Calculus of Variations and Geometric Measure Theory
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M. Colombo

Flows of non-smooth vector fields and degenerate elliptic equations

created by colombom on 30 Jul 2015

[BibTeX]

Phd Thesis

Inserted: 30 jul 2015
Last Updated: 30 jul 2015

Year: 2015

Abstract:

The Vlasov-Poisson system is a classical model in physics used to describe the evolution of particles under their self-consistent electric or gravitational field. The existence of classical solutions is limited to dimensions d <4 under strong assumptions on the initial data, while weak solutions are known to exist under milder conditions. However, in the setting of weak solutions it is unclear whether the Eulerian description provided by the equation physically corresponds to a Lagrangian evolution of the particles.

The main results of this thesis show that weak solutions of Vlasov-Poisson are Lagrangian and obtain global existence of weak solutions under minimal assumptions on the initial data. In order to obtain this property of solutions, we develop in the thesis several general tools concerning the connection between the Lagrangian and the Eulerian structure of transport equations with non-smooth vector fields.

In 1989, Di Perna and Lions exploited this connection showing that Sobolev regularity for vector fields in Rd, 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. We present a notion of maximal flow for non-smooth vector fields which allows for finite-time blow up of the trajectories; existence and uniqueness of this maximal flow holds under only local assumptions on the vector field.

The connection between Eulerian and Lagrangian point of view appears also in the semigeostrophic system, a model used in meteorology to describe athmospheric-ocean flows. In this context, Lagrangian solutions exist thanks to a change of variable that reduces the problem to a dual system. In the thesis, we discuss the existence of Eulerian (distributional) solutions for the semigeostrophic system in the physical space.

Finally, we present a traffic problem that can be formulated both in Lagrangian terms and as a very degenerate elliptic equation; we provide a regularity result for the gradient of solutions of this equation.

Keywords: degenerate elliptic equation, Continuity/transport equation, Traffic models, Vlasov-Poisson sysyem, Semigeostrophic system, Lagrangian/Eulerian solution


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