Depending on the scale of description, the equations modeling the movement of fluids are very different: Newton's laws describe the evolution of the elementary particles constituting a fluid, whereas Euler or Navier-Stokes equations describe the macroscopic behaviour of a fluid. I will discuss two different, though not unrelated questions related to this. On the one hand one can be interested in the modeling that gives rise to these equations, and in proving that one can pass from one scale of description to the other in a rigorous way (this is in relation with Hilbert's sixth problem). On the other hand solving the equations, and describing their solutions is a challenging problem which is widely open in many situations. I will discuss some results, more or less recent, in both directions.

November 07, 2023, 16:00h (coffee and cakes from 15:30h). Room BA 10A, 10th floor, Getreidemarkt 9

Modern technologies and biological systems, such as temperature-responsive polymers and lipid rafts, take advantage of engineered inclusions, or natural heterogeneities of the medium, to obtain novel composite materials with specific physical properties. To model such situations by using a variational approach based on the gradient theory, the potential and the wells have to depend on the spatial position, even in a discontinuous way, and different regimes should be considered. In the case where the scale of the small heterogeneities is of the same order of the scale governing the phase transition, the interaction between homogenization and the phase transitions process leads to an anisotropic interfacial energy. In the case where the heterogeneities of the material are of a larger scale than that of the diffuse interface between different phases, it is noted that there is no macroscopic phase separation and thermal fluctuations play a role in the formation of nanodomains. In addition, a characterization of the large-scale limiting behavior of viscosity solutions to nondegenerate and periodic Eikonal equations in half-spaces is given.

June 23, 2022, 15:20h. Kontaktraum, 6th floor, Gusshausstraße 25-27

The lecture will discuss some mathematical problems of liquid crystals, including a new proof that stationary solutions to the Onsager equation with the Maier-Saupe interaction are radially symmetric, some remarks on the singular bulk potential introduced by Majumdar and the author, and a description of joint work with Lu Liu on exterior problems in the 2D one-constant Oseen-Frank theory.

December 05, 2019, 16:00h. Boecklsaal, Stiege 1, 1st floor, Karlsplatz 13

We will first present the mathematical theory of Diffusion and Heat Propagation as seen from the classical point of view of Partial Differential Equations. This will allow us to mention concepts of great beauty and relevance in what follows. We will then introduce our favorite topic of Nonlinear Diffusion, embodied in equations like the Porous Medium Equation, and focused on the existence and properties of the very interesting geometrical objects called Free Boundaries. We will complete the presentation with a review of personal work on diffusion equations involving long distance interactions in the form of Fractional Laplacian Operators. This topic has been intensely developed in the last decade and is still in full bloom.

May 17, 2018, 16:00h. Getreidemarkt 9, 11th floor, TUtheSky

Cardiovascular diseases unfortunately represent one of the leading causes of death in Western countries. Mathematical models allow the description of the blood motion in the human circulatory system, as well as the interplay between electrical, mechanical and fluid-dynamical processes occurring in the heart. This is a classical environment where multiphysics processes have to be addressed. Appropriate numerical strategies can be devised to allow for an effective description of the fluid in large and medium size arteries, the analysis of physiological and pathological conditions, and the simulation, control and shape optimization of assisted devices or surgical prostheses. This presentation will address some of these issues and a few representative applications of clinical interest.

June 06, 2017, 16:00h. Getreidemarkt 9, 11th floor, TUtheSky

We present several stability results concerning smooth solutions of the compressible Euler and Navier-Stokes system in the class of measure-valued solutions. We show that strong and measure-valued solutions of these systems coincide as soon as the strong solution exists. The question if any measure-valued solution can be generated by a sequence of weak solutions will be also addressed.

May 31, 2016, 16:00h. Getreidemarkt 9, 11th floor, TUtheSky

A rubber band constrained to remain on a manifold evolves by trying to shorten its length, eventually settling on some minimal closed geodesic, or collapsing entirely. It is natural to try to consider a noisy version of such a model where each segment of the band gets pulled in random directions. Trying to build such a model turns out to be surprisingly difficult and generates a number of nice geometric insights, as well as some beautiful algebraic and analytical objects.

June 17, 2015, 16:00h. Getreidemarkt 9, 11th floor, TUtheSky

I discuss some recent progress in the study of stable self- similar blowup in energy-supercritical wave equations. In particular, I present a proof for the stability of the so- called ODE blowup for the energy-supercritical focusing wave equation without symmetry assumptions. This talk is based on joint work with Birgit Schörkhuber.

December 16, 2014, 14:00h. Wiedner Hauptstr. 8-10, green area, SEM 101C