The purpose of this paper is to mathematically investigate the formation of a plasma sheath near the surface of walls immersed in a plasma, and to analyze qualitative information of such a sheath layer. In the case of planar wall, Bohm proposed a criterion on the velocity of the positive ion for the formation of sheath, and several works gave its mathematical validation. It is of more interest to analyze the criterion for the nonplanar wall. In this paper, we study the existence and asymptotic stability of stationary solutions for the Euler-Poisson equations in a domain of which boundary is drawn by a graph. The existence and stability theorems are shown by assuming that the velocity of the positive ion satisfies the Bohm criterion at infinite distance. What most interests us in these theorems is that the criterion together with a suitable necessary condition guarantees the formation of sheaths as long as the shape of walls is drawn by a graph.
The main purpose of this paper is to provide an effective procedure to study rigorously the relationship between unipolar and bipolar Euler-Poisson system in the perspective of mass. Based on the fact that the mass of an electron is far less than that of an ion, we amplify this property by letting $m_e/m_i\rightarrow 0$ and using two different singular limits to illustrate it, which are zero-electron mass limit and infinity-ion mass limit. We use the method of asymptotic expansion to handle the problem and find that the limiting process from bipolar to unipolar system is actually the process o…
This paper studies the two-dimensional Euler-Poisson equations associated with either attractive or repulsive forces. We mainly study the Riccati system that governs the flow's gradient. Under a suitable condition, it is shown that the Euler-Poisson system admits global smooth solutions for a large set of initial configurations. This paper is a continuation of our former work [8].
We prove shock formation for 3D Euler-Poisson system for electron fluid plasma. The shock solution we construct is of large initial data, also compactly supported during the lifespan. In addition, the blowup time and location can be computed explicitly.
We prove the unique existence of supersonic solutions of the Euler- Poisson system for potential flow in a three-dimensional rectangular cylinder when prescribing the velocity and the strength of electric field at the entrance. Overall, the main framework is similar to [1], but there are several technical differences to be taken care of vary carefully. And, it is our main goal to treat all the technical differences occurring when one considers a three dimensional supersonic solution of the steady Euler-Poisson system.
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.
We construct a special $C^{1,α}$ blow up solution to the three dimensional system modeling electro-hydrodynamics, which is strongly coupled with incompressible Euler equation and Nernst-Planck-Poisson equation. Our construction lies on the framework established in Elgindi [11] and relies on a special solution to variant spherical Laplacian.
Weak asymptotic methods for 3-D self-gravitating pressureless fluids. Application to the creation and evolution of solar systems from the fully nonlinear Euler-Poisson equations
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 [Auchmuty and Beals, Arch. Ration. Mech. Anal., 1971] to the non-isentropic case. Compared with the previous result of variational rotating solutions in non-isentropic case [Wu, Journal of Differential Equations, 2015], to keep the constraint of a prescribed finite total mass, the author establishes a new variational structure the non-isentropic Euler-Poisson equations.
We show the continuity of the flow map for quasilinear symmetric hyperbolic systems with general right--hand sides in different functional setting, including weighted Sobolev spaces $H_{s,δ}$. An essential tool to achieve the continuity of the flow map is a low regularity energy estimate. We then apply these results to the Euler--Poisson system which describes various systems of physical interests.
The Euler Poisson equations describe important physical phenomena in many applications such as semiconductor modeling and plasma physics. This paper is to advance our understanding of critical threshold phenomena in such systems in the presence of different forces. We identify critical thresholds in two damped Euler Poisson systems, with and without alignment, both with attractive potential and spatially varying background state. For both systems, we give respective bounds for subcritical and supercritical regions in the space of initial configuration, thereby proving the existence of a critic…
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. Ou…
In this paper we consider the Euler-Poisson system (describing a plasma made of ions with a negligible ion temperature) on the Sobolev spaces $H^s(\R^3)$, $s > 5/2$. Using a geometric approach we show that for any time $T > 0$ the corresponding solution map, $(ρ_0,u_0) \mapsto (ρ(T),u(T))$, is nowhere locally uniformly continuous. On the other hand it turns out that the trajectories of the ions are analytic curves in $\R^3$.
By introducing a new averaged quantity with a fast decay weight to perform Sideris's argument (Commun Math Phys, 1985) developed for the Euler Equations, we extend the formation of singularities of classical solution to the 3D Euler Equations established in Sideris (1985) and Makino et al. (Jpn J Appl Math, 1986) for the initial data with compactly supported disturbances to the spherically symmetric solution with general initial data in Sobolev space. Moreover, we also prove the formation of singularities of the spherically symmetric solutions to the 3D Euler-Poisson Equations, but remove the …
The nonlinear Schrödinger (NLS) equation can be derived as a formal approximation equation describing the envelopes of slowly modulated spatially and temporarily oscillating wave packet-like solutions to the ion Euler-Poisson equation. In this paper, we rigorously justify such approximation by giving error estimates in Sobolev norms between exact solutions of the ion Euler-Poisson system and the formal approximation obtained via the NLS equation. The justification consists of several difficulties such as the resonances and loss of regularity, due to the quasilinearity of the problem. These dif…
In this note we derive the descriptions of the system of Euler-Poisson equations which governs the hydrodynamic evolution of gaseous stars in various co-ordinate systems. This note does not contain essentially new results for astrophysicists, but mathematically rigorous derivations cannot be found in the literatures written by physicists so that it will be useful to prepare details of rather stupidly honest derivations of the equations in various flames when we are going to push forward the mathematical research of the problem.
We consider several pressureless variants of the compressible Euler equation driven by nonlocal repulsionattraction and alignment forces with Poisson interaction. Under an energy admissibility criterion, we prove existence of global measure-valued solutions, i.e., very weak solutions described by a classical Young measure together with appropriate concentration defects. We then investigate the evolution of a relative energy functional to compare a measure-valued solution to a regular solution emanating from the same initial datum. This leads to a (partial) weak-strong uniqueness principle.
This paper deals with the existence of BV solution for the Euler-Poisson system endowed with a $γ$ pressure law. More precisely, we prove the existence of weak solution in the BV framework with arbitrary large initial data when $γ=1+2ε$ satisfies a smallness condition. We use the Glimm scheme combined with a splitting method as introduced in [Poupaud, Rascle and Vila, J. Differential Equations, 1995]. Existence of BV solution of 1-D isentropic Euler equation for large data and $γ=1+2ε$ is proved in [Nishida and Smoller, Comm. Pure Appl. Math, 1973]. Due to the presence of electric field, the d…
We consider the Cauchy problem for the barotropic Euler system coupled to Helmholtz or Poisson equations, in the whole space. We assume that the initial density is small enough, and that the initial velocity is close to some reference vector field u0 such that the spectrum of Du0 is bounded away from zero. We prove the existence of a global unique solution with (fractional) Sobolev regularity, and algebraic time decay estimates. Our work extends the papers by D. Serre and M. Grassin [20, 21, 43] dedicated to the compressible Euler system without coupling and integer regularity exponents.
The hydrodynamic evolution of self-gravitating gaseous stars is governed by the Euler-Poisson equations. We study the structure of the linear approximation of barotropic perturbations around spherically symmetric equilibria based on functional analytic tools. In contrast to folklore, we show that the spectrum of the linearized operator for general perturbations is not of the Sturm-Liouville type unless the perturbations are restricted. In particular, we prove that it is of the Sturm-Liouville type for irrotational perturbations.
