Flows of non-smooth vector fields and applications to PDEs

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.


This seminar is organized within the PRIN 2012 Research project «Equazioni alle derivate parziali di tipo ellittico e parabolico: aspetti geometrici, disuguaglianze collegate, e applicazioni - Partial Differential Equations and Related Analytic-Geometric Inequalities» Grant Registration number 2012TC7588_003, funded by MIUR – Project coordinator Prof. Filippo Gazzola