The seminar will take place on Monday, March 19, 2018 at 9:00 am in K4. Erika Maringová will give a lecture “Up to the boundary Lipschitz regularity for variational problems”.

Abstract: We prove the existence of a regular solution to a wide class of convex, variational integrals. Via technique of construction of the barriers we show that the solution is Lipschitz up to the boundary. For the linear growth case [1], we identify the necessary and sufficient condition to existence of solution; in the case of superlinear growth [2], we provide the sufficient one. The result is not restricted to any geometrical assumption on the domain, only its regularity plays the role. The talk will be based on two works,

[1] L. Beck, M. Bulíček, and E. Maringová.  Globally Lipschitz minimizers for variational problems with linear growth, accepted to ESAIM: COCV in 2017.

[2] M. Bulíček, E. Maringová, B. Stroffolini and A. Verde. A boundary regularity result for minimizers of variational integrals with nonstandard growth, accepted to Nonlinear Analysis in 2018.

The seminar will take place on Monday, March 5, 2018 at 9:00 am in K4. Tomáš Los will give a lecture “On three-dimensional flows of internal pore pressure activated Bingham fluids”.

Abstract: We are concerned with a system of partial differential equations describing internal flows of homogeneous incompressible fluids of Bingham type with activated boundary conditions.The Bingham activation threshold depends on internal pore pressure in the material, which is governed by an advection-diffusion equation. This model may be suitable for description of certain class of granular water-saturated materials. By suitably extending recent approaches by Chupin and Math ́e and Bulíček and Málek (see also a closely related work by Maringova and Zabensky), we prove long time and large data existence of weak solutions.
This is a joint work with A. Abbatiello, J. Málek, and O. Souček.

The seminar will take place on Monday, March 5, 2018 at 9:00 am in K4. Anna Abbatiello will give a lecture “On the existence of classical solution to the steady flows of generalized Newtonian fluid with concentration dependent power-law index”.

Abstract: Steady flows of an incompressible synovial fluid are described by a coupled system, consisting of the generalized Navier – Stokes equations and convection – diffusion equation with diffusivity dependent on the concentration and the shear rate. Cauchy stress behaves like power-law fluid with the exponent depending on the concentration. It makes the problem much more difficult than the standard model for power-law fluid in the analysis of the system of PDEs, since the variable exponent space W1,p(x) is a priori unknown. We investigate the question of the existence of a classical solution for the two dimensional periodic case.
This is a joint work with M. Bulíček and P. Kaplický.

[1] A. Abbatiello, M. Bulíček and P. Kaplický, On the existence of classical solution to the steady flows of generalized Newtonian fluid with concentration dependent power-law index, forthcoming.

The seminar will take place on Monday, February 26, 2018 at 9:00 am in K4. Prof. Zdeněk Strakoš will continue with the lecture “Decomposition into subspaces and operator preconditioning”.

Abstract: We will consider linear equations in the abstract infinite-dimensional Hilbert space setting with bounded, coercive and self-adjoint operators, which can represent, e.g., boundary value problems formulated via partial differential equations.
Efficient numerical solution procedures often incorporate transformation of the original problem using preconditioning. Motivated, in particular, by the works of Faber, Manteuffel, Parter, Oswald, Dahmen, Kunoth and Rude published in the early 90’s, we will present an abstract formulation of operator preconditioning based on the idea of decomposition of a Hilbert space into a finite number of (infinite-dimensional) subspaces, by formulating the main results using the concepts of norm equivalence and spectral equivalence of infinite-dimensional operators. Its goal is to describe in a concise way the common principles behind various adaptive multilevel and domain decomposition techniques using infinite-dimensional function spaces.