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# PDEs in fluid dynamics and material science

Friday, 5th February 2021

**Online Symposium**

For information on how to access the event, please contact Henning Reinken via: henning.reinken(at)itp.tu-berlin.de

Guests are welcome!

# Programme

15:00 - 15:50 | Resonance phenomena and construction of metamaterialsAgnes Lamacz-Keymling (Universität Duisburg-Essen) |

15:50 - 16:00 | Coffee Break |

16:00 - 16:50 | Solving ill posed problems Eduard Feireisl (Czech Academy of Sciences/TU Berlin) |

16:50 | Discussion |

# Abstracts

**Resonance phenomena and construction of metamaterials**

**Agnes Lamacz-Keymling** (Universität Duisburg-Essen)

In the first setting we analyze the Helmholtz equation in a complex domain where a sound absorbing structure at a part of the boundary is modelled by a periodic geometry with periodicity *ε* > 0. A resonator volume of thickness *ε* is connected with thin channels (opening *ε*^{3}) with the main part of the macroscopic domain. We analyze solutions in the limit* ε* → 0 to find that the effective system can describe sound absorption.

The second part of the talk focuses on the mathematical analysis of negative index materials which is connected to a study of singular limits in Maxwell’s equations. We present a result on homogenization of the time harmonic Maxwell’s equations in a complex geometry. The homogenization process is performed in the case that many (order *η*^{−3}) small (order *η*^{1}), flat (order *η*^{2}) and highly conductive (order *η*^{−3}) metallic split-rings are distributed in a domain Ω ⊂ *R*^{3}. We determine the effective behavior of this metamaterial in the limit *η* → 0. We show that even though both original materials (metal and void) have the same positive magnetic permeability *μ*_{0} > 0, the effective Maxwell system exhibits, depending on the frequency, a negative magnetic response. Furthermore, we demonstrate that combining the split-ring array with thin, highly conducting wires can effectively provide a negative index metamaterial.

**Solving ill posed problems**

**Eduard Feireisl** (Czech Academy of Sciences/TU Berlin)

Some fundamental equations of continuum fluid mechanics are ill posed in the class of global-in-time solutions. Still numerical experiments yield in many cases unique results. We discuss the situation from the point of view of analysis. We introduce several concepts of generalized solutions and study convergence of their consistent approximations, in particular certain numerical methods. A new concept of S-convergence is developed to describe the limit even if the approximate solutions exhibit oscillatory behavior. Possible scenarios that may give rise to turbulence will be shortly discussed.