Robert T. Glassey

Singularity formation in a collisionless plasma could occur only at high velocities

The global existence problem is studied for regular solutions of the relativistic Vlasov-Maxwell equations. If it is assumed that the plasma density vanishes a priori for velocities near the speed of light, then regular solutions with arbitrary initial data exist in all of space and time. This assumption is either postulated for a solution or is arranged for all solutions through a modification of the equations themselves.

Absence of shocks in an initially dilute collisionless plasma

The Cauchy Problem for the relativistic Vlasov-Maxwell equations is studied in three space dimensions. It is assumed that the initial data satisfy the required constraints and have compact support. If in addition the data have sufficiently smallC2 norm, then a uniqueC1 solution to this system is shown to exist on all of spacetime.

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.

On certain global solutions of the cauchy problem for the (classical) coupled Klein-Gordon-Dirac equations in one and three space dimensions

If the data of the Dirac spinor field is suitably chosen, global solutions of the Klein-Gordon-Dirac Equations with Yukawa coupling are shown to exist for which the meson field remains free. In one space dimension, conditions are found under which the spinor field does not decay uniformly to zero, thus precluding a scattering theory in the H1 (IR1) norm. In three space dimensions, the existence of free Hn(103) (n≧0) asymptotic spinors is established and a scattering theory is developed.

Asymptotic stability of the relativistic Maxwellian via fourteen moments

Abstract Consider a relativistic Maxwellian distribution of matter in equilibrium. It is shown that small perturbations which vanish at spatial infinity and are governed by the relativistic Boltzmann equation converge to the equilibrium as t → ∞, under appropriate conditions on the scattering kernel. The convergence is proved in a Sobolev space of arbitrarily high order.

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.

Decay of classical Yang-Mills fields

The classical Yang-Mills equations in four-dimensional Minkowski space are invariant under the conformal group. The resulting conservation laws are explicitly exhibited in terms of the Cauchy data at a fixed time. In particular, it is shown that, for any finite-energy solution of the Yang-Mills equations, the local energy tends to zero ast→∞.

Asymptotic behavior of solutions to certain nonlinear Schrödinger-Hartree equations

AbstractThe asymptotic behavior of solutions to the Cauchy problem for the equation