International Cooperation


Grant: GF1733849L
from 01/01/2017
to 31/12/2019
Grantor: Austrian Science Foundation (FWF)  Czech Science Foudation

Filters, Ultrafilters and Connections with Forcing


Objectives:
The project falls within the scope of Set Theory & Foundations of Mathematics, specifically Combinatorial Set Theory and Forcing. We will investigate new combinatorial and forcing methods for constructing ultrafilters with special properties in different models of Set Theory.It will use these methods to answer independence questions about ultrafilters (no Ppoints with a large continuum or small character filters), structural questions about filters (which ultrafilters/filters contain towers) and questions about the related cardinal invariants (independence number, free sequence number). It is known that current methods cannot answer some of these questions so the project will have to come up with novel ideas. The methods used will include forcing iterations, diamondlike constructions, preservation theorems and methods from descriptive set theory.



Grant: 7AMB17FR053
from 01/01/2017
to 31/12/2018
Grantor: Ministry of Education, Youth and Sports  MŠMT

Dynamics of mutlicomponent fluids


The goal of the project is studying qualitative properties of a particular class of the socalled energetically weak solutions to complex system of the NavierStokesFourier type as well as Coupling of these systems with the phase transition equations of the CahnHilliard or AllenCahn type. We plan to investigate these problems also in unbounded physical domains in appropriate classes of uniformly bounded functions.
The main goal is obtaining new results in the following directions:
• applications of the relative entropy methods and the consequences concerning stability of the socalled dissipative solutions
• singular limits, in particular the sharp interface limits with rigorous mathematical justification
• longtime dynamics, with a particular emphasis on the existence of bounded absorbing sets, asymptotic compactness of greajectories and the relevant questions concerning the attractors and their structure and complexity



Grant: CNR1608
from 05/04/2016
to 31/12/2018
Grantor: Czech Academy of Sciences / CNR Italy

Phenomenological modeling of polymeric smart foams with behavior controlled by the magnetic field


The aim of the collaboration is to develop and test experimentally a reliable mathematical model for magnetic foams taking into account hysteresis effects in the material, with the goal to provide, as output, parameters needed to obtain optimal performance in both "passive" (structural reinforcement and gradient properties) and "active" modes (actuation and control through magnetic field) of lightweight porous polymeric structures reinforced with aligned magnetic particles. The average distribution, size and shape of particles, their magnetic characteristics, and mechanical properties of the foam will be considered as processing parameters. Homogenization techniques will be adopted in order to derive simple constitutive relationships for low computational requirements. The developed constitutive model will be phenomenological, while the governing equations will be based on the magnetic and mechanical balance laws. The development and validation of the model will be performed by comparing predictions with experiments.



Grant: GF1634860L
from 01/01/2016
to 31/12/2018
Grantor: Austrian Science Foundation (FWF)  Czech Science Foudation

Logic and Topology in Banach spaces


Objectives:
The project is devoted to the study of topological and geometric properties of Banach spaces and their duals, aiming at a better understanding of their structure. Properties of the weak topology often imply important geometric properties of the Banach space in question. On the other hand, geometric properties of the Banach space often give information about its weak topology. Similar statements are true for duals of Banach spaces with the weakstar topology. We are going to explore this interplay in detail. The main project goals are:
1. Developing new tools for constructing and studying Banach spaces, using techniques from set theory and category theory.
2. Exploring different types of networks and related concepts in weak topologies, determining connections with renorming theory.
Results of Goal 1 will lead to new examples, settling some of the problems concerning interplay between geometric and topological properties of nonseparable Banach spaces. Goal 2 will lead to a better understanding of the weak topology and its relations to the geometric structure of a Banach space.



