Poisson System Research Articles

Isomorphisms of Poisson systems over locally compact groups

Isomorphisms of Poisson systems over locally compact groups

On the Existence of Global Compactly Supported Weak Solutions of the Vlasov–Poisson System with an External Magnetic Field

On the Existence of Global Compactly Supported Weak Solutions of the Vlasov–Poisson System with an External Magnetic Field

Error analysis of Fourier–Legendre and Fourier–Hermite spectral-Galerkin methods for the Vlasov–Poisson system

We conduct a rigorous error analysis of the spectral-Galerkin methods for the 1D-1V Vlasov–Poisson system with the velocity variable in both finite and infinite domains. The estimates significantly improve the very limited existing results. We also provide numerical results to demonstrate the effectiveness of the analysed methods.

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.

On Liouville theorems of a Hartree–Poisson system

AbstractIn this paper, we are concerned with the non-existence of positive solutions of a Hartree–Poisson system: \begin{equation*} \left\{ \begin{aligned} &-\Delta u=\left(\frac{1}{|x|^{n-2}}\ast v^p\right)v^{p-1},\quad u \gt 0\ \text{in} \ \mathbb{R}^{n},\\ &-\Delta v=\left(\frac{1}{|x|^{n-2}}\ast u^q\right)u^{q-1},\quad v \gt 0\ \text{in} \ \mathbb{R}^{n}, \end{aligned} \right. \end{equation*} where $n \geq3$ and $\min\{p,q\} \gt 1$. We prove that the system has no positive solution under a Serrin-type condition. In addition, the system has no positive radial classical solution in a Sobolev-type subcritical case. In addition, the system has no positive solution with some integrability in this Sobolev-type subcritical case. Finally, the relation between a Liouville theorem and the estimate of boundary blowing-up rates is given.

Variable-moment fluid closures with Hamiltonian structure

Based on ideas due to Scovel–Weinstein, I present a general framework for constructing fluid moment closures of the Vlasov–Poisson system that exactly preserve that system’s Hamiltonian structure. Notably, the technique applies in any space dimension and produces closures involving arbitrarily-large finite collections of moments. After selecting a desired collection of moments, the Poisson bracket for the closure is uniquely determined. Therefore data-driven fluid closures can be constructed by adjusting the closure Hamiltonian for compatibility with kinetic simulations.

Global Solution of the 3D Relativistic Vlasov–Poisson System for a Class of Large Data

Global Solution of the 3D Relativistic Vlasov–Poisson System for a Class of Large Data

Variational radial sheath solutions for a kinetic model of a cylindrical Langmuir probe

We study qualitative and quantitative properties of the solutions of a Vlasov–Poisson system modeling the interaction between a plasma and a cylindrical Langmuir probe. In particular, we exhibit a class of radial sheath solutions for which the electrostatic potential is increasing and strictly concave with a strong variation in the vicinity of the probe which scales as the inverse of the Debye length. These solutions are proven to exist provided the incoming distributions of particles from the plasma verify the so‐called generalized Bohm condition of plasma physics. It extends our previous studies.

Existence of infinitely many weak solutions to Kirchhoff–Schrödinger–Poisson systems and related models

In this article we investigate the existence of infinitely many weak solutions for Kirchhoff–Schrödinger–Poisson systems via the critical point theory. We obtain infinitely many large energy solutions and negative energy solutions of the system with superlinear nonlinearities by using the fountain theorem and the dual fountain theorem, respectively. Moreover, we establish the existence of infinitely many weak solutions of the problem with sublinear nonlinearities by applying the genus theory introduced by Krasnolsel’skii [Topological Methods in the Theory of Nonlinear Integral Equations (The Macmillan Company, New York, 1964)]. In particular, we do not use the classical Ambrosetti–Rabinowitz condition.

Local wellposedness of the relativistic Vlasov–Poisson equation in Besov space

We obtain the local existence and uniqueness of solution to the relativistic Vlasov–Poisson system in Besov space, extending the non-relativistic result to the relativistic case. Weaker regularity is required for the initial datum by optimizing the commutator estimates related to the field. Commutator estimates associated with relativistic velocity are proved by dyadic decomposition of the frequency space and the decreasing properties of Bessel potential.

Scattering and Asymptotic Behavior of Solutions to the Vlasov–Poisson System in High Dimension

Scattering and Asymptotic Behavior of Solutions to the Vlasov–Poisson System in High Dimension

Mountain-pass type solution for planar Schrödinger–Poisson systems with critical exponential growth

In this paper,we considered the following planar Schrödinger–Poisson system −Δu+V(x)u+ϕu=f(x,u),x∈R2,Δϕ=u2,x∈R2,where V∈C(R2,[0,∞)) is axially symmetric and f∈C(R2×R,R) has critical exponential growth in the sense of Trudinger-Moser at t=±∞. We can convert the above system to the integro-differential equation with logarithmic convolution potential. So that we prove the existence of an axially symmetric mountain-pass type solution by some new useful estimates on logarithmic convolution potential. Our results improve the ones of Chen and Tang (2020) and some other related results in literature.

