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Analysis of generalized Stokes system with implicitly given Cauchy stress tensor
Bulušek, Petr ; Bulíček, Miroslav (advisor) ; Kaplický, Petr (referee)
The goal of the thesis was a qualitative analysis of system of partial differential equations describing simplified steady incompressible fluid flow with implicitly given Cauchy stress tensor. In chapter 2 one can find issues regarding generalization of constitutive relations for the Cauchy stress tensor. It was necesarry to get familiar with mathematical tools used for proving the existence of weak solutions of such studied equations. In chapter 3 we study stress tensor given as a continuous function of velocity gradient satisfying some restrictive conditions and prove the existence of weak solution. In chapter 4 detailed proof is presented for implicitly given stress tensor leading to the so called maximal monotone r-graph. Both cases are ilustrated on concrete models.
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Slabá řešení pro třídu nelineárních integrodiferenciálních rovnic
Soukup, Ivan ; Bárta, Tomáš (advisor) ; Kaplický, Petr (referee)
Title: Weak solutions for a class of nonlinear integrodifferential equations Author: Ivan Soukup Department: Department of mathematical analysis Supervisor: RNDr. Tomáš Bárta, Ph.D. Supervisor's e-mail address: tomas.barta@mff.cuni.cz Abstract: The work investigates a system of evolutionary nonlinear partial integrodifferential equations in three dimensional space. In particular it stud- ies an existence of a solution to the system introduced in [1] with Dirichlet boundary condition and initial condition u0. We adopt the scheme of the proof from [9] and try to avoid the complications rising from the integral term. The procedure consists of an approximation of the convective term and an ap- proximation of the potentials of both nonlinearities using a quadratic function, proving the existence of the approximative solution and then returning to the original problem via regularity of the approximative solution and properties of the nonlinearities. The aim is to improve the results of the paper [1]. 1
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Spaces of functions with fractional derivatives on interval
Lopata, Jan ; Kaplický, Petr (advisor) ; Hencl, Stanislav (referee)
In literature we can find a variety of ways to introduce Sobolev space W1,1 on bounded and open interval. In this thesis we will put them in context. We will show that completion of set of function with continuous first derivative, the space of functions with weak derivative and space of absolutely continuous functions are isometrically isomorphic. Furthemore, we will demonstrate that the Sobolev space W1,∞ is isometrically isomorphic to space of Lipschitz functions. We will also show several trivial and nontrivial embeddings for Besov spaces. Finnaly, we will examine the question, whether functions from Besov space are, given some parameters, included in set of continuous functions. 1
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Solution of Poiseuille and plane Couette flow associated with the dynamic boundary conditions
Vejvoda, Martin ; Málek, Josef (advisor) ; Kaplický, Petr (referee)
In the presented work we study the effect of dynamic boundary conditions on Couette and Poisseuille flows that represent two types of flow between two parallel impervious plates. In the firts part, the Navier-Stokes equations are considered describing flows of an incompressible Newtonian fluid, and dynamic boundary conditions in general three- dimensional setting. Then we look at how our problem reduces in the simplified geome- trical setting. In the second part, we study several selected problems, some of them are supported by numerical simulations. 1
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On convergence of a series
Procházka, Antonín ; Zelený, Miroslav (advisor) ; Kaplický, Petr (referee)
in English This text is devoted to the series, whose n-th term is defined by (−1)n |sin n| /n. The goal of this work is to prove convergence of this series. The solution uses standard convergence tests, the theory of Fourier Series and findings about approximation of number π. 1
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Banach-Tarski paradox
Klůjová, Jana ; Zelený, Miroslav (advisor) ; Kaplický, Petr (referee)
In the present work we study the Banach-Tarski Paradox and other paradoxical decompositions of sets, groups and semigroups. These decompositions are described especially using free groups and semigroups. We can construct such groups using words made of letters. We study both finite and denumerable paradoxical decomposition. Further we deal with equidecomposability, which we need for a proof of Banach-Tarski Paradox. We present a proof of Banach-Schröder-Berstein Theorem.
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