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

On the incompressible limit for some tissue growth models

Tomasz Debiec
University of Warsaw
Tuesday, 13. December 2022 - 9:00 to 10:00
I will discuss some approaches to mathematical modelling of living tissues, with application to tumour growth. In particular, I will describe recent results on to the incompressible limit of a compressible model, which builds a bridge between density-based description and a geometric free-boundary problem by passing to the singular limit in the pressure law.

The talk is divided in two parts. First, I discuss the rate of convergence of solutions of a general class of nonlinear diffusion equations of porous medium type to solutions of a Hele-Shaw-type problem. Then, I shall present a two-species tissue growth model — the main novelty here is the coupling of both species through the so-called Brinkman law which is typically used in the context of visco-elastic media, where the velocity field is linked to the total population pressure via an elliptic equation.

Regularity and Convergence to Equilibrium for a Navier-Stokes-Cahn-Hilliard System with Unmatched Densities

Helmut Abels
University of Regensburg
Tuesday, 15. November 2022 - 9:00 to 10:00
We study the initial-boundary value problem for an incompressible Navier-Stokes-Cahn-Hilliard system with non-constant density proposed by Abels, Garcke and Grün in 2012. This model arises in the diffuse interface theory  for binary mixtures of viscous incompressible fluids. This system is a generalization of the well-known model H in the case of fluids with unmatched densities. In three dimensions, we prove that any global weak solution (for which uniqueness is not known) exhibits a propagation of regularity in time and stabilizes towards an equilibrium state as time tends to infinity. Our analysis hinges upon the following key points: a novel global regularity result (with explicit bounds) for the Cahn-Hilliard equation with divergence-free velocity belonging only to the Leray-Hopf class, the energy dissipation of the system, the separation property for large times, a weak strong uniqueness type result, and the Lojasiewicz-Simon inequality.

On blowup for the supercritical quadratic wave equation

Elek Csobo
University of Innsbruck
Tuesday, 8. November 2022 - 10:15 to 11:15
See the attached file.

On the well-posedness of an inviscid fluid-structure interaction model

Amjad Tuffaha
American University of Sharjah
Tuesday, 8. November 2022 - 9:00 to 10:00
We consider the Euler equations on a domain with free moving interface. The motion of the interface is governed by a 4th order linear Euler-Bernoulli beam equation. The fluid structure interaction  dynamics are realized through normal velocity matching of the fluid and the structure in addition to the aerodynamic forcing due to the fluid pressure.
We derive a-priori estimates and construct local-in-time solutions to the system in the Sobolev space H^r, with r>5/2. We also establish uniqueness in the Sobolev space H^r with r>3. An important consequence of the existence theorem is that the Taylor-Rayleigh instability does not occur. This is joint work with Igor Kukavica.