Geometry and 

University of Innsbruck

Seminar on Analysis and Numerics of PDEs

The schedule includes talks in the seminar series as well as closely related events at the University of Innsbruck. If you would like to be added to our mailing list, please contact Heiko Gimperlein.

Unless noted otherwise, the seminar takes place in Room 609 in the building of the mathematics departments, Technikerstrasse 13, 6020 Innsbruck. From the city center you reach the campus in less than 15 minutes by tram lines 2, 5 (direction Technik West) or bus lines K (direction Kranebitten), T (direction Technik West) to the stop Technik.

Date Speaker and title
22/01/24 Roland Donninger (Wien)
15/01/24 Helmut Harbrecht (Basel)
11/12/23 Michael Feischl (TU Wien)
20/11/23 Gianluca Ceruti (Innsbruck)
17/10/23 Phan Thanh Nam (Munich)
12:30 Critical mass in nuclear fission and isoperimetric inequalities
I will discuss the connection from the critical mass in nuclear fission described via the liquid drop model and the classical isoperimetric inequalities. In particular, I will address some recent results on the existence/nonexistence of minimizers, as well as several open questions. The talk is based on joint work with Rupert Frank.
04/07/23 Michael Oberguggenberger (Innsbruck)
12:30 Solutions to semilinear wave equations of very low regularity
The talk reports about recent joint work with Heiko Gimperlein.
First, we exhibit new phenomena concerning wellposedness and propagation of singularities for semilinear wave equations with p-th power nonlinearity for initial data of very low Sobolev-regularity. In one space dimension, solutions whose singular support propagates along any ray outside the light cone are obtained. These solutions exist for any Sobolev exponent s < 1/2 in space, while the singular support of any solution of higher regularity is known to be contained in the light cone.
Second, we establish a new wellposedness result for semilinear wave equations with Sobolev data whose Fourier transform is supported in a half-line. The result improves the wellposedness results for Sobolev data without the support condition and, in some cases, gives Sobolev-wellposedness for certain s < 0.
Extensions to higher space dimensions as well as some observations illuminating the phenomena are given.
16/05/23 Erich Wehrle (Collins Aerospace), industrial seminar
12:30, HSB6 Design optimization including dynamics and nonlinearity
25/04/23 Ruma Maity (Aalto, Helsinki)
12:30 Finite element methods for reduced Landau-De Gennes minimization problems of nematic liquid crystals and ferronematic systems
21/03/23 Akansha Sanwal (Innsbruck)
12:30 Low regularity well-posedness for dispersion generalised KP-I equations
The talk concerns new well-posedness results for the dispersion generalised KP-I equations in R2 with initial data in anisotropic Sobolev spaces. For strong dispersion, we show global well-posedness in L2. This is achieved by exploiting transversality in the resonant case via bilinear Strichartz estimates and nonlinear Loomis-Whitney inequality. For small dispersion, the equations cannot be solved by Picard iteration and we use frequency-dependent time localisation. This is based on joint work with Robert Schippa (Karlsruhe Institute of Technology, Germany).
14/03/23 Gissell Estrada-Rodriguez (UPC Barcelona / Oxford)
12:30 Macroscopic limits of kinetic equations for swarming
24/01/23 Maciej Maliborski (Vienna)
13:15 Soliton resolution for critical equivariant Yang-Mills equation outside a ball
Inspired by the recent work of Jendrej and Lawrie on soliton resolution for equivariant wave maps, we consider the 4-dimensional equivariant Yang-Mills equation outside a ball, for which solutions exist globally. We provide numerical evidence supporting the soliton resolution conjecture for our toy model: asymptotically, the solution approaches a superposition of rescaled kinks and radiation. Based on joint work with Piotr Bizon and Bradley Cownden.
11/01/23 Rolf Stenberg (Aalto University)
16:30 Stabilized Finite Element Methods for Contact Problems
16/12/22 Lukas Eigentler (Dundee / Bielefeld)
14:00 Modelling dryland vegetation patterns
Vegetation patterns are a ubiquitous feature of semi-arid regions and are a prime example of a self-organisation principle in ecology. In this talk, I present bifurcation analyses of two PDE models to (i) investigate the effects of nonlocal seed dispersal, and (ii) identify a mechanism that enables species coexistence despite competition for a limiting resource. First, I present a nonlocal model in which plant dispersal is modelled by a nonlocal convolution term, motivated by empirical data. Asymptotic analysis of the model is possible due to a scale difference between plant dispersal and water transport. I show that a condition for pattern onset in the model can be derived analytically, which indicates that long-range seed dispersal inhibits the onset of spatial patterns. Results on pattern existence and stability, obtained via a numerical continuation method, further show a change in the type of stability boundaries in the pattern's stability regions as dispersal distance is varied. This suggests increased resilience of patterns to reductions in precipitation due to long dispersal distances. Stability results further propose a resolution of a mismatch between previous mathematical models predicting movement of vegetation patterns and some field studies reporting stationary patterns.
