University of Pisa

Title: Around the transport equation with non-smooth velocity: uniqueness, mixing and loss of regularity

The (linear) transport equation associated to a given, divergence-free velocity field appears as part of more complicated (systems of) equations that describe the dynamics of incompressible fluids (for example Euler and Navier-Stokes equations); typically, in these context, the velocity field is no longer given, but is an unknown, and its regularity is not granted a priori. It is therefore of interest to study the transport equation associated to non-smooth velocity fields. In these lectures I will focus on the following issues: existence and uniqueness of solutions, and, if time allows, "mixing" and loss of regularity.

Concerning the "mixing" phenomenon, the starting point is a conjecture by A. Bressan which states that, under certain assumptions, the "mixing scale" of the flow associated to the velocity field decays at most exponentially in time. Despite the fact that this conjecture has been proved almost fifteen years ago in some relevant cases (by G. Crippa and C. De Lellis) there are relatively few examples of flows which actually exhibit such exponential decay. I will illustrate some of these examples, and the connection to similar problem in discrete dynamical systems.

Finally, a strictly related issue is the loss of regularity of solutions (of the transport equation): for non-smooth velocity fields it may happen that even for smooth initial data the solution has no regularity (in the space variable) for every positive time.

SISSA

Title: Topology of level sets of harmonic functions

The purpose of this course is to show how, combining algebraic topology and real analytic geometry, one can deduce interesting results on the structure of level sets of harmonic functions. We will start by defining the fundamental class and the local dimension, and then discuss classical examples in dimension two. We will show that level sets of a non constant harmonic function on R^2 cannot have bounded components (a global result) and that at each singular point the number of branches of the level set is even (a local result). We will then generalize these constructions to higher dimensions.

University of Pisa

Title: Evolution by curvature of planar networks

I will discuss the analysis of the motion by curvature of networks of curves in the plane, presenting the state of the art about existence and uniqueness of solutions, singularity formation and asymptotic behavior of the flow.