Weak geodesic flow on a semidirect product and global solutions to the periodic Hunter–Saxton system

Geodesic flows on semidirect-product Lie groups: geometry of singular measure-valued solutions

The EPDiff equation (or dispersionless Camassa-Holm equation in 1D) is a well known example of geodesic motion on the Diff group of smooth invertible maps (diffeomorphisms). Its recent two-component extension governs geodesic motion on the semidirect product Diff sF , where F denotes the space of scalar functions. This paper generalizes the second construction to consider geodesic motion on Dif...

Geodesic Deviation and Weak-Field Solutions

Global weak solutions to the Nordström-Vlasov system

The Nordström-Vlasov system is a Lorentz invariant model for a selfgravitating collisionless gas. We establish suitable a-priori-bounds on the solutions of this system, which together with energy estimates and the smoothing effect of “momentum averaging” yield the existence of global weak solutions to the corresponding initial value problem. In the process we improve the continuation criterion ...

Global weak solutions to the relativistic Vlasov-Maxwell system revisited

In their seminal work [3], R. DiPerna and P.-L. Lions established the existence of global weak solutions to the Vlasov-Maxwell system. In the present notes we give a somewhat simplified proof of this result for the relativistic version of this system, the main purpose being to make this important result of kinetic theory more easily accessible to newcomers in the field. We show that the weak so...

Global weak solutions for a periodic two-component μ-Hunter-Saxton system

This paper is concerned with global existence of weak solutions for a periodic twocomponent μ-Hunter-Saxton system. We first derive global existence for strong solutions to the system with smooth approximate initial data. Then, we show that the limit of approximate solutions is a global weak solution of the two-component μ-Hunter-Saxton system. 2000 Mathematics Subject Classification: 35G25, 35L05