Duyckaerts Mini-school Schedule
|Monday, February 13||
|Tuesday, February 14||
|Wednesday, February 15||
Thomas Duyckaerts, Dynamics of the energy critical wave equations:
These lectures concern the energy-critical focusing nonlinear wave equation. It is conjectured that any solution of this equation that is bounded in the energy space is asymptotically the sum of a finite number of decoupled solitons and a radiation term. My goal is to prove this conjecture for radial solutions in space dimension 3 and to give partial results in the general case.
This is based on joint works with Hao Jia, Carlos Kenig and Frank Merle.
The lectures will start at an elementary level and will be accessible to non-specialist.
Hao Jia, Universality of blow up for small energy wave maps
We will introduce a recent application of the channel of energy argument to the energy critical wave maps into the sphere. The main issue is to eliminate the so-called “null concentration of energy”. We will explain why this is an important issue in the wave maps. Combining the absence of null concentration with suitable coercive property of energy near traveling waves, we show a universality property for the blow up of wave maps with energy that are just above the co-rotational wave maps. Difficulties with extending to arbitrarily large wave maps will also be discussed. This is joint work with Duyckaerts, Kenig and Merle.
Jacek Jendrej, Construction of concentrating bubbles for the energy-critical focusing wave equation
In the first part I will consider the problem of constructing a type II blow-up solution with a prescribed weak limit at the blow-up time (the so-called asymptotic profile). It turns out that the speed of concentration is related to the behavior of the asymptotic profile at the blow-up point.
In the second part, I will talk about radial solutions which approach in infinite time a sum of two energy bubbles. One of the bubbles is steady, whereas the other concentrates at a specific rate.
Giuseppe Negro, Maximizers for a Strichartz norm of small solutions to the energy-critical NLW
This is a report on a work in progress (PhD thesis directed by Thomas Duyckaerts and Keith Rogers). We consider the energy-critical NLW in spatial dimension 5. For small values of $\delta>0$, we define a functional $J=J(\delta)$ as the supremum of the spacetime $L^4$ norm over all solutions having energy equal to $\delta$. We show that $J(\delta)$ is indeed an attained maximum and we give a Taylor expansion of it around $\delta = 0$. The first order term of such an expansion turns out to be related to the best constant of the 5-dimensional energy-Strichartz inequality found by Bez \& Rogers.