# Junior Research Group Singularity formation in nonlinear PDEs

The mathematical description of dynamical processes in terms of time-dependent partial differential equations (PDEs) plays a fundamental role in many areas of science. In many models nonlinearities appear naturally due to self-reinforcing processes. If the nonlinear effects dominate smoothing mechanisms such as dissipation or dispersion singularities may form in finite time. In such a scenario, also referred to as 'blowup', the amplitude of the solution diverges or discontinuities are formed. On the one hand, this may indicate certain limitations in the underlying modelling assumptions. However, in many cases mathematical singularities correspond to physical phenomena such as shock waves in hydrodynamics, light focussing in nonlinear fibre optics or the formation of black holes in gravitational collapse. This motivates the mathematical study of the details of singularity formation in nonlinear PDEs.

Our group focuses on the investigation of blowup dynamics in nonlinear wave equations and heat flows in the so-called energy supercritical case. We mainly use tools from functional analysis, operator theory and spectral analysis as well as ODE methods.

## Current members

Birgit Schörkhuber (head)

Elek Csobo (Postdoc)

## Recent preprints

I. Glogic, B. Schörkhuber

Nonlinear stability of homothetically shrinking Yang-Mills solitons in the equivariant case.

arXiv preprint 2019

I. Glogic, M. Maliborski and B. Schörkhuber

Threshold for blowup for the supercritical cubic wave equation

arXiv preprint 2019

I. Glogic, B. Schörkhuber.

Co-dimension one stable blowup for the supercritical cubic wave equation.

arXiv preprint 2018.

P. Biernat, R. Donninger and B. Schörkhuber.

Hyperboloidal similarity coordinates and a globally stable blowup profile for supercritical wave maps.

Accepted in Int. Math. Res. Not. arXiv preprint 2017.

## Events at KIT

23.-27.09 2019: Minisymposium *Analysis of Wave propagation* DMV Jahrestagung (organized by R. Mandel and B. Schörkhuber)

04.07.2019: Talk by Ziping Rao (University of Vienna) "Optimal blowup stability for the energy critical wave equation", 9h45 - 10h45, SR 3.61

12.02.2019: Talk by Elek Csobo (TU Delft) "Orbital stability of a Klein-Gordon equation with Dirac delta potentials", 11h00 - 12h00, SR 2.67

25.10.2018: Talk by Hatem Zaag (Paris 13 University) "Blow-up for the Complex Ginzburg-Landau in some critical case"

23.07.2018: Minisympsium Nonlinear dispersive equations - blowup, solitons and long-time behavior at the* Conference on Mathematics of Wave Phenomena*, Department of Mathematics, KIT (organized by B. Schörkhuber)

Name | Tel. | |
---|---|---|

Elek Csobo | elek.csobo@kit.edu | |

M.Sc. David Lichti | +49 721 608 48728 | david.lichti@kit.edu |

Dr. Birgit Schörkhuber | +49 721 608 46197 | birgit.schoerkhuber@kit.edu |