# Registered Data

Contents

- 1 [CT038]
- 1.1 [02581] Well-Posedness and smoothness of geometric flows with nonlinear boundary conditions
- 1.2 [02692] Discovering extremal domains via shape optimization for passive tracers
- 1.3 [00486] Singularity formation in the Keller-Segel system
- 1.4 [01080] About Thermistor Problem: numerical study using Discrete Duality Finite Volume
- 1.5 [02566] A spherically symmetric and steady flow describing the motion of a viscous gaseous star

# [CT038]

## [02581] Well-Posedness and smoothness of geometric flows with nonlinear boundary conditions

**Session Date & Time**: 5D (Aug.25, 15:30-17:10)**Type**: Contributed Talk**Abstract**: Geometric flows are geometric evolution equations often depicting physical phenomena. We consider a class of geometric flows of order $2m \in 2\mathbb{N}$ describing evolving $n$-manifolds attached to fixed hypersurfaces with some nonlinear boundary conditions. We modify the theory of Maximal Regularity to accommodate quasilinear parabolic PDEs with such boundary conditions. For initial conditions in $W_p^{2m – \frac{2m}{p}}, p\geq \max\{2m, \frac{n}{2m}\}$) we show well-posedness and instantaneous smoothing of the solution on a maximal interval of existence.**Classification**:__35Kxx__,__35Qxx__**Author(s)**:**Daniel Goldberg**(Technion-Israel Institute of Technology)

## [02692] Discovering extremal domains via shape optimization for passive tracers

**Session Date & Time**: 5D (Aug.25, 15:30-17:10)**Type**: Contributed Talk**Abstract**: Work in passive tracers investigates how properties of a tracer distribution depend on boundary conditions and properties of the underlying fluid flow. We apply shape optimization to discover extremal domains for Poiseuille flow informed by analytic predictions of spatial moments - such as mean, effective diffusivity, skewness - derived in prior work. With this combination of asymptotic formulas and numerical study, we find and report on surprising nonlinear behavior depending on shape parameters.**Classification**:__35Kxx__,__90C90__**Author(s)**:**Manuchehr Aminian**(California State Polytechnic University Pomona)

## [00486] Singularity formation in the Keller-Segel system

**Session Date & Time**: 5D (Aug.25, 15:30-17:10)**Type**: Contributed Talk**Abstract**: The talk will give an up-to-date result on singularity formation in the Keller-Segel system in $\mathbb{R}^d$. For $d \geq 2$, there exist blowup solutions that are of Type II with finite mass. Blowup rates are completely quantized. There also exist Type I blowup solutions with infinite mass for $d = 3, 4$. The constructed solution is asymptotically self-similar with a logarithmic correction to its profile which is either radial or non-radial.**Classification**:__35M10__,__35K40__,__35K55__,__35K57__**Author(s)**:**Van Tien Nguyen**(National Taiwan University)

## [01080] About Thermistor Problem: numerical study using Discrete Duality Finite Volume

**Session Date & Time**: 5D (Aug.25, 15:30-17:10)**Type**: Contributed Talk**Abstract**: We propose a DDFV for a coupled nonlinear parabolic-elliptic equations. The system is known as a generalization of the Thermistor problem which models a temperature dependent electrical resistor. We first establish some a prior estimates satisfied by the sequences of approximate solutions. Then, it yields the compactness of these sequences. Passing to the limit in the numerical scheme, we finally obtain that the limit of the sequence of approximate solutions is a weak solution to the problem under study.**Classification**:__35M30__,__35K92__,__35J46__,__65N08__**Author(s)**:**Manar Lahrache**(Moulay Ismail University, Faculty of Science, Meknes, Morocco)

## [02566] A spherically symmetric and steady flow describing the motion of a viscous gaseous star

**Session Date & Time**: 5D (Aug.25, 15:30-17:10)**Type**: Contributed Talk**Abstract**: We consider a system of equations describing a spherically symmetric $n$-dimensional motion of a gaseous star, whose gas is viscous, heat-conducting, self-gravitating and bounded by the free-surface, and flows around a central rigid sphere. We discuss first unique existence of the solution to the corresponding stationary problem, and next do a large-time behaviour of the flow, under a certain restricted but physically plausible condition on parameters and initial data.**Classification**:__35M33__,__35Q30__,__35R35__,__35Q85__,__76N10__**Author(s)**:**Morimichi Umehara**(University of Miyazaki)