# Oberseminar Analysis Summer 2013

## Organizers: B. Zwicknagl, S. Conti, H. Koch, S. Müller, B. Niethammer, M. Rumpf, B. Schlein, C. Thiele, J. López-Velázquez,

- Thursday, April 18, 2013, 2:15 p.m., Lipschitz-Saal

Chiara SaffirioIn this talk we introduce the problem of the validity of the Boltzmann equation. After a brief review, we focus on the case of a classical system of N particles interacting by means of a short range potential. We show that, in the low--density limit, the system behaves as predicted by the associated Boltzmann equation. This is an extension of the unpublished thesis by King (appeared after the well known result of Lanford for a system of hard spheres). Our analysis applies to any stable and smooth potential. The results presented are obtained in collaboration with M. Pulvirenti and S. Simonella.**Validity of the Boltzmann equation for short range potential**

- Thursday, May 2, 2013, 2:15 p.m., Lipschitz-Saal

Francesco di Plinio (Università di Roma Tor Vergata)**Endpoint behavior of modulation invariant singular integrals.**One of the main difficulties arising in the treatment of the Carleson

operator, and of the related bilinear Hilbert transform, acting onfunction spaces close to L^1, is that the usual Calderón-Zygmunddecomposition fails to be effective, due to the modulation invarianceproperties of both operators. In this talk, we present severalendpoint (near L^1 and L^1 x L^2) bounds for both the Carlesonoperator and the Walsh analogue of the bilinear Hilbert transform based on amultifrequency Calderon-Zygmund decomposition first introduced by Nazarov, Oberlin and Thiele. In particular, we discuss recent progress towards the solution of Konyagin's conjecture on almost everywhere convergence of lacunary Fourier and Walsh-Fourier series of functions in the Orlicz space L\log\log L(T). Partly joint work with Ciprian Demeter. - Thursday, May 16, 2013, 2:15 p.m., seminar room 2.040

Michael Helmers**Interfaces in discrete forward-backward diffusion equations**

We study the motion of interfaces in a diffusive lattice equation with

bistable nonlinearity and derive a free boundary problem with

hysteresis that describes the macroscopic evolution in the parabolic

scaling limit. To this end, we first present numerical results and

heuristic arguments for general bistable nonlinearities and discuss

the phenomena that appear for different types of initial data. Then we

rigorously justify the limit dynamics for single-interface data and a

piecewise affine nonlinearity. - Thursday, June 20, 2013, 2:15 p.m., Lipschitz-Saal

Martin Bock (Theoretical Biology, Bonn University)In this interactive talk I give an introduction to several physical**On multi-phase flow**

models of multi-phase flow. These models describe a fluid with several

components, with special focus on systems in the strongly dissipative

limit. First, I propose the Alt/Dembo model of the cytoplasm of

biological cells. Second, I generalize to N phases and arrive at a set

of equations initially proposed by Drew/Segel in the 70s. Third, I

construct the Navier-Stokes equation from mass and momentum conservation

with the help of non-equilibrium thermodynamics as proposed by

DeGroot/Mazur. Time permitting, I will sketch how similar thermodynamic

principles give rise to multi-phase models with additional hydrodynamic

variables like polarity. - Thursday, June 27, 2013, 2:15 p.m., seminar room 1.008

Juan Luis Vazquez (Universidad Autónoma de Madrid)Much recent research in the area of elliptic and parabolic equations has**The theory of fractional heat and porous medium equations**

been devoted to study the eect of replacing the Laplace operator, and its

usual variants, by a fractional Laplacian operator or other similar nonlocal

operators. Linear and nonlinear models are involved. I will describe recent progress made by me and collaborators on the topic of

nonlinear fractional heat equations, in particular when the

nonlinearities are of porous medium and fast diffusion type. The results cover existence and uniqueness of solutions, boundedness, regularity and continuous dependence, positivity and Harnack estimates, and symmetrization. Special attention is given to the construction of fractional Barenblatt solutions and asymptotic behaviour. - Thursday, July 4, 2013, 2:15 p.m., seminar room 2.040

Michael Goldman (MPI MiS, Leipzig)**Regularity and Strict Convexity of Homogenized Interfacial Energies**

In this talk I would like to present a recent joint work with A. Chambolle and M. Novaga about the differentiability and strict convexity properties of the stable norm. This function arises in the process of homogenization of interfacial energies in periodic media. We will see that the differentiability of the stable norm depends on the existence of gaps in the lamination made of some particular minimizers of these interfacial energies, the so-called plane-like minimizers. Our analysis heavily relies on the notion of calibrations for this problem. - Thursday, July 11, 2013, 2:15 p.m., seminar room 1.008

YuNing Liu (Universitaet Regensburg)**Single input controllability of a simplified fluid-structure interaction model**

We study a controllability problem for a simplified one

dimensional model for the motion of a rigid body in a viscous

fluid. The control variable is the

velocity of the fluid at one end. One of the novelties brought in with

respect to the existing literature consists in the fact that we use a

single scalar control. Moreover, we introduce a new methodology, which

can be used for other nonlinear parabolic systems, independently of

the techniques previously used for the linearized problem. This

methodology is based on an abstract argument for the null

controllability of parabolic equations in the presence of source terms

and it avoids tackling linearized problems with time dependent

coefficients.

- Thursday, July 18, 2013, 2:15 p.m., Lipschitz-Saal

Michael Bildhauer (Universitaet des Saarlandes)We discuss several variants of the "Total Variation Regularization**Denoising and Inpainting in Image Analysis:**

Variants of the Total Variation Regularization

Model" used both for Denoising and for Inpainting problems in image analysis. The main features are the investigation of the analytic properties of solutions and uniqueness results both for the solution and for the dual problem. - Thursday, July 18, 2013, 3:30 p.m., Lipschitz-Saal

Oliver Schnuerer (Universitaet Konstanz)We study graphical mean curvature flow of complete solutions defined**Mean curvature flow without singularities**on subsets of Euclidean space. We obtain smooth long time

existence. The projections of the evolving graphs also solve mean

curvature flow. Hence this approach allows to smoothly flow through

singularities by studying graphical mean curvature flow with one

additional dimension.

We present joint work with Mariel Sáez.