SHORT Course: On the ODE associated to nonsmooth vector fields

Monday 6 May 2024, 00:00 → Friday 17 May 2024, 00:00 Europe/Rome

Ex-ISEF/Building-Main Lecture Hall (GSSI)

Lecturer 
Stefano Modena stefano.modena@gssi.it
 

Timetable and workload
• May 14, 9am - 11am;
• May 15, 9am - 11am;
• May 16, 9am - 11am;
• May 20, 9am - 11am;
• May 21, 3pm - 5pm;
• May 24, 9am - 11am *.
* All lectures will be held at ex-ISEF MLH, except for the one on May 24 that will take place in Room A. For further information, please contact apde_aq@gssi.it
 

Course description and outcomes 
For “smooth" vector fields $u(t,x)$, it is well known that the associated ODE $\gamma’(t) = u(t, \gamma(t))$ has a unique solution for every initial datum $\gamma(0) = x$. If $u$ is not smooth, this is in general not true anymore. However, if the vector field $u$ has some weak differentiability property (e.g. it is Sobolev), it is still possible to solve “uniquely” (in a suitable sense) the ODE associated to $u$, by means of the so called "regular Lagrangian flow". Aim of the course is to provide an introduction to the theory of regular Lagrangian flows, with a particular focus on the regularity and stability estimates proven by Crippa and De Lellis in 2008.