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

Variational-Hemivariational Inequalities with Applications to Contact Mechanics

Justyna Ogorzaly
Jagiellonian University, Krakow
Tuesday, 5. April 2022 - 9:00 to 10:00
We will present the existence and uniqueness results for the special classes of nonlinear variational-hemivariational inequalities. Then, we will consider concrete contact problems and we will show how these problems lead to the different type of variational-hemivariational inequalities.

Two-phase compressible/incompressible Navier-Stokes system with inflow-outflow boundary conditions

Aneta Wróblewska-Kamińska
Institute of Mathematics, Polish Academy of Sciences
Tuesday, 22. March 2022 - 9:00 to 10:00
I will show proof of the existence of a weak solution to the compressible Navier-Stokes system with singular pressure that explodes when density achieves its congestion level. This is a quantity whose initial value evolves according to the transport equation. We then prove that the “stiff pressure" limit gives rise to the two-phase compressible/incompressible system with congestion constraint describing the free interface. We prescribe the velocity at the boundary and the value of density at the inflow part of the boundary of a general bounded C2 domain. For the positive velocity flux, there are no restrictions on the size of the boundary conditions, while for the zero flux, a certain smallness is required for the last limit passage. This result is based on a work with Milan Pokorný and Ewelina Zatorska.
M. Pokorný, A. Wróblewska-Kamińska, E. Zatorska. Two-phase compressible/incompressible Navier–Stokes system with inflow-outflow boundary conditions. arXiv:2202... more

About a 1D Green-Naghdi model with vorticity and surface tension for surface waves

Colette Guillopé
Paris-East Créteil University
Tuesday, 15. March 2022 - 10:15 to 11:15
The Green-Naghdi model is currently the most well-known model used for numerical simulations of waterfront streams, even in setups that incorporate vanishing depth (at the shoreline) and wave breaking. Regardless of their many favorable circumstances, the Green-Naghdi equations specially take into consideration neglected rotational effects, which are significant for wind-driven waves, waves riding upon a sheared current, waves near a ship, or tsunami waves approaching a shore. The Green-Naghdi system is first rewritten as an equivalent system by using an adequate change of unknowns. We show that solutions to the model here considered, with voracity and surface tension, may be obtained by a standard Picard iterative scheme. No loss of regularity is involved with respect to the initial data. Moreover solutions exist at the same level of regularity as for 1st order hyperbolic symmetric systems, i.e. with a regularity in... more

Poroelasticity Interacting with Stokes Flow

Boris Muha
University of Zagreb
Tuesday, 15. March 2022 - 9:00 to 10:00
We consider the interaction between an incompressible, viscous fluid modeled by the dynamic Stokes equation and a multilayered poroelastic structure which consists of a thin, linear, poroelastic plate layer (in direct contact with the free Stokes flow) and a thick Biot layer. The fluid flow and the elastodynamics of the multilayered poroelastic structure are fully coupled across a fixed interface through physical coupling conditions (including the Beavers-Joseph-Saffman condition), which present mathematical challenges related to the regularity of associated velocity traces. We prove existence of weak solutions to this fluid-structure interaction problem with either (i) a linear, dynamic Biot model, or (ii) a nonlinear quasi-static Biot component, where the permeability is a nonlinear function of the fluid content (as motivated by biological applications). The proof is based on constructing approximate solutions through Rothe... more