SUMMER & WINTER SCHOOLS

# 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)

\begin{equation}
\partial_t \rho + \dive (\rho \mathbf b) = 0, \quad u_t + \mathbf b \cdot \nabla u = 0,
\end{equation}

and their relations with the ODE

\begin{equation}
\frac{d}{dt} x = \mathbf b(t,x).
\end{equation}

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.