This year's lecture focused on transport equations (linear and non-linear) corresponding to non-regular vector fields, and the associated ordinary differential equations. This is a classic subject, dating back to the work of Joseph Liouville (professor at the Collège de France) in the 19th century. R.J. Di Perna and P.-L. Lions were responsible for the first extension of classical theory to fields of vectors that are not very regular (with integrable first derivatives and bounded divergence) (the so-called " de Di Perna-Lions " theory). Since then, various authors have attempted to weaken the regularity assumptions, and the main aim of the lecture has been to highlight a few situations where it is possible to deal with vector fields that are less regular but with geometric assumptions such as monotonicity, anti-monotonicity, growth..
The (linear) transport equations studied in the lecture are the non-conservative equations.