Workshop 1

This workshop will take place

from February 15th to February 16th 2019 at
Université Paris Pierre et Marie Curie (Paris 6)

It will consist in 3 mini-courses given by:

• Dorothee Frey : Mild solutions of the Navier-Stokes equations.

Navier-Stokes equations are one of the most prominent models for the description of viscous fluid flows. The global existence and smoothness of strong solutions to Navier-Stokes equations in three dimensions is still unsolved, and is one of the seven Millennium Prize problems. We will focus in this lecture on so-called mild solutions, and establish well-posedness results in Lebesgue and Sobolev spaces. The proofs make use of various fundamental methods of harmonic analysis, and will provide a first encounter with e.g. Sobolev embeddings, heat kernels, Riesz transforms, BMO spaces and commutators, some of which will be studied in full detail in the Master class later on.

• Victor Lie : Hilbert transform along curves.

• Keith Rogers : Pointwise convergence to initial data. Slides.

I will review recent progress for Carleson's question for the Schrödinger equation $i\partial_t u +\Delta u=0$ with initial data $u_0$ in the Sobolev space $H^s(\mathbb{R}^n)$. That is,  for which $s$ can we be sure that $u(x,t)\to u_0(x)$ as $t\to 0$ for almost every $x\in \mathbb{R}^n$. First I will present examples, due to Bourgain, Lucà and myself, which show that $s\ge \frac{1}{2} - \frac{1}{2(n+1)}$ is necessary. We will see that spread out interference-type behaviour becomes a problem when $n\ge 2$. I will then present maximal estimates, due to Du, Guth, Li and Zhang, which show that $s>\frac{1}{2} - \frac{1}{2(n+1)}$ is sufficient. Loosely speaking, these are a consequence of Strichartz-type estimates that improve for spread out solutions. We will also consider the fractal dimension version of the problem and the analogous questions for other PDE.

Map of the campus Paris 6 : Map

***** Program *****

Friday, February 15th (Room 103 in "Aile 24/25")
09:30 - 10:00: Welcome (+coffee)
10:00 - 12:00: Keith Rogers I
12:00 - 14:00:  Lunch
13:45 - 15:45: Keith Rogers II

15:45 - 16:15:  Coffee break
16:15 - 18:15: Victor Lie I

Saturday, February 16th (Amphiteatre Astier)
8:45 - 9:00:  Small coffee break
9:00 - 11:00: Victor Lie II
11:00 - 13:00:  Lunch
13:00 - 14:45: Dorothee Frey I
14:45 - 15:15:  Coffee break
15:15 - 17:00: Dorothee Frey II