# Linear transport and ODEs for weakly regular vector fields

## Lecturer

Stefano Bianchini, SISSA

## Location

601 Pao Yue-Kong Library

## Time

• June 12, 2017 15:00-17:00
• June 13, 2017 09:00-11:00
• June 15, 2017 09:30-11:30

## Abstract

These lectures concern the PDEs of linear transport (in conservative or advective form)

$$\partial_t \rho + \dive (\rho \mathbf b) = 0, \quad u_t + \mathbf b \cdot \nabla u = 0,$$

and their relations with the ODE

$$\frac{d}{dt} x = \mathbf b(t,x).$$

The aim is to show that as in the classical/regular case, a synergy between the two formulations allows to develop an advanced theory about flows generated by weakly regular vector fields.

We will try to cover the following subjects:

• classical formulations and Lagrangian representations,
• the renormalization theory,
• singular integrals and explicit compactness estimates,
• a Lagrangian formulation of uniqueness and the Bressan conjecture.