Dr. Maria Colombo, ETH Zürich, Vortrag: Flows of non-smooth vector fields and applications to PDEs

Thursday, 07.01.2016 16:30 im Raum M5

Mathematik und Informatik

The classical Cauchy-Lipschitz theorem shows existence and uniqueness of the flow of any sufficiently smooth vector field in R^d. In 1989, Di Perna and Lions proved that Sobolev regularity for vector fields, 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 give an overview of the topic and we introduce 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.

Angelegt am Thursday, 27.08.2015 14:17 von shupp_01
Geändert am Monday, 04.01.2016 14:57 von shupp_01
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