Grants


Grant: Neuron Impuls 24/2016
from 01/01/2017
to 31/12/2019
Grantor: Neuron Fund for Support of Science

Guaranteed bounds of eigenvalues and eigenfunctions of differential operators


We will propose new a posteriori error estimates for eigenvalue problems of symmetric elliptic partial differential operators. We will prove their reliability and local efficiency. We will use them in the adaptive finite element method for reliable error estimates of the size of the error in eigenvalues and eigenfunctions.



Grant: Neuron Impuls Junior
from 01/01/2017
to 30/06/2018
Grantor: Neuron Fund for Support of Science

Mathematical analysis of hyperbolic conservation laws


Objectives:
The goal of the project is to deepen present knowledge concerning existence of weak solutions for hyperbolic systems of partial differential equations arising from conservation laws in multiple space dimensions, study of uniqueness and nonuniqueness of entropic weak solutions, analysis of admissibility criteria (maximization of entropy production or inviscid limit) or even designing other suitable admissibility criteria.
A convenient test problem for admissibility criteria is the so called Riemann problem, i.e. problem with initial data consisting of a jump discontinuity separating two constant states. As such initial data are onedimensional, one can use standard theory to find a onedimensional selfsimilar solution which is unique in the class of BV functions. In a broader class of functions (in particular in the class of essentialy bounded functions), other weak solutions may exist as is in the case of compressible Euler equations. In the framework of this project I will study also Riemann problems for various hyperbolic systems concerning uniqueness or nonuniqueness of weak solutions in the class of essentialy bounded functions and in the case of nonuniqueness, admissibility criteria will be tested.



Grant: GA1700941S
from 01/01/2017
to 31/12/2019
Grantor: Czech Science Foundation

Topological and geometrical properties of Banach spaces and operator algebras II


Objectives:
We wish to investigate the structure of Banach spaces, C*algebras and Jordan algebras and their relationship. Main topics include quantitative approach to Banach spaces, various methods of separable reduction, decompositions of Banach spaces to smaller subspaces, integral representation of affine Baire functions, descriptive properties of weak topologies, small sets in Banach spaces and Polish groups, universal spaces in various categories of Banach spaces, operators and their numerical ranges, structure of abelian subalgebras of a C*algebra, of associative subalgebras of a Jordan algebra and related structures, different types of order in operator algebras, representation of morphisms on various substructures of operator algebras, Bell's inequalities and quantum correlations. We wish to focus especially on those problems where the mentioned areas intersect each other and by solving them to contribute to clarification of connections among various areas of functional analysis.



Grant: GA1701747S
from 01/01/2017
to 31/12/2019
Grantor: Czech Science Foundation

Theory and numerical analysis of coupled problems in fluid dynamics


Objectives:
The project is focused on several important fields of today's rapidly developing mathematical fluid mechanics. The aim is to derive a series of results, from new regularity criteria, stability and robustness analysis of solutions, up to the low Mach and high Reylolds limits in a compressible fluid interacting with a solid structure. Beside the qualitative analysis of flow problems, a part of the project is the development and analysis of new, accurate and robust numerical methods for the solution of important and topical models of fluid dynamics. The attention will be paid to the development and analysis of high order methods for the solution of nonstationary nonlinear partial differential equations and compressible flow, based on the discontinuous Galerkin method. Particularly we have hpversions in mind. These methods will be applied to the numerical solution of fluidstructure interaction and multiphase flow. Another subject is the study of flow model with slip boundary conditions.



Grant: GJ1701694Y
from 01/01/2017
to 31/12/2019
Grantor: Czech Science Foundation

Mathematical analysis of partial differential equations describing inviscid flows


Objectives:
The investigator and his team will develop the theory of Camillo De Lellis and Lászlo Székelyhidi which allows to prove surprising results for incompressible Euler equations in multiple space dimensions. They will focus mainly on development and applications of the theory in the field of compressible flow, both in the isentropic case and in the case of full system of partial differential equations. They will study and propose criteria to choose "physical" solutions among the infinitely many weak solutions of appropriate systems of equations. The investigator and his team will maintain already established scientific cooperations (De Lellis, Chiodaroli, Wiedemann) and establish new ones. The results of the project will be presented on international conferences and will be published as articles in impacted journals.



