Young PDEs Mini Symposium: April 29, 2022

The Young PDEs Mini Symposium will take place on April 29 at Boltzmanngasse 9A, in the Schrödinger Lecture Hall of the Erwin Schrödinger International Institute for Mathematics and Physics. It aims to bring together PhD students and postdoctoral researchers in the field of partial differential equations. There will be six 20 minute presentations in English each followed by an equally long collegial discussion. Snacks will be provided.

The Young PDEs Mini Symposium is organized by Thomas Körber with assistance from Michael Eichmair. Among the Young PDE honoris causa, there will be Wilhelm Schlag from Yale University and Roland Donninger, David Fajman, and Michael Eichmair from the University of Vienna.


8:30-9:10 Liam Urban

Nonlinear stability of Big Bang formation for FLRW solutions with hyperbolic spatial geometry within the Einstein scalar field system

The Strong Cosmic Censorship conjecture, in the context of cosmological spacetimes, posits that, for generic initial data, the Kretschmann curvature scalar blows up where causal geodesics become incomplete. In my talk, I will outline an approach (based on joint work with D. Fajman) to verify this conjecture in presence of Einstein scalar field matter for initial data close to that of Friedman-Lemaître-Robertson-Walker (FLRW) spacetimes with hyperbolic spatial geometry. This takes the form of showing the past nonlinear stability of the blow-up exhibited by such FLRW solutions (the Big Bang formation) within a system of hyperbolic and elliptic geometric PDEs. Crucially and in contrast to previous results, we actually do not rely on the specific spatial geometry to a very large degree, potentially making our approach viable for other, more involved Einstein systems.

9:10-9:50 Matthias Ostermann

Stable blowup in the whole space for geometric wave equations

A universal phenomenon among partial differential equations of wave-type is the spontaneous breakdown of their solutions. This non-linear effect may be encountered in evolution equations via blowup formation from smooth initial data in finite time. Such blowup profiles are known in closed form for, e.g., the co-rotational Wave Maps Equation and the equivariant Yang-Mills Equation in supercritical space dimensions. The central question for their role in the dynamics of the equations concerns their stability, i.e., if this blowup also arises for all “near” initial data. Until recently, stability results have only been available in the region of the backwards lightcone of the singularity. The first global stability result for co-rotational wave maps from (1 + 3)-dimensional Minkowski spacetime into the three-sphere was proved by P. Biernat, R. Donninger and B. Schörkhuber. One of the key insights is the construction of hyperboloidal similarity coordinates. In my joint work with R. Donninger, we have now expanded the methods to establish a new stability theory for blowup in the Yang-Mills equation in the whole space. This result holds in all odd space dimensions. A better understanding of the geometric and analytic interplay of hyperboloidal similarity coordinates with different equations of wave-type will be relevant for further progress and the problem whether a well-defined continuation of the Cauchy evolution after the singularity is possible.

9:50-10:30 David Wallauch

Strichartz estimates and blowup stability for energy critical nonlinear wave equations

Energy critical nonlinear wave equations exhibit many fascinating phenomena, among them finite time blowup of solutions which start from smooth and compactly supported initial data. Whenever one has a solution which exhibits finite time blowup, a natural question that arises is the stability of the singular behavior under perturbations of the initial data. In this talk, I will present the derivation of Strichartz estimates for radial wave equations with a potential in similiarity coordinates. These estimates are then used to establish the asymptotic blowup stability of the ODE blowup profile of the energy critical wave equation with a power nonlinearity under radial perturbations in dimensions 3≤ d≤ 6 at the lowest possible regularity.

10:30-11:10 Maciej Maliborski

Characteristic approach to the soliton resolution

I will present a toy model for studying the soliton resolution phenomenon. The soliton resolution conjecture states that global-in-time generic solutions of nonlinear dispersive wave equations resolve for late times into a superposition of decoupled nonlinear bound states (solitons) and radiation. Our main objective is to illustrate the advantages of employing outgoing null (or asymptotically null) foliations in analyzing the relaxation processes due to the dispersal of energy by radiation. Based on results from joint work with Piotr Bizoń and Bradley Cownden.

11:10-11:50 Thomas Körber

Uniqueness of large stable constant mean curvature spheres in asymptotically flat 3-manifolds

Stable constant mean curvature spheres encode important information on the asymptotic geometry of initial data sets for isolated gravitational systems. By the work of C. Nerz, the asymptotic region of such an initial data set is foliated by large stable constant mean curvature spheres. In this talk, I will present a short proof (joint with M. Eichmair) that, in the case where the initial data set has non-negative scalar curvature, the leaves of this foliation are the only large stable constant mean curvature spheres that enclose the center of the initial data set.

11:50 - 12:30 Irfan Glogić

Stable finite-time aggregation for the supercritical Keller-Segel model

We consider a parabolic-elliptic system which arises in modeling the bacterial growth or stellar dynamics. This model, which goes under the name of Keller-Segel, is known to admit a closed form self-similar blowup solution in all mass-supercritical dimensions. In this talk, I will report on my joint work in progress with Birgit Schörkhuber, in which we establish global nonlinear stability of the aforementioned blowup profile. This solves a more than two decades old open problem.

Young PDE speakers

Irfan Glogić (he/him)

Irfan Glogic

Thomas Körber (he/him)

PC Thomas Körber

Maciej Maliborski (he/him)

PC Maciej Maliborski

Matthias Ostermann (he/him)

PC Matthias Ostermann

Liam Urban (he/they)

PC Liam Urban

David Wallauch (he/him)

PC David Wallauch

Young PDE honoris causa

Roland Donninger (he/him)

PC Roland Donninger

David Fajman (he/him)

PC David Fajman

Michael Eichmair (he/him)

PC Michael Eichmair

Wilhelm Schlag (he/him)

PC Wilhelm Schlag


Please contact Thomas Körber (he/him) with any questions related to the Young PDEs Mini Symposium.

Uni Wien
Uni Wien