Grant: 7AMB16PL060
from 01/01/2016
to 31/12/2017
Grantor: Ministry of Education, Youth and Sports  MŠMT

Flow of viscous fluid in time dependent domain


Objectives:
G1. Global existence of weak solution of full system in timedependent domain.
Goals: Our goals is the study of the global existence of a weak solution in the case of a bounded or an exterior domains for the Navier or Dirichlet type of boundary conditions. We will use the method introduced in E. Feireisl: Dynamics of viscous compressible fluids, 2004 for fixed domain and we will also apply the penalization method.
G2: Relative entropy inequality.
Goals: We will focus on the derivation of the relative entropy inequality in the case of timedependent domain. We will use the results from fixed domain introduced in E. Feireisl and A. Novotný: Weakstrong uniqueness for the full NavierStokesFourier system, 2012.
G3. Singular limits.
Goals We shall study the singular limits in the regime of low Mach number. From this follows that the limit system (target system) is the system of incompressible flow in the timedependent domain. We will use the results from E. Feireisl, O. Kreml, Š. Nečasová, J. Neustupa, J. Stebel: Incompressible limits of fluids excited by moving boundaries, 2014, where barotropic case was studied, and also the results for fixed domains.
G4: Weak solution of viscous flow around rotating rigid body and his asymptotic behavior.
Goals: We will focus on the problem of existence of weak solution in weighted Lorentz spaces to get the asymptotic behavior of flow around a body. Because of the lack of the regularity in L^{p} spaces it is necessary to go into more complicated structure of Lorentz spaces, where the integrability of convective term is satisfied. Moreover, we would like to study the relative entropy inequality in the case of compressible flow around rotating body.



Grant: 7AMB16AT035
from 01/01/2016
to 31/12/2017
Grantor: Ministry of Education, Youth and Sports  MŠMT

Performance and thermodynamic aspects of incrementally nonlinear constitutive equations of the rate type


Objectives:
Although first models of hypoplasticity were proposed some decades ago, their properties have not yet been sufficiently understood, neither from the viewpoint of general applicability to large classes of technical problems, nor from the viewpoint of their mathematical wellposedness and numerical complexity. In special cases, however, there is evidence of their strong modeling potential for describing irreversible deformations, in particular in the case of granular or porous solids. The goal of this project is to exploit the complementary knowledge and experience of its participants, and classify the individual models within a larger thermomechanical context. The models will be investigated both analytically and numerically, and the results will be compared with known experimental data published in the literature, which will enable us to decide about their applicability for solving practical problems.



Grant: GF1534700L
from 01/01/2015
to 31/12/2017
Grantor: Austrian Science Foundation (FWF)  Czech Science Foudation

The continuum, forcing and large cardinals


Objectives:
The work shall study interactions between large cardinals and forcing and also with regard to properties of the continuum. The work shall be divided into three subtopics:
(1) First subtopic is called "Combinatorial properties of cardinals and the continuum function". The focus shall be on the impact of several combinatorial properties on the continuum function (a function which maps kappa to the size of the powerset of kappa). As Easton showed, without extra assumptions, there is very little ZFC can prove about the continuum function on regulars. However, once we start to consider large cardinals, or in general combinatorial properties often formulated in the context of large cardinals (tree property, square principles, etc.), suddenly the situation is much more interesting. Subtopic 1 studies these connections.
(2) Second subtopic is called"Combinatorial characteristics of the continuum and the Mathias forcing". This subtopic focuses on the real line from the point of certain combinatorial characteristics which shall be studied by means of the Mathias forcing. This forcing is determined by a filter on the powerset of omega. Thus it is natural to study the connections between the combinatorial properties of filters and the properties of the real line. This subtopic combines set theory and topology.
(3) The third subtopic is called "New developments in the template iterations." Template iterations is a modern forcing method with the potential to show new results regarding the combinatorial characteristics of the real line. Among the main problems to be studied are: obtaining cardinal characteristics with singular values, application of template iterations on nondefinable posets, and finally the application of the method with the mixed finite and countable support. These new approaches will be useful in obtaining new results.



Grant: 7AMB15AT035
from 01/01/2015
to 31/12/2016
Grantor: Ministry of Education, Youth and Sports  MŠMT

Topics in set theory: Traces of large cardinals, variants of Hechler's theorem, and ultrafilters on countable sets


Objectives:
The goal of the project is to expand the collaboration between specialists working in different areas of Set Theory. We shall, in particular, try to exploit the complementary expertise of the Czech and Austrian teams while solving problems on the borders of traditional areas of research (large cardinals, forcing, infinitary combinatorics, topology) and encourage mutual crossfertilization and inspiration sharing. The main goal is to encourage new members of both Czech and Austrian research teams to interact in join international research, to explore new possibilities for scientific collaboration, and to introduce international contacts enabling applications for funding with more ambitious join projects in future years.