Grant: GA1727844S
from 01/01/2017
to 31/12/2019
Grantor: Czech Science Foundation

Generic objects


Objectives:
An object can be called generic if it occurs typically, in the sense that its copies can be found in every residual set in an appropriate space of objects. The aim of the project is to study generic objects appearing in several areas of mathematics, finding new tools for constructing and detecting such objects, and exploring their combinatorial structure. Wellknown examples of generic objects in model theory are Fraisse limits. Generic structures occur also naturally in topology (e.g. the Cantor set) and various areas of mathematical analysis (e.g. generic Banach spaces). Cohen's settheoretic forcing offers a strong tool of constructing generic objects, usually needed for specific purposes and not related to Fraisse limits. One of our objectives is to explore links between the settheoretic method of forcing and modeltheoretic methods for constructing universal homogeneous structures.



Grant: GA1603230S
from 01/01/2016
to 31/12/2018
Grantor: Czech Science Foundation (GAČR)

Thermodynamically consistent models for fluid flows: mathematical theory and numerical solution


Objectives:
Mathematical and numerical analysis and numerical solution of fluid flows belong to the most often studied problems of the theory of partial differential equations and their numerical solution. During the last decades, a big progress has been achieved in these fields which enables us to study models of complex fluids including the possibility to consider their dependence on temperature. This project is focused on the study of such models of fluid thermodynamics and mechanics with the aim to extend the knowledge in the field of the theoretical analysis of the corresponding systems of partial differential equations and numerical analysis of the methods for their solution. Computational simulations using specific numerical methods will be performed to support the analytical results concerning the wellposedness of the model problems and qualitative properties of their solutions. The proposed projects assumes a tight collaboration of specialists in these fields which is an important prerequisite for further development of mathematical and computational fluid thermodynamics.



Grant: GA1607378S
from 01/01/2016
to 31/12/2018
Grantor: Czech Science Foundation (GAČR)

Nonlinear analysis in Banach spaces


Objectives:
We plan to investigate problems concerning uniformly continuous and Lipschitz mappings between Banach spaces and their possible applications in other areas of mathematics, such as theoretical computer science, differential equations etc. This project is devoted to the following aspects of the subject:
a) Uniformly continuous and coarse mappings
b) Lipschitz isomorphism
c) Lipschitz free spaces
d) Linear and descriptive properties



Grant: GA1502532S
from 01/01/2015
to 31/12/2017
Grantor: Czech Science Foundation (GAČR)

Modular and Decentralized Control of DiscreteEvent and Hybrid Systems with Communication


Objectives:
Discreteevent systems represent an important class of dynamical systems with discrete state spaces and eventdriven dynamics. For large systems, methods of hierarchical, modular, and decentralized decentralized supervisory control have been proposed. Since a solution to modular and decentralized supervisory control may not exist without communication between controllers, coordination control has been proposed as a form of decentralized control with supervisors communicating via coordinators. In this project we will study computationally efficient solutions to coordination supervisory control of large automata with product structure based on multilevel communication structure. Both logical automata and those stemming from discretizations will be considered. Decentralized supervisory control of automata without a priori known modular (product) structure will also be investigated. The motivations for investigating new efficient methods are that communications are sometimes lost or delayed and the original product structure is often lost after discretization.



Grant: GA1512227S
from 01/01/2015
to 31/12/2017
Grantor: Czech Science Foundation (GAČR)

Analysis of mathematical models of multifunctional materials with hysteresis


Objectives:
The project topic is mathematical modeling, analysis, and numerical simulations of processes taking place in multifunctional materials with hysteresis. The results will include (1) a rigorous derivation of systems of ordinary and partial differential equations based on physical principles and experimentally verified constitutive relations, (2) proofs of existence, and possibly also uniqueness and stability of solutions to the equations, and (3) their numerical approximation including error bounds. The main applications will involve piezoelectric and magnetostrictive materials used as sensors, actuators, and energy harvestors, as well as thermoelastoplastic materials subject to material fatigue. The presence of hysteresis makes all these steps challenging, also because hysteresis nonlinearities are nondifferentiable, which creates difficulties both in the analysis and in the numerics. New algorithms will have to be developed to treat the problems in maximal complexity.
