Arbeitsgruppe Analysis und Partielle Differentialgleichungen
S5B1  Graduate Seminar on Advanced Topics in PDE (winter term 2014/2015)
Instructors: Prof. Dr. Herbert Koch,
Prof. Dr. Christoph Thiele,
Dr. Roland Donninger
Course Description:
A first organizatorial meeting will be Friday October 10, 2014.

October 17, 2014: Polona Durcik (University of Bonn)

Title: A proof of the A_2 theorem.

Abstract: We present an alternative proof of the A_2 theorem given by A. Lerner, F.
Nazarov. Their approach is based on pointwise estimates of CalderonZygmund
operators with dyadic sparse operators.
 October 24, 2014: Gennady Uraltsev (University of Bonn)

Title: Quadratic Carleson in the Walsh case.

Abstract:
We will present V. Lie's proof of the weak L^2 boundedness of the quadratic Carleson
operator revisited in the Walsh case. We hope that this approach clarifies the main
ideas of the proof while simplifying some technical estimates. Finally, we will
highlight the additional machinery needed to prove strong L^2 bounds directly.
 October 31, 2014:
Shaoming Guo (University of Bonn)

Title: Geometric propf of Bourgain's L^2 bounds of the maximal operator along
analytic vector fields.

Abstract:
We will apply the timefrequency decomposition initiated by
Lacey and Li to provide a geometric proof of Bourgain's L^2 bounds of the
maximal operator along analytic vector fields.
 November 7, 2014: Michal Warchalski (University of Bonn)

Title: Wittwer's inequality via outer measure spaces.
 Abstract: We will present a proof of a generalization
of Wittwer's inequality with an arbitrary reference measure given by
Christoph Thiele, Sergei Treil and Alexander Volberg. For this we will
use embeddings into outer measure spaces and concavity arguments.
 November 14, 2014:
Christian Zillinger (University of Bonn)

Title: Linear inviscid damping for monotone shear flows.
 Abstract: We will present a proof of linear stability, scattering and
damping for monotone shear flow solutions to the 2D Euler equations both
in an infinite and finite periodic channel. A particular focus will be
on the additional boundary effects arising in the latter setting.
 November 21, 2014:

The first speaker [2:15 pm]: Damiano Foschi (Universita di Ferrara)

Title: Local wellposedness of semilinear Schrodinger equations under minimal
smoothness assumptions for the nonlinearity
 Abstract:
The problem of local wellposedness for semilinear Schrodinger equations $ i u_t + \Delta u = f(u) $ is well understood for smooth nonlinearities. When we consider powerlike nonlinear terms of the form $ f(u) = u^{p1} u $ the degree of the power is also a measure of the smoothness (near zero) of the nonlinearity. A simple scaling argument can show that local wellposedness for the initial value problem with data in the Sobolev space $ H^s $ requires that $ p \leq 1 + 4/(n2s)_+ $. This scaling condition alone usually is not sufficient. Known results require also some lower bound for $ p $: Cazenave and Weissler (1990) proved LWP with $ p > \floor{s} + 1 $; arguments of Ginibre, Ozawa and Velo (1994) allowed to relax the condition to $ p > s $; Pecher (1997) improved to $ p > s1 $ when $2 < s < 4$, and $ p > s2 $ when $s \geq 4$; recently Uchizono and Wada (2012) obtained LWP with $p < s/2$ when $ 2 < s < 4 $. We will show that these lower bounds for $ p $ are not yet optimal. For example when $ s=4 $ we will show how to obtain LWP for $ p > 3/2 $.

The second speaker [3:15 pm]:
Bartosz Trojan (University of Wroclaw)

Title: Bourgain's logarithmic lemma: 2parameter case.
 Abstract:
We discuss 2parameter generalization of Bourgain's logarithmic lemma arising
in the context of pointwise ergodic theory.
 November 28, 2014: Mariana Smit Vega Garcia

Title: New developments in the lower dimensional obstacle problem
 Abstract:
We will describe the Signorini, or lowerdimensional obstacle problem,
for a uniformly elliptic, divergence form operator $L = $
div$(A(x)\nabla)$ with Lipschitz continuous coefficients. We will give
an overview of this problem and discuss some recent developments,
including the optimal regularity of the solution and the regularity of
the free boundary. This is joint work with Nicola Garofalo and Arshak
Petrosyan.
 December 5, 2014: Pavel ZorinKranich (University of Bonn)
 December 12, 2014: Wenhui Shi (University of Bonn)

Title: A higher order boundary Harnack inequality.
 Abstract:
We will present a higher order boundary Harnack inequality
for harmonic functions and show its application to the free boundary
problems. This is a method due to De Silva and Savin.
 January 16, 2015: Pawel Biernat (University of Bonn)

Title: Formal
construction of singular solutions to harmonic map heat flow.
 Abstract:
Heat flow for harmonic maps is known to produce finitetime
singularities from smooth initial data. These singular solutions
arise for a large class of initial data and present a major
obstacle in solving the heat flow equation for arbitrarily large
times. I will show how to (formally) construct such singular
solutions using matched asymptotics and how to determine their
blowup rate (the speed with which the singularity forms).
 January 23, 2015: Emil Wiedemann (University of Bonn)

Title: Weak Solutions for the 2D Stationary Euler Equations.
 Abstract:
We present recent work by A. Choffrut and L. Szekelyhidi
on the stationary Euler equations. It is proved that in
any L^2neighbouhood of a smooth solution, there exist
infinitely many weak solutions.
Surprisingly, this is true even in two dimensions.
 January 30, 2015: Mariusz Mirek (University of Bonn)

Title: Recent developments in discrete harmonic analysis.
 Abstract:
In recent times  particularly the last two decades  discrete
analogues in harmonic analysis have gone through a period of
considerable changes and developments. This is due in part to Bourgain's
pointwise ergodic theorem for the squares on L^p, (p>1). The main aim of
this talk is to discuss recent developments in discrete harmonic analysis.
We will be mainly concerned with the discrete maximal functions and
singular integral operators along polynomial mappings. We will also
discuss twoparameter discrete analogues. All the results are subjects of
the ongoing projects with Elias M. Stein, Bartosz Trojan and Jim Wright.

Title: VW Car
 Abstract:
I will motivate and present a version of Bourgain's
multifrequency lemma with two bounded rvariation hypotheses due to
Oberlin.
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