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Seminar on Partial Differential Equations

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

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.

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.

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