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Maximal flows of non-smooth vector fifields and the Vlasov-Poisson system

  • Maria Colombo (University of Zurich)
A3 01 (Sophus-Lie room)

Abstract

In 1989, Di Perna and Lions showed that Sobolev regularity for vector fields in R^d, with bounded divergence and a growth assumption, is sufficient to establish existence, uniqueness and stability of a generalized notion of flow, consisting of a suitable selection among the trajectories of the associated ODE. Their theory relies on a growth assumption on the vector field which prevents the trajectories from blowing up in finite time; in particular, it does not apply to fast-growing, smooth vector fields. In this seminar we present a notion of maximal flow for non-smooth vector fields which allows for finite-time blow up of the trajectories. We show existence and uniqueness under only local assumptions on the vector field and we apply the result to a kinetic equation, the Vlasov-Poisson system, where we describe the solutions as transported by a suitable flow in the phase space. This allows, in turn, to prove existence of weak solutions for general initial data.

Katja Heid

MPI for Mathematics in the Sciences Contact via Mail

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