slideshow 3

Seminar on Partial Differential Equations

usually takes place each Tuesday at 09:00 in IM, rear building, ground floor
Chair: Šárka Nečasová, Milan Pokorný

Problems of linearized Navier-Stokes equations in frequency domain

Viktor Hruška
Czech Technical University in Prague
Tuesday, 1. November 2022 - 10:15 to 11:15
For aeroacoustics applications, it is very tempting to work with linearized equations in frequency domain. Not only are the solutions simpler in overall, but also some variables are defined solely in the frequency domain (such as impedance and related quantities). In quiescent media, the frequency domain calculations enjoy well-deserved popularity. However, great caution must be taken when applying the same mathematical steps to linearized Navier-Stokes equations, although technically there is no apparent difficulty. The talk will present a specific case of the method failure: despite the fact that the acoustic quantities are indeed small, the hydrodynamics cannot be governed by the linearized equations. The final part of the talk will be a discussion of some papers that use the linearized equations.

On the derivation of viscoelastic models for Brownian suspensions

Richard Höfer
Institut de Mathématiques de Jussieu
Tuesday, 1. November 2022 - 9:00 to 10:00
We consider effective properties of suspensions of inertialess, rigid, anisotropic, Brownian particles in Stokes flows. Recent years have seen tremendous progress regarding the rigorous justification of effective fluid equations for non-Brownian suspensions, where the complex fluid can be described in terms of an effective viscosity. In contrast to this (quasi-)Newtonian behavior, anisotropic Brownian particles cause an additional elastic stress on the fluid. A  rigorous derivation of such visco-elastic systems starting from particle models is completely missing so far. In this talk I will present first results in this direction starting from simplified microscopic models where the particles evolve only due to rotational Brownian motion and cause a Brownian torque on the fluid. In the limit of infinitely many small particles with vanishing particle volume fraction, we rigorously obtain an elastic stress on the fluid in... more

Analysis of generalized Aw-Rascle type model

Nilasis Chaudhuri
Imperial College
Tuesday, 25. October 2022 - 9:00 to 10:00
In this talk we consider the multidimensional generalization of the Aw-Rascle system for vehicular traffic. For a large class of initial data and the periodic boundary conditions, we prove the existence of a global-in-time measure-valued solution. Moreover, using the relative energy technique, we show a weak-strong uniqueness result. Next, we analyse the similar generalization in one dimensional setting by considering the offset function is a gradient of a singular function of the density and the resulting system of PDEs can be used to model traffic or suspension flows with the maximal packing constraint taken into account. We study the so-called 'hard congestion limit' and show the convergence of solutions towards a weak solution of a hybrid free-congested system.

Convergence of shape design solutions for the Navier-Stokes equations

John Sebastian Simon
Kanazawa University
Tuesday, 11. October 2022 - 9:00 to 10:00
We investigate the behavior of dynamic shape design problems for fluid flow at large time horizon. In particular, we shall compare the solutions of a dynamic shape optimization problem with that of a stationary problem and show that the solution of the former converges to that of the latter. The convergence of domains is based on the $L^\infty$-topology of their corresponding characteristic functions which is closed under the set of domains satisfying the cone property. Lastly, a numerical example is provided to show the occurrence of such convergence.