Mathematisches Institut der Universität Bonn

Arbeitsgruppe Analysis und Partielle Differentialgleichungen

S5B1 - Graduate Seminar on Advanced Topics in PDE

Winter term 2014/2015

Instructors

Talks

• 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 Calderon-Zygmund 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 time-frequency 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 well-posedness for semilinear Schrodinger equations $ i u_t + \Delta u = f(u) $ is well understood for smooth nonlinearities. When we consider power-like nonlinear terms of the form $ f(u) = |u|^{p-1} 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/(n-2s)_+ $. 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 > s-1 $ when $2 < s < 4$, and $ p > s-2 $ 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: 2-parameter case.
  Abstract: We discuss 2-parameter 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 lower-dimensional 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 Zorin-Kranich (University of Bonn)
Title: Variational Walsh Carleson
Abstract: I will motivate and present a version of Bourgain's multi-frequency lemma with two bounded r-variation hypotheses due to Oberlin.
• 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 finite-time 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 blow-up 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^2-neighbouhood 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 two-parameter discrete analogues. All the results are subjects of the ongoing projects with Elias M. Stein, Bartosz Trojan and Jim Wright.