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TU Berlin

Inhalt des Dokuments

Lectures

Fatiha Alabau-Boussouira (Université Paul Verlaine-Metz, France)

Control of coupled systems of hyperbolic PDE's and applications



Ľubomír Baňas (Heriot Watt University, Edinburgh, Great Britain)

Numerical methods for dynamical micromagnetism



Barbara Niethammer (Universität Bonn, Germany)

Analysis of coagulation equations



Felix Otto (Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany)

Optimal estimates in stochastic homogenization


 


Abstracts

 

Fatiha Alabau-Boussouira: Control of scalar and coupled systems of hyperbolic PDE's and applications


An abstract infinite dimensional control system consists of an inhomogeneous evolution equation, for which the inhomogeneity depends on a time-dependent function u -called the control- with values in a suitable functional space. Given an initial state and a final desired state in an appropriate functional space, we may search for the existence of a control u, such that the trajectory of the controlled system starting from the initial state reaches the final state at a given time T>0. Control theory deals in particular with the questions of existence, construction and properties of such controls. If the control theory for scalar evolution PDE's has been intensively studied since the eighties, the control theory of coupled systems of evolution PDE's is, by contrast, much less developed.

 We shall present in the first part of the course some motivations on the control of coupled wave systems, by a reduced number of controls, that is when the number of controls is strictly less than the number of unknowns (or equations) of the system. Such questions naturally raise in the building of robust controls, insensitive to small unknown perturbations of the initial data for scalar evolution equations. It also raises in the question of existence of simultaneous controls for systems coupled in parallel. The first part of the course will present the control of scalar wave equations. We shall then present some recent results on the control of coupled systems and applications.


(Homepage of Fatiha Alabau-Boussouira)

 

Ľubomír Baňas: Numerical methods for dynamical micromagnetism


  1. Introduction to micromagnetism, the Landau-Lifshitz-Gilbert (LLG) equation. Overview of numerical methods for the LLG equation

  2. Maxwell-Landau-Lifshitz-Gilbert equations. LLG with magnetostriction

  3. Harmonic heat flow and wave map equations

  4. Stochastic LLG equation and the augmented LLG equation for thermal recording.

(Homepage of Ľubomír Baňas)

 

Barbara Niethammer: Analysis of classical coagulation equations

  1. Smoluchowski's coagulation equation: modeling aspects and basic properties

  2. Global existence results for unbounded kernels

  3. Scale invariance and the scaling hypothesis

  4. Existence of self-similar solutions

 

(Homepage of Barbara Niethammer)

     

Felix Otto: Optimal estimates in stochastic homogenization


In many applications, one has to solve an elliptic equation with coefficients that vary on a length scale much smaller than the domain size. We are interested in a situation where the coefficients are characterized in statistical terms: Their statistics are assumed to be translation invariant and to decorrelate over large distances. In this situation, the solution operator behaves -on large scales- like the solution operator of an elliptic problem with  homogeneous, deterministic coefficients! Hence the relation between the statistics of the coefficients and the value of the homogenized coefficients is of practical importance.

 Theory provides a formula for the homogenized coefficients, based on the construction of a "corrector", which defines harmonic coordinates. This formula has to be approximated in practise. In this course, we give sharp estimates on the corrector that allow to assess these approximations. We also give some sharp estimates on the homogenization error itself. These estimates use tools from statistical mechanics (spectral gap, logarithmic Sobolev inequality) and elliptic regularity theory (De Giorgi, Nash). This is joint work with A. Gloria, S. Neukamm, and D. Marahrens.


(Homepage of Felix Otto)

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