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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

On the passage from nonlinear to linearized viscoelasticity

Martin Kružík
Institute of Information Theory and Automation, Czech Academy of Sciences
Tuesday, 30. April 2019 - 9:00 to 10:00
We formulate a quasistatic nonlinear model for nonsimple viscoelastic materials at a finite-strain setting in the Kelvin‘s-Voigt‘s rheology where the viscosity stress tensor complies with the principle of time-continuous frame-indifference. We identify weak solutions in the nonlinear framework as limits of time-incremental problems for vanishing time increment. Moreover, we show that linearization around the identity leads to the standard system for linearized viscoelasticity and that solutions of the nonlinear system converge in a suitable sense to solutions of the linear one. The same property holds for time-discrete approximations and we provide a corresponding commutativity result. Main tools used are rigidity estimates and gradient flows in metric spaces. This is a joint work with M. Friedrich (Munster).

On Payne's nodal set conjecture for the p-Laplacian

Vladimir Bobkov
University of West Bohemia
Tuesday, 16. April 2019 - 9:00 to 10:00
The Payne conjecture asserts that the nodal set of any second eigenfunction of the zero Dirichlet Laplacian intersects the boundary of the domain. We prove this conjecture for the p-Laplacian assuming that the domain is Steiner symmetric. (In particular, the domain can be a ball.) The talk is based on the joint work with S. Kolonitskii.

Analysis of a degenerate and singular volume-filling cross-diffusion system modeling biofilm growth

Nicola Zamponi
Charles University
Tuesday, 9. April 2019 - 9:00 to 10:00
We analyze the mathematical properties of a multi-species biofilm cross-diffusion model together with very general reaction terms and mixed Dirichlet-Neumann boundary conditions on a bounded domain. This model belongs to the class of volume-filling type cross-diffusion systems which exhibit a porous medium-type degeneracy when the total biomass vanishes as well as a superdiffusion-type singularity when the biomass reaches its maximum cell capacity. The equations also admit a very interesting non-standard entropy structure. We prove the existence of global-in-time weak solutions, study the asymptotic behavior and the uniqueness of the solutions, and complement the analysis by numerical simulations that illustrate the theoretically obtained results.

Getting familiar with the General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC)

Petr Pelech
Charles University
Tuesday, 2. April 2019 - 9:00 to 10:00
One common feature of new emerging technologies is the fusion of the very small (nano) scale and the large scale engineering. The classical enviroment provided by single scale theories, as for instance by the classical hydrodynamics, is not anymore satisfactory. It is the main goal of GENERIC to provide a suitable framework for developing and formulating new thermodynamic models [1]. As an inevitable consequence, the mathematical nature of these new models is different. For instance, the governing equations cannot be written as conservation laws and hence finding a new suitable mathematical structure is necessary. A possible solution seems to be given by the so-called Symmetric Hyperbolic Thermodynamical Consistent (SHTC) equations [2], for which local well-posedness is known [3-6]. There are also numerical computations based on the discontinuous Galerkin method [7], however, a rigorous mathematical analysis of the global-in-time existence has still not been developed.

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