On Classical Solutions to the First Mixed Problem for the Vlasov–Poisson System in an Infinite Cylinder

Doklady Mathematics - The first mixed problem for the Vlasov–Poisson system in an infinite cylinder is considered. This problem describes the kinetics of charged particles in a...     

Stationary Spherically Symmetric Solutions of the Vlasov–Poisson System in the Three-Dimensional Case

Doklady Mathematics - We consider the three-dimensional stationary Vlasov–Poisson system of equations with respect to the distribution function of the gravitating matter $$f =...     

Gravitational Collapse and the Vlasov–Poisson System

A self-gravitating homogeneous ball of a fluid with pressure zero where the fluid particles are initially at rest collapses to a point in finite time. We prove that this gravitational collapse can be approximated arbitrarily closely by suitable solutions of the Vlasov–Poisson system which are known to exist globally in time.     

Stable Ground States for the Relativistic Gravitational Vlasov–Poisson System

We consider the three dimensional gravitational Vlasov–Poisson (GVP) system in both classical and relativistic cases. The classical problem is subcritical in the natural energy space and the stability of a large class of ground states has been derived by various authors. The relativistic problem is critical and displays finite time blow up solutions. Using standard concentration compactness techniques, we however show that the breaking of the scaling symmetry allows the existence of stable relativistic ground states. A new feature in our analysis which applies both to the classical and relativistic problem is that the orbital stability of the ground states does not rely as usual on an argument of uniqueness of suitable minimizers—which is mostly unknown—but on strong rigidity properties of the transport flow, and this extends the class of minimizers for which orbital stability is now proved.     

Stability for the Gravitational Vlasov–Poisson System in Dimension Two

We consider the two dimensional gravitational Vlasov–Poisson system. Using variational methods, we prove the existence of stationary solutions of minimal energy under a Casimir type constraint. The method also provides a stability criterion of these solutions for the evolution problem.     

Propagation of Density-Oscillations in Solutions to the Compressible Navier-Stokes-Poisson System

Concerning a bounded sequence of finite energy weak solutions to the compressible Navier-Stokes-Poisson system （denoted by CNSP）, which converges up to extraction of a subsequence, the limit system may not be the same system. By introducing Young measures as in [6, 15], the authors deduce the system （HCNSP） which the limit functions must satisfy. Then they solve this system in a subclass where Young measures are convex combinations of Dirac measures, to give the information on the propagation of density-oscillations. The results for strong solutions to （CNSP） （see Corollary 6.1） are also obtained.     

On the Nonexistence of Global Weak Solutions to the Navier–Stokes–Poisson Equations in ℝ N

In this paper we prove nonexistence of stationary weak solutions to the Euler–Poisson equations and the Navier–Stokes–Poisson equations in ? ^N , N ≥ 2, under suitable assumptions of integrability for the density, velocity and the potential of the force field. For the time dependent Euler–Poisson equations we prove nonexistence result assuming additionally temporal asymptotic behavior near infinity of the second moment of density. For a class of time dependent Navier–Stokes–Poisson equations in ? ^N this asymptotic behavior of the density can be proved if we assume the standard energy inequality, and therefore the nonexistence of global weak solution follows from more plausible assumption in this case.     

Existence,Uniqueness and Asymptotic Behavior for the Vlasov–Poisson System with Radiation Damping

We investigate the Cauchy problem for the Vlasov–Poisson system with radiation damping.By virtue of energy estimate and a refined velocity average lemma, we establish the global existence of nonnegative weak solution and asymptotic behavior under the condition that initial data have finite mass and energy. Furthermore, by building a Gronwall inequality about the distance between the Lagrangian flows associated to the weak solutions, we can prove the uniqueness of weak solution when the initial data have a higher order velocity moment.     

From the Vlasov–Poisson equation with strong local alignment to the pressureless Euler–Poisson system

This article provides a rigorous justification on a hydrodynamic limit from the Vlasov–Poisson system with strong local alignment to the pressureless Euler–Poisson system for repulsive dynamics.     

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...     

Regularity Results for the Spherically Symmetric Einstein–Vlasov System

The spherically symmetric Einstein–Vlasov system is considered in Schwarzschild coordinates and in maximal-isotropic coordinates. An open problem is the issue of global existence for initial data without size restrictions. The main purpose of the present work is to propose a method of approach for general initial data, which improves the regularity of the terms that need to be estimated compared to previous methods. We prove that global existence holds outside the center in both these coordinate systems. In the Schwarzschild case we improve the bound on the momentum support obtained in Rein et al. (Commun Math Phys 168:467–478, 1995) for compact initial data. The improvement implies that we can admit non-compact data with both ingoing and outgoing matter. This extends one of the results in Andréasson and Rein (Math Proc Camb Phil Soc 149:173–188, 2010). In particular our method avoids the difficult task of treating the pointwise matter terms. Furthermore, we show that singularities never form in Schwarzschild time for ingoing matter as long as 3m ≤ r. This removes an additional assumption made in Andréasson (Indiana Univ Math J 56:523–552, 2007). Our result in maximal-isotropic coordinates is analogous to the result in Rendall (Banach Center Publ 41:35–68, 1997), but our method is different and it improves the regularity of the terms that need to be estimated for proving global existence in general.     

Moment identities for Poisson–Skorohod integrals and application to measure invariance

We present a moment identity on the Poisson space that extends the Skorohod isometry to arbitrary powers of the Skorohod integral. Applications of this identity are given to the invariance of Poisson measures under intensity preserving random transformations. To cite this article: N. Privault, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Global Weak Solutions of Initial Boundary Value Problem for Boltzmann－Poisson System with Absorbing Boundary

This paper deals with the initial boundary value problem for the Boltzmann-Poisson system, which arises in semiconductor physics, with absorbing boundary. The global existence of weak solutions is proved by using the stability of velocity averages and the compactness results on L1-theory under weaker conditons on initial boundary values.     

Global Solutions for the Two-Component Camassa–Holm System

We prove existence of a global conservative solution of the Cauchy problem for the two-component Camassa–Holm (2CH) system on the line, allowing for nonvanishing and distinct asymptotics at plus and minus infinity. The solution is proven to be smooth as long as the density is bounded away from zero. Furthermore, we show that by taking the limit of vanishing density in the 2CH system, we obtain the global conservative solution of the (scalar) Camassa–Holm equation, which provides a novel way to define and obtain these solutions. Finally, it is shown that while solutions of the 2CH system have infinite speed of propagation, singularities travel with finite speed.     

Weak Solutions for Stochastic FitzHugh–Nagumo Equations

This article studies the existence of weak solutions for a stochastic version of the FitzHugh–Nagumo equations. The random elements are introduced through initial values and forcing terms of associated Cauchy problem, which may be white noise in the time. Moreover there is a dependence of a stochastic parameter.