On some variational Properties of the Allen-Cahn Action Functional in the sharp Interface Limit
Mittwoch, 11.5.11, 16:15-17:15, Hörsaal II, Albertstr. 23b
We will address the problem of existence, uniqueness and regularity of minimizers of the Allen-Cahn action functional in the sharp interface limit. In the smooth case, a Hamiltonian interpretation for the problem will be given and some related conserved quantities will be deduced.\n
On averaged equations for turbulent flows, with applications to Magnetohydrodynamics
Mittwoch, 18.5.11, 16:15-17:15, Hörsaal II, Albertstr. 23b
I will make a short review of some continuous approximations to the Navier-Stokes equations. Next, I will present some recent results about approximate deconvolution models, derived with ideas similar to image processing. Finally, I will show the rigorous convergence of solutions towards those of the averaged fluid equations and applications to filtered Magnetohydrodynamics equations.
Legendrian knots and nonalgebraic contact Anosov flows on 3-manifolds
Mittwoch, 6.7.11, 16:15-17:15, Hörsaal II, Albertstr. 23b
We describe a surgery construction in a neighborhood of a transverse Legendrian knot that gives rise to new contact structures preserved by Anosov flows. In particular, this includes examples on many hyperbolic 3-manifolds, and it gives contact Anosov flows that are not quasigeodesic. As a byproduct, this also yields quasigeodesic pseudo-Anosov flows.
Vollständige Ricci-flache Kählermetriken auf offenen komplexen Flächen
Dienstag, 12.7.11, 16:15-17:15, Raum 127, Eckerstr. 1
Ich werde eine neue Konstruktion von nichtkompakten vollständigen Ricci-flachen 4-Mannigfaltigkeiten vorstellen, die auf der Geometrie elliptisch gefaserter algebraischer Flächen beruht. Eine weitere Zutat ist die Asymptotik uniform beschränkter Lösungen der komplexen Monge-Ampere-Gleichung. Die resultierenden Geometrien beinhalten beispielsweise Familien von sogenannten ALG-Räumen, deren Existenz von Physikern vorhergesagt wurde. Ich möchte auch kurz besprechen, inwieweit diese Konstruktion - zusammen mit anderen, schon bekannten - vermutlich alle 4-dimensionalen Beispiele abdeckt.
Numerical schemes for short wave long wave interaction equations
Mittwoch, 13.7.11, 16:15-17:15, Hörsaal II, Albertstr. 23b
We prove convergence of some finite volume type numerical schemes for short wave long wave interaction equations. These are systems of coupled nonlinear PDEs consisting of a Schroedinger equation for the short waves and a hyperbolic conservation law modeling the long waves. We use compensated compactness along with energy estimates. We present some computations and a numerical study of some open problems. This is joint work with M. Figueira of CMAF-UL.\n
The Gradient Flow Of O’Hara’s Knot Energies
Mittwoch, 20.7.11, 16:15-17:15, Hörsaal II, Albertstr. 23b
All of us know how hard it can be to decide whether the cable spaghetti lying in front of us is really knotted or whether the knot vanishes into thin air after pushing and pulling at the right strings. In this talk we approach this problem using gradient flows of a family of energies introduced by O’Hara in 1991. We will see that this allows\nus to transform any closed curve into a special set of representatives - the stationary points of these energies - without changing the type of knot. We prove longtime existence and smooth convergence to stationary points for\nthese evolution equations.\n
The Newtonian Limit of Geometrostatics
Dienstag, 26.7.11, 16:15-17:15, Raum 127, Eckerstr. 1
Geometrostatics is the geometric theory of static isolated relativistic systems modeling for example static stars or black holes in our universe. It is governed by a system of (mainly) elliptic PDEs relating a 3-dimensional Riemannian metric and a positive function describing the geometry of space at any given point of time and the passage of time, respectively. While these PDEs have been intensively studied in the last century from both analytic and physical viewpoints, their geometric nature had not been exploited to its full extent. We will present a number of new insights into geometrostatics that mainly rely on a geometric approach but are also interrelated with analytic and physical aspects of the theory. For example, we will discuss geodesics of the corresponding Lorentzian spacetimes, uniqueness questions, mass and center of mass of geometrostatic systems, and last but not least their Newtonian limit -- the question of whether and how these\nrelativistic systems have Newtonian counterparts.
Regularity for degenerate elliptic Monge-Ampere equations
Mittwoch, 3.8.11, 17:15-18:15, Hörsaal II, Albertstr. 23b
It is well known that C^{1,1} convex solutions of elliptic Monge-Ampere equations are smooth. A natural question is whether this holds for degenerate elliptic Monge-Ampere equations. In this talk, we will review known results and present new ones. Assumptions of our results are phrased in terms of the set where degeneracy occurs. \n