Jack Schaeffer

Global existence of smooth solutions to the vlasov poisson system in three dimensions

Recently Plaffelmoser has shown that solutions of the Vlasov–Poisson system in three dimensions remain smooth for all time. This paper establish the same existence theorem by a simpler method

The classical limit of the relativistic Vlasov-Maxwell system

Solutions of the relativistic Vlasov-Maxwell system of partial differential equations are considered in three space dimensions. The speed of light,c, appears as a parameter in this system. For smooth Cauchy data, classical solutions are shown to exist on a time interval that is independent ofc. Then, using an integral representation for the electric and magnetic fields due to Glassey and Strauss [6], conditions are given under which solutions of the relativistic Vlasov-Maxwell system converge in pointwise sense to solutions of the non-relativistic Vlasov-Poisson system at the asymptotic rate of 1/c, asc tends to infinity.

Global existence for the relativistic Vlasov-Maxwell system with nearly neutral initial data

Global classical solutions to the initial value problem for the relativistic Vlasov-Maxwell equations are obtained in three space dimensions. The initial distribution of the various species may be large, provided that the total positive charge nearly cancels the total negative charge.

On symmetric solutions of the relativistic Vlasov-Poisson system

Spherically symmetric solutions to the Cauchy problem for the relativistic Vlasov-Poisson system are studied in three space dimensions. If the energy is positive definite (the plasma physics case), global classical solutions exist. In the case of indefinite energy, “small” radial solutions exist in the large, but “large” data solutions (those with negative energy) will blow-up in finite time.

A regularity theorem for solutions of the spherically symmetric Vlasov-Einstein system

We show that if a solution of the spherically symmetric Vlasov-Einstein system develops a singularity at all then the first singularity has to appear at the center of symmetry. The main tool is an estimate which shows that a solution is global if all the matter remains away from the center of symmetry.

Time decay for solutions to the linearized Vlasov equation

Abstract Time decay for solutions to the initial-value problem for the linearized Vlasov equation is studied. Here Ex = ρ = ∫ gdv and f(v2 ) ≥ 0 is to be sufficiently smooth and strictly decreasing. The initial value for g is to be suitably smooth and small at infinity. When f1 (v2 ) → 0 as |v| → ∞ at an algebraic rate, it is shown that ρ → 0 at an algebraic rate as t → ∞ in both the L2 and maximum norms. When f is a Gaussian, the decay rate is logarithmic. The field E is also shown to decay in the maximum norm for both generic classes of fs. Similar results are obtained in three dimensions for spherically symmetric data. When f has compact support, no decay of the density in L 2(R1) is possible for data of compact support.

Critical collapse of collisionless matter : A numerical investigation

In recent years the threshold of black hole formation in spherically symmetric gravitational collapse has been studied for a variety of matter models. In this paper the corresponding issue is investigated for a matter model significantly different from those considered so far in this context. We study the transition from dispersion to black hole formation in the collapse of collisionless matter when the initial data is scaled. This is done by means of a numerical code similar to those commonly used in plasma physics. The result is that for the initial data for which the solutions were computed, most of the matter falls into the black hole whenever a black hole is formed. This results in a discontinuity in the mass of the black hole at the onset of black hole formation.

On time decay rates in landau damping

In a previous publication the authors have obtained time decay rates for solutions to the initial-value problem for the linearized Vlasov equation (in one space dimension).

The Vlasov Poisson System with Steady Spatial Asymptotics

Abstract A collisionless plasma is modeled by the Vlasov-Poisson system in three space dimensions. A fixed background of positive charge, which is independent of time and space, is assumed. The situation in which mobile negative ions balance the positive charge as |x|→∞ is considered. Hence the total positive charge and the total negative charge are infinite. Smooth solutions with appropriate asymptotic behavior are shown to exist locally in time. Conditions for continuation of these solutions are also established.

Convergence of a Difference Scheme for the Vlasov--Poisson--Fokker--Planck System in One Dimension

A finite difference scheme is proposed for the periodic one-dimensional Vlasov--Poisson--Fokker--Planck system which is related to the scheme in [C.\ Cheng and G.\ Knorr, {\em J. Comp.\ Phys}., 22 (1976), pp. 330--351]. The density