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 Saffirio 
  Validity of the Boltzmann equation for short range potential
  In 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 This is an extension of the unpublished thesis by King (appeared after the well known result of Lanford for a system of hard spheres).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.
  
  

• 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 on
  function spaces close to L^1, is that the usual Calderón-Zygmund
  decomposition fails to be effective, due to the modulation invariance
  properties of both operators. In this talk, we present several
  endpoint (near L^1 and L^1 x L^2) bounds for both the Carleson
  operator and the Walsh analogue of the bilinear Hilbert transform based on a 
  multifrequency 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)
  On multi-phase flow
  In this interactive talk I give an introduction to several physical
  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)
  The theory of fractional heat and porous medium equations
  Much recent research in the area of elliptic and parabolic equations has
  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)
  Denoising and Inpainting in Image Analysis:
  Variants of the Total Variation Regularization
  We discuss several 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)
  Mean curvature flow without singularities
  We study graphical mean curvature flow of complete solutions defined

  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.