TOPOLOGIES OF CONTINUITY FOR SKEW-PRODUCT FLOWS INDUCED BY CARATHEODORY ODEs, AND APPLICATIONSMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

The aim of this talk is to show how to apply tools of non-autonomous dynamical systems to Carathéodory differential equations, i.e. to study problems of the type

x^'=f(t,x), x(t_0 )=x_0 (1)

where

f:R×R^N→R^N is Borel measurable, for every compact set K⊂R^N there exists a real-valued function m^K (∙)∈L_loc^1 such that for almost every t∈R, one has |f(t,x)|≤m^K (t), for all x∈K for almost every t∈R, f(t,∙ ) is continuous.

The Greek and cosmopolitan mathematician Constantin Carathéodory proved that, under the previous assumptions, any problem like (1) admits a solution in an extended sense, that is, an absolutely continuous function defined on an interval I⊂R containing t_0, so that (1) holds almost everywhere in I.

In particular, we show how to define a continuous skew-product flow in order to study the qualitative behavior of the solutions. The idea of skew-product flow dates back to the groundbreaking work of Bebutov (1941) and it has become a fundamental tool in the study of non-autonomous ordinary differential equations. In order to retrieve a group structure on the evolution of a non-autonomous system (immediate for autonomous systems, where one can build a flow using the solutions), the idea is to jointly track the solution and the evolution of the vector field in time. If these two components depend continuously on time, initial vector field and initial data, then the obtained map defines a continuous skew-product flow.

The study of the topologies of continuity for a skew-product flow generated by a Carathéodory differential equations is a classical question which was initially posed by Miller and Sell (1968, 1970), and then treated by many authors but, despite its potential interest, the classic theory has not been conveniently developed in the field of non-linear differential equations.

Such result opens several directions of research. We will present a theoretical applications from Longo et al. (2017) and a motivational example inspired by Rasmussen et al. (2016).

This is a joint work with Prof. Sylvia Novo and Prof. Rafael Obaya.