Date 
Speaker and title 
 
 
22/01/24  Roland Donninger (Wien) 
12:30  Selfsimilar blowup for wave maps 
 
 Wave maps are the fundamental example of geometric wave equations and they have been studied extensively in the last 30 years. One particular line of research is concerned with singularity formation in finite time which often manifests itself by the presence of selfsimilar solutions. I will present recent progress in the study of the stability of selfsimilar wave maps at the critical regularity level. This is based on joint work with David Wallauch. 
 
 
15/01/24  Helmut Harbrecht (Basel) 
11:30  Wavelet compressed, modified Hilbert transform
in the spacetime discretization of the heat equation 
 
 
11/12/23  Michael Feischl (TU Wien) 
11:30  Stochastic collocation for dynamic micromagnetism

 
 We consider the stochastic LandauLifschitzGilbert equation, an SPDE model for
dynamic micromagnetism. We first convert the problem to a (highly nonlinear) PDE
with parametric coefficients using the DossSussmann transform and the LevyCiesielsky
parametrization of the Brownian motion. We prove analytic regularity of the parameter
tosolution map and estimate its derivatives. These estimates are used to prove convergence
rates for piecewisepolynomial sparse grid methods. Moreover, we propose novel
timestepping methods to solve the underlying deterministic equations. 
 
 
27/11/23  Simon Blatt (Salzburg) 
11:30  Analyticity of solutions to fractional partial differential equations 
 

In this talk, we explore the topic of analyticity of solutions to elliptic equations in a fresh context, tracing back to Hilbert's 19th problem. While Bernstein's work in 1904 yielded early results for classical elliptic partial differential equations, fractional and nonlocal equations have remained less explored, with limited results available, mostly restricted to special cases like the HartreeFock and Boltzmann equations.
We will present known results before delving into our recent discoveries, spanning from unique fractional equations related to knot energies to semilinear and completely nonlinear integrodifferential equations, encompassing sminimal surfaces. Central to our findings is the use of novel refined estimates that account for the longrange interactions within these equations. 
 
 
20/11/23  Gianluca Ceruti (Innsbruck) 
11:30  A retraction perspective on dynamical lowrank approximation 
 
 The availability of more powerful hardware has opened up new opportunities for exploring the timeevolution of highdimensional problems that describe phenomena of interest. However, from a discrete perspective, tackling these challenges comes with increased computational and storage complexity, leading to bottlenecks. This makes timedependent model order reduction strategies highly attractive in this context. Among these strategies, variational approaches like DiracFrenkel have garnered significant interest. In this contribution, we delve into one such approach: Dynamical LowRank (DLR) approximation. We provide an overview of the main elements, examine the limitations in terms of time accuracy, and introduce a retraction perspective on the topic. We demonstrate how retractions can be employed to explore wellknown DLR techniques, such as the KSL and KLS algorithms, and to devise novel approaches. In particular, we show that these retractions are wellsuited for constructing curve approximations on manifolds, with a local truncation error of order three. This introduces new perspectives on addressing current timeaccuracy constraints in recent robust DLR algorithms. 
 
 
10/11/23  Workshop on PDEs and Uncertainty Quantification 
 Claudia Garetto (London), Michael Kunzinger (Wien), Edoardo Patelli (Glasgow) 
 
 
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 pth power nonlinearity for initial data of very low
Sobolevregularity. 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
halfline. The result improves the wellposedness results for Sobolev
data without the support condition and, in some cases, gives
Sobolevwellposedness 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 LandauDe Gennes minimization problems of nematic liquid crystals and ferronematic systems 
 


21/03/23  Akansha Sanwal (Innsbruck) 
12:30  Low regularity wellposedness for dispersion generalised KPI equations 
 
 The talk concerns new wellposedness results for the dispersion generalised KPI equations in R2 with initial data in anisotropic Sobolev spaces. For strong dispersion, we show
global wellposedness in L2. This is achieved by exploiting transversality in the resonant case via bilinear Strichartz estimates and
nonlinear LoomisWhitney inequality. For small dispersion, the equations cannot be solved by Picard iteration and we use
frequencydependent time localisation.
This is based on joint work with Robert Schippa (Karlsruhe Institute of Technology, Germany).

 
 
14/03/23  Gissell EstradaRodriguez (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 YangMills equation outside a ball 
 
 Inspired by the recent work of Jendrej and Lawrie on soliton resolution for equivariant wave maps, we consider the 4dimensional equivariant YangMills 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 semiarid regions and are a prime example of a selforganisation 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 longrange 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 selforganisation 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 singlespecies 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 singlespecies solution branches as the singlespecies 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. 
 
 
06/12/22  Lukas 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  Multidomain spectral methods for dispersive PDEs 
 

We discuss numerical approaches for various nonlinear dispersive PDEs, in particular the nonlinear Schrödinger equation, the Kortewegde Vries equation and the BenjaminOno equation. In particular we are interested in numerical methods on the compactified real line thus avoiding nonreflecting 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 semilinear dispersive problems with nonsmooth 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 (ParisSaclay), 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 twodimensional 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 1sphere, and the surrounding fluids satisfy the incompressible Stokes equations in timedependent 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 finitedimensional 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 noncontact with the outer boundary.

 
 
 
 