel faster than the ion-sound speed. In contrast, we show that the two-dimensional and three-dimensional models do not admit nontrivial irrotational spatially localized traveling waves in the L1 space for any traveling velocity and for general pressure laws. Our results provide theoretical evidence for the stability of line solitary waves in multi-dimensional Euler–Poisson flows. We derive some Pohozaev type identities associated with the energy and density integrals. This approach is extended to prove the nonexistence of irrotational multi-dimensional solitary waves for the two-species Euler–Poisson system for ions and electrons.

ZK‐type limit for the Euler–Poisson system with a realistic electrostatic potential in dimension three

We explore the Zakharov–Kuznetsov‐type limit for the three‐dimensional Euler–Poisson systems with external magnetic fields when the plasma has a realistic electrostatic potential. It reveals that both homogeneous and inhomogeneous Zakharov–Kuznetsov‐type equations can be obtained from the Euler–Poisson system through the application of the Gardner–Morikawa transformation. By analyzing the remaining system, we employ the energy method within Sobolev spaces to provide estimates, leading to the emergence of Zakharov–Kuznetsov‐type dynamics over a time interval of the order .

Global-in-time error estimates of non-relativistic limits for Euler–Maxwell system near non-constant equilibrium

It was proved that Euler–Maxwell systems converge globally-in-time to Euler–Poisson systems near non-constant equilibrium states when the speed of light c→∞. In this paper, we establish the global-in-time error estimates between smooth solutions of Euler–Maxwell systems and those of Euler–Poisson systems near non-constant equilibrium states. The main difficulty lies in the singularity of the error variable for the electric field E, so that more careful estimates for the time derivatives of error variables should be established. The proof takes good advantage of the anti-symmetric structure of the error system and an induction argument on the order of the derivatives.

Smooth Subsonic and Transonic Spiral Flows with Nonzero Vorticity to Steady Euler–Poisson System in Concentric Cylinders

Smooth Subsonic and Transonic Spiral Flows with Nonzero Vorticity to Steady Euler–Poisson System in Concentric Cylinders

High Order Asymptotic Preserving and Classical Semi-implicit RK Schemes for the Euler–Poisson System in the Quasineutral Limit

High Order Asymptotic Preserving and Classical Semi-implicit RK Schemes for the Euler–Poisson System in the Quasineutral Limit

The zero-electron-mass limit of the bipolar non-isentropic Euler-Poisson system

We study the zero-electron-mass limit of the bipolar non-isentropic Euler-Poisson system. It is known that ions and electrons are relatively independent in plasmas so that they may have several temperatures at the same time. Moreover, since the velocity of the electrons is larger, so that the heat conductivity can be regarded as infinite. In this paper, we assume that the electrons are isothermal. The study of the convergence of this limit is based on the asymptotic analysis and we prove that the limiting process is actually decoupling and the limiting system is the unipolar non-isentropic Euler-Poisson system for ions.

Stability of transonic shocks to the Euler-Poisson system with varying background charges

Stability of transonic shocks to the Euler-Poisson system with varying background charges

Formation of singularities in plasma ion dynamics

We study the formation of singularity for the Euler–Poisson system equipped with the Boltzmann relation, which describes the dynamics of ions in an electrostatic plasma. In general, it is known that smooth solutions to nonlinear hyperbolic equations fail to exist globally in time. We establish criteria for C 1 blow-up of the Euler–Poisson system, both for the isothermal and pressureless cases. In particular, our blow-up condition for the pressureless model does not require that the gradient of velocity is negatively large. In fact, our result particularly implies that the smooth solutions can break down even if the gradient of initial velocity is trivial. For the isothermal case, we prove that smooth solutions leave C 1 class in a finite time when the gradients of the Riemann functions are initially large.

An asymptotic preserving and energy stable scheme for the Euler-Poisson system in the quasineutral limit

An asymptotic preserving (AP) and energy stable scheme for the Euler-Poisson (EP) system under the quasineutral scaling is designed and analysed. Appropriate stabilisation terms are introduced in the convective fluxes of mass and momenta, and the gradient of the electrostatic potential which lead to the dissipation of mechanical energy and consequently the entropy stability of solutions. The time discretisation is semi-implicit in nature, whereas the space discretisation uses a finite volume framework on a marker and cell (MAC) grid. The numerical resolution of the fully-discrete scheme involves two steps: the solution of a linear elliptic problem for the potential and an explicit evaluation for the density and velocity. The proposed scheme possesses several physically relevant attributes, such as the positivity of density, entropy stability and the consistency with the weak formulation of the continuous EP system. The AP property of the scheme, i.e. the boundedness of the mesh parameters with respect to Debye length and its consistency with the quasineutral limit system, is demonstrated. The results of numerical case studies are presented to substantiate the robustness and efficiency of the proposed method.