Multiplicity of positive solutions for the fractional Schrödinger–Poisson system with critical nonlocal term

This paper deals with the following fractional Schrödinger–Poisson system: [Formula: see text] with multiple competing potentials and a critical nonlocal term, where [Formula: see text], [Formula: see text] or [Formula: see text], and [Formula: see text] is the fractional critical exponent. By combining the Nehari manifold analysis and the Ljusternik–Schnirelmann category theory, we establish how the coefficient [Formula: see text] of the nonlocal critical nonlinearity affects the number of positive solutions. We propose a new relation between the number of positive solutions and the category of the global maximal set of [Formula: see text].

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.

Ion-acoustic waves in weakly ionized plasmas with charge-exchange collisions

Ion-neutral charge-exchange collisions in plasmas of laboratory, space, and astrophysical origins are fundamental to understanding wave dissipation and wave generation phenomena. This paper implements a charge-exchange collision operator in the Boltzmann–Poisson system equations for a weakly ionized plasma. When considering an electric field perturbation, the governing kinetic equations provide significant results concerning the plasma conductivity and the dielectric function, appearing in simple, sensible forms. The present analysis reveals a backward wave propagation phenomenon at maximum conductivity when the wavenumber of the plasma wave is smaller than the reciprocal of the ion-neutral collisions mean free path. In addition, it is shown that ion-neutral coupling resulting from charge-exchange collisions enhances ion-acoustic waves below and beyond the ion plasma frequency and leads to the onset of a fundamental instability that overcomes Landau damping under certain circumstances. The collisionless model is recovered as a limiting case, i.e., in the asymptotic limit of a long mean free path.

Solutions of a Schrödinger–Kirchhoff–Poisson system with concave–convex nonlinearities

Solutions of a Schrödinger–Kirchhoff–Poisson system with concave–convex nonlinearities

A Multilevel Active-Set Preconditioner for Box-Constrained Pressure Poisson Solvers

Efficiently solving large-scale box-constrained convex quadratic programs (QPs) is an important computational challenge in physical simulation. We propose a new multilevel preconditioning scheme based on the active-set method and combine it with modified proportioning with reduced gradient projections (MPRGP) to efficiently solve such QPs arising from pressure Poisson equations with non-negative pressure constraints in fluid animation. Our method employs a purely algebraic multigrid method to ensure the solvability of the coarser level systems and to merge only algebraically-connected components, thereby avoiding performance degradation of the preconditioner. We present a filtering scheme to efficiently apply our multilevel preconditioning only to unconstrained subsystems of the pressure Poisson system while reusing the hierarchy constructed per simulation step. We demonstrate the effectiveness of our method over previous approaches in various examples.

The KP limit of a reduced quantum Euler–Poisson equation

AbstractIn this paper, we consider the derivation of the Kadomtsev–Petviashvili (KP) equation for cold ion‐acoustic wave in the long wavelength limit of a two‐dimensional reduced quantum Euler–Poisson system under different scalings for varying directions in the Gardner–Morikawa transform. It is shown that the types of the KP equation depend on the scaled quantum parameter . The KP‐I is derived for , KP‐II for , and the dispersiveless KP (dKP) equation for the critical case . The rigorous proof for these limits is given in the well‐prepared initial data case, and the norm that is chosen to close the proof is anisotropic in the two directions, in accordance with the anisotropic structure of the KP equation as well as the Gardner–Morikawa transform. The results can be generalized in several directions.

On the stability of homogeneous equilibria in the Vlasov–Poisson system on

The goal of this article is twofold. First, we investigate the linearized Vlasov–Poisson system around a family of spatially homogeneous equilibria in (the unconfined setting). Our analysis follows classical strategies from physics (Binney and Tremaine 2008 Galactic Dynamics (Princeton University Press); Landau 1946 Acad. Sci. USSR. J. Phys. 10 25–34; Penrose 1960 Phys. Fluids 3 258–65) and their subsequent mathematical extensions (Bedrossian et al 2022 SIAM J. Math. Anal. 54 4379–406; Degond 1986 Trans. Am. Math. Soc. 294 435–53; Glassey and Schaeffer 1994 Transp. Theory Stat. Phys. 23 411–53; Grenier et al 2021 Math. Res. Lett. 28 1679–702; Han-Kwan et al 2021 Commun. Math. Phys. 387 1405–40; Mouhot and Villani 2011 Acta Math. 207 29–201). The main novelties are a unified treatment of a broad class of analytic equilibria and the study of a class of generalized Poisson equilibria. For the former, this provides a detailed description of the associated Green’s functions, including in particular precise dissipation rates (which appear to be new), whereas for the latter we exhibit explicit formulas. Second, we review the main result and ideas in our recent work (Ionescu et al 2022 (arXiv:2205.04540)) on the full global nonlinear asymptotic stability of the Poisson equilibrium in .

Quasi-neutral limit for the full Euler–Poisson system in one-dimensional space

In this paper, we consider the quasi-neutral limit of the full Euler–Poisson system in one-dimensional space when the Debye length tends to zero. Due to the observation that the full Euler–Poisson system is Friedrich symmetrizable, we can obtain uniform estimates by applying the pseudo-differential energy estimates. It is shown that for well-prepared initial data the strong solution of the full Euler–Poisson system converges strongly to the compressible Euler equations in small time interval.