Second, I reveal that the vegetation's self-organisation principle also acts as a coexistence mechanism. I present a multispecies model for two plant species that interact with a sole limiting resource. A stability analysis of the system's single-species patterns, performed through a calculation of their essential spectra, provides an insight into the onset of coexistence states. I show that coexistence solution branches bifurcate off single-species solution branches as the single-species states lose their stability to the introduction of a second species. Moreover, I present a comprehensive existence and stability analysis to establish key conditions, including a balance between the species' local competitive abilities and their colonisation abilities, for species coexistence in the model.
06/12/22Lukas Neumann (Innsbruck)
13:15 Initial boundary value problems for 1D pressureless gas dynamics
It is well known that solutions to the pressureless Euler system develop singularities in finite time also for very smooth initial data. Analytical difficulties generated by this fact become even more severe if one looks at the initial boundary value problem. We extended a potential method originally proposed by F. Huang and Z. Wang to the boundary value problem in the quarterspace. This allows to construct solutions that satisfy the inflow boundary conditions in an appropriate sense. If time allows we will discuss extensions to dual boundaries and uniqueness issues specific to the boundary value problem.
This talk is based on joint work with Abhrojyoti Sen, Michael Oberguggenberger and Manas. R. Sahoo.
22/11/22 Heiko Gimperlein (Innsbruck, inaugural lecture)
16:00 Anomalous diffusion and complex materials:
From models to simulations in space and time
15/11/22 Christian Klein (Bourgogne)
13:15 Multi-domain spectral methods for dispersive PDEs
We discuss numerical approaches for various nonlinear dispersive PDEs, in particular the nonlinear Schrödinger equation, the Korteweg-de Vries equation and the Benjamin-Ono equation. In particular we are interested in numerical methods on the compactified real line thus avoiding non-reflecting boundary conditions. As examples we consider the stability of breathers and the solutions for initial data not vanishing at infinity.
08/11/22 Alexander Ostermann (Innsbruck)
13:15 Bourgain techniques for error estimates at low regularity
Standard numerical integrators such as splitting methods or exponential integrators suffer from order reduction when applied to semi-linear dispersive problems with non-smooth initial data. In this talk, we focus on the cubic nonlinear Schrödinger equation with periodic boundary conditions. For such problems, we present and analyze (filtered) Fourier integrators that exhibit superior convergence rates at low regularity. Numerical examples illustrating the analytic results will be given.
This is joint work with Frederic Rousset (Paris-Saclay), Katharina Schratz (Sorbonne, Paris), Yifei Wu (Tianjin University, China) and Fangyan Yao (South China University, Guangzhou)
11/10/22 Sebastien Court (Innsbruck)
12:15 Feedback stabilization of a surface tension system modeling the motion of a two-dimensional soap bubble
The aim of this work is to design a feedback operator for stabilizing in infinite time horizon a system modeling the interactions between a viscous incompressible fluid and the deformation of a soap bubble. The latter is represented by an interface separating a bounded domain of $\R^2$ into two connected parts filled with viscous incompressible fluids. The interface is a smooth perturbation of the 1-sphere, and the surrounding fluids satisfy the incompressible Stokes equations in time-dependent domains. The mean curvature of the surface defines a surface tension force which induces a jump of the normal trace of the Cauchy stress tensor. The response of the fluids is a velocity trace on the interface, governing the time evolution of the latter, via the equality of velocities. The data are assumed to be sufficiently small, in particular the initial perturbation, that is the initial shape of the soap bubble is close enough to a circle. The control function is a surface tension type force on the interface. We design it as a finite-dimensional feedback operator, which enables us to define a control operator that stabilizes locally the soap bubble to a circle with an arbitrary exponential decay rate, up to non-contact with the outer boundary.