Convergent discretizations for the Nernst–Planck–Poisson system

We propose and compare two classes of convergent finite element based approximations of the nonstationary Nernst–Planck–Poisson equations, whose constructions are motivated from energy versus entropy decay properties for the limiting system. Solutions of both schemes converge to weak solutions of the limiting problem for discretization parameters tending to zero. Our main focus is to study qualitative properties for the different approaches at finite discretization scales, like conservation of mass, non-negativity, discrete maximum principle, decay of discrete energies, and entropies to study long-time asymptotics.     

Decay of the two-species Vlasov–Poisson–Boltzmann system

We establish the time decay rates of the solution to the Cauchy problem for the two-species Vlasov–Poisson–Boltzmann system near Maxwellians via a refined pure energy method. The total density of two species of particles decays at the optimal algebraic rate as the Boltzmann equation, but the disparity between two species and the electric field decay at an exponential rate. This phenomenon reveals the essential difference when compared to the one-species Vlasov–Poisson–Boltzmann system or the Navier–Stokes–Poisson equations in which the electric field decays at the optimal algebraic rate, and compared to the Vlasov–Boltzmann system in which the disparity between two species decays at the optimal algebraic rate. Our achievement heavily relies on a reformulation of the problem which well displays the cancelation property of the two-species system, and our proof is based on a family of scaled energy estimates with minimum derivative counts and interpolations among them without linear decay analysis.     

On the Hamilton–Poisson realizations of the integrable deformations of the Maxwell–Bloch equations

In this note, we construct integrable deformations of the three-dimensional real valued Maxwell–Bloch equations by modifying their constants of motions. We obtain two Hamilton–Poisson realizations of the new system. Moreover, we prove that the obtained system has infinitely many Hamilton–Poisson realizations. Particularly, we present a Hamilton–Poisson approach of the system obtained considering two concrete deformation functions.     

Stability of the nonconstant stationary solution to the Poisson–Nernst–Planck–Navier–Stokes equations

We study the three-dimensional Cauchy problem of the Poisson–Nernst–Planck–Navier–Stokes equations. We first show that the corresponding stationary system has a unique semi-trivial solution under a general doping profile. Under initial small perturbations around such the semi-trivial stationary solution and small doping profile, we obtain the unique global-in-time solution to the non-stationary system. Moreover, we prove the asymptotic convergence of the solution toward the semi-trivial stationary solution as time tends to infinity.     

Asymptotic results for the two-parameter Poisson–Dirichlet distribution

The two-parameter Poisson–Dirichlet distribution is the law of a sequence of decreasing nonnegative random variables with total sum one. It can be constructed from stable and gamma subordinators with the two parameters, α$\alpha$ and θ$\theta$, corresponding to the stable component and the gamma component respectively. The moderate deviation principle is established for the distribution when θ$\theta$ approaches infinity, and the large deviation principle is established when both α$\alpha$ and θ$\theta$ approach zero.     

A generalized Fourier–Hermite method for the Vlasov–Poisson system

BIT Numerical Mathematics - A generalized Fourier–Hermite semi-discretization for the Vlasov–Poisson equation is introduced. The formulation of the method includes as special cases the...     

A singular Cauchy problem for the Euler–Poisson–Darboux equation

We consider a singular Cauchy problem for the Euler–Poisson–Darboux equation of Fuchsian type in the time variable with ramified Cauchy data. In this paper we establish an expansion of the solutions in a series of hypergeometric functions and then investigate the nature of the singularities of the solutions.     

The Convergence of the Navier–Stokes–Poisson System to the Incompressible Euler Equations

ABSTRACT

The combining quasineutral and inviscid limit of the Navier–Stokes–Poisson system in the torus 𝕋 ^d , d ≥ 1 is studied. The convergence of the Navier–Stokes–Poisson system to the incompressible Euler equations is proven for the global weak solution and for the case of general initial data.     

Irrotational approximation to the one-dimensional bipolar Euler–Poisson system

Under the assumptions that initial data have sufficiently small total variation and that the initial data are supersonic (or are subsonic respectively), we prove that in any bounded domain the L¹$L^1$ norm of the difference between the local solutions of the one-dimensional bipolar Euler–Poisson system and the potential flow system of the one-dimensional bipolar Euler–Poisson system with the same initial data can be bounded by the cube of the total variation of the initial data.     

Stabilization and stability for the spherically symmetric Navier–Stokes–Poisson system

We consider global behaviour of viscous compressible flows with spherical symmetry driven by gravitation and an outer pressure, outside a hard core. For a general state function $p=p\left(\rho \right)$, we present global-in-time bounds for solutions with arbitrarily large data. For non-decreasing $p$, the $\omega$-limit set for the density $\rho$ is studied. For increasing $p$, uniqueness and static stability of the stationary solutions (including variational aspects) are investigated. Moreover, stabilization rate bounds toward the statically stable solutions are given and their nonlinear dynamical stability is shown.     

Schrödinger–Poisson systems in the 3-sphere

We investigate nonlinear Schrödinger–Poisson systems in the 3-sphere. We prove existence results for these systems and discuss the question of the stability of the systems with respect to their phases. While, in the subcritical case, we prove that all phases are stable, we prove in the critical case that there exists a sharp explicit threshold below which all phases are stable and above which resonant frequencies and multi-spikes blowing-up solutions can be constructed. Solutions of the Schrödinger–Poisson systems are standing waves solutions of the electrostatic Maxwell–Schrödinger system. Stable phases imply the existence of a priori bounds on the amplitudes of standing waves solutions. Unstable phases give rise to resonant states.     

Cauchy problems with modified conditions for the Euler–Poisson–Darboux equations in the hyperbolic space

In this note, we give the solutions of the Cauchy problems for the Euler–Poisson–Darboux equations (EPD) with modified conditions in the hyperbolic space with application to the wave equation.