Zero-electron-mass and quasi-neutral limits for bipolar Euler–Poisson systems

Zero-electron-mass and quasi-neutral limits for bipolar Euler–Poisson systems

Non-uniqueness of transonic shock solutions to Euler–Poisson system with varying background charges

Non-uniqueness of transonic shock solutions to Euler–Poisson system with varying background charges

Global convergence of a two-fluid non-isentropic Euler–Poisson system in one space dimension

In this paper, we study the global in time quasi-neutral limit for a two-fluid non-isentropic Euler–Poisson system in one space dimension. We prove that the system converges to the non-isentropic Euler equations as the Debye length tends to zero. This problem is studied for smooth solutions near the constant equilibrium state. To prove this result, we establish uniform energy estimates and various dissipation estimates with respect to the Debye length and the time. These estimates allow to pass to the limit to obtain the limit system by compactness arguments. In addition, the global convergence rate is obtained by use of stream function technique.

Structural Stability of Radial Interior Subsonic Steady-States to n-D Euler-Poisson System of Semiconductor Models with Sonic Boundary

Structural Stability of Radial Interior Subsonic Steady-States to n-D Euler-Poisson System of Semiconductor Models with Sonic Boundary

Sonic–supersonic–shock–subsonic–sonic solution to Euler–Poisson system with varying background charges

In this paper, steady transonic shock solutions to the Euler–Poisson system are investigated with respect to varying background charge. More precisely, the density of the background charge is prescribed as a piecewise constant function whose discontinuity is determined only by the shock front. The Euler–Poisson system with varying background charge can describe the motion of the electrons in the special semiconductor devices. We establish the well-posedness of sonic–supersonic–shock–subsonic–sonic solutions, with sonic physical boundary conditions at the upstream and downstream respectively.

NLS approximation of the Euler-Poisson system for a cold ion-acoustic plasma

In the previous paper Liu and Pu (2019) [17], we proved the nonlinear Schrödinger (NLS) approximation for the Euler-Poisson system for a hot ion-acoustic plasma, where the appearance of resonances and the loss of derivatives of quadratic terms are the main difficulties. Note that when the ion-acoustic plasma is hot, the Euler-Poisson system is Friedrich symmetrizable, and the linear term can provide a derivative to compensate the loss of derivative induced by quadratic terms after diagonalizing the linearized system. When the ion-acoustic plasma is cold, as considered in the present paper, the situation is very different from that in the previous paper. The Euler-Poisson system becomes a pressureless system, so the linear operator has no regularity, and the quadratic terms still lose a derivative in the diagonalized system. This fact makes it more difficult to prove the NLS approximation of Euler-Poisson system for a cold ion-acoustic plasma. In this paper, we take advantage of the special structure of the pressureless Euler-Poisson system and the normal-form transformation to deal with the difficulties caused by resonances, especially the difficulties caused by derivative loss, in order to prove the NLS approximation.

Global convergence in non-relativistic limits for Euler-Maxwell system near non-constant equilibrium

In this paper, we study the global-in-time convergence of non-relativistic limits from Euler-Maxwell systems to Euler-Poisson systems near non-constant equilibrium states by letting the reciprocal of the speed of light ν:=1/c→0. In previous studies, the dissipative estimates for the electric field E appear to be singular and the orders of singularities for divE and ∇×E are different, hence a div-curl decomposition should be considered in our case, and this makes it unclear the preservation of the anti-symmetric structure of the system and the global-in-time L2–estimate of the smooth solutions. To overcome these difficulties, we find a strictly convex entropy of the system that is valid for the case of non-constant equilibrium to obtain the global L2–estimate and use some induction arguments to close the estimates. It is worth mentioning that in our proof, very careful and accurate estimates of solutions are needed and we show that the electric field E is actually non-singular, which is vital and necessary for our proof.

Global error estimates in zero-relaxation limit of Euler–Poisson system for ion dynamics

The zero-relaxation limit of Euler–Poisson systems for ion dynamics in a slow time scale leads to drift-diffusion equations. This fact was justified in previous works. This paper concerns global error estimates between solutions of Euler–Poisson systems and that of drift-diffusion equations. In the proof, we use Sobolev inequalities and employ a multiplier related to the electric potential.