Séminaire Bourbaki
# Jonathan Hickman — Pointwise convergence for the Schrödinger equation , after Xiumin Du and Ruixiang Zhang

→
Europe/Paris

Hermite (IHP)
### Hermite

#### IHP

Description

For an initial datum f ∈ L2 (Rn ), consider the linear Schrödinger equation

?

iut − ∆x u = 0,

(x, t) ∈ Rn × R.

u(x, 0) = f (x)

In 1980, Carleson asked which additional conditions on f guarantee

(⋆)

lim u(x, t) = f (x)

for almost every x ∈ Rn .

t→0

More precisely, what is the minimal Sobolev regularity index s such that (⋆) holds

whenever f ∈ H s (Rn) ?

Whilst the n = 1 case was fully understood by the early 1980s, in higher

dimensions the situation is much more nuanced. Nevertheless, a recent series of

dramatic developments brought about an almost complete resolution of the

problem. First Bourgain 2016 produced a subtle counterexample demonstrating that

pointwise convergence can fail for certain f ∈ H s (Rn ) with s < n/(2n+2)

. Complementing

this, convergence was then shown to hold for s > n/(2n+2)

when n = 2 in a landmark

2(n+1)paper of Du, Guth and Li 2017 and later in all dimensions in equally important work

of Du and Zhang 2019.

This seminar will explore the positive result of Du and Zhang 2019. The argument

combines sophisticated modern machinery from harmonic analysis such as the

multilinear Strichartz estimates of Bennett, Carbery and Tao 2006 and the l2

decoupling theory of Bourgain and Demeter 2015. However, equally important are

a variety of elementary guiding principles, rooted in Fourier analysis, which govern

the behaviour of solutions to the Schrödinger equation. The talk will focus on these

basic Fourier analytic principles, building intuition and presenting a powerful toolbox

for tackling problems in modern PDE and harmonic analysis.