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Konjugovaná funkce
Bathory, Michal ; Opic, Bohumír (advisor) ; Bulíček, Miroslav (referee)
Using interpolation methods, new results on the boundedness of quasilinear joint weak type operators on Lorentz-Karamata (LK) spaces are established. LK spaces generalize many function spaces introduced before in literature, for example, the generalized Lorentz- Zygmund spaces, the Zygmund spaces, the Lorentz spaces and, of course, the Lebesgue spaces. The focus is mainly on the limiting cases of interpolation, where the spaces involved are, in certain sense, very close to the endpoint spaces. The results contain both necessary and sufficient conditions for the boundedness of the given operator on LK spaces. The complete characterization of embeddings of LK spaces is also included and the optimality of achieved results is then discussed. Finally, we apply our results to the conjugate function operator, which is known to be bounded on $L_p$ only if $1<p<\infty.$ Powered by TCPDF (www.tcpdf.org)
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Analysis of unsteady flows of incompressible heat-conducting rate-type viscoelastic fluids with stress-diffusion
Bathory, Michal ; Bulíček, Miroslav (advisor) ; Feireisl, Eduard (referee) ; Süli, Endré (referee)
We prove a global-in-time and large-data existence of a suitable weak solution to a system of partial differential equations describing an unsteady flow of homogeneous incom- pressible viscoelastic rate-type fluid. The material parameters are continuous functions of temperature and, in particular, the dependence of the shear modulus is assumed to be linear. It is shown that studied models obey the fundamental laws of thermodynamics. The key step towards the existence proof is derivation of the balance of entropy. This in- equality is paramount in the analysis and as its consequence, we obtain sufficient a priori estimates, positivity of temperature and also regularity of the elastic deformation. The second part of the thesis deals with the existence analysis for the isothermal case, however using a completely different method, which is of independent interest. 1
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Analysis of unsteady flows of incompressible heat-conducting rate-type viscoelastic fluids with stress-diffusion
Bathory, Michal ; Bulíček, Miroslav (advisor)
We prove a global-in-time and large-data existence of a suitable weak solution to a system of partial differential equations describing an unsteady flow of homogeneous incom- pressible viscoelastic rate-type fluid. The material parameters are continuous functions of temperature and, in particular, the dependence of the shear modulus is assumed to be linear. It is shown that studied models obey the fundamental laws of thermodynamics. The key step towards the existence proof is derivation of the balance of entropy. This in- equality is paramount in the analysis and as its consequence, we obtain sufficient a priori estimates, positivity of temperature and also regularity of the elastic deformation. The second part of the thesis deals with the existence analysis for the isothermal case, however using a completely different method, which is of independent interest. 1
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Analysis of unsteady flows of incompressible heat-conducting rate-type viscoelastic fluids with stress-diffusion
Bathory, Michal ; Bulíček, Miroslav (advisor)
We prove a global-in-time and large-data existence of a suitable weak solution to a system of partial differential equations describing an unsteady flow of homogeneous incom- pressible viscoelastic rate-type fluid. The material parameters are continuous functions of temperature and, in particular, the dependence of the shear modulus is assumed to be linear. It is shown that studied models obey the fundamental laws of thermodynamics. The key step towards the existence proof is derivation of the balance of entropy. This in- equality is paramount in the analysis and as its consequence, we obtain sufficient a priori estimates, positivity of temperature and also regularity of the elastic deformation. The second part of the thesis deals with the existence analysis for the isothermal case, however using a completely different method, which is of independent interest. 1
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Analysis of unsteady flows of incompressible heat-conducting rate-type viscoelastic fluids with stress-diffusion
Bathory, Michal ; Bulíček, Miroslav (advisor) ; Feireisl, Eduard (referee) ; Süli, Endré (referee)
We prove a global-in-time and large-data existence of a suitable weak solution to a system of partial differential equations describing an unsteady flow of homogeneous incom- pressible viscoelastic rate-type fluid. The material parameters are continuous functions of temperature and, in particular, the dependence of the shear modulus is assumed to be linear. It is shown that studied models obey the fundamental laws of thermodynamics. The key step towards the existence proof is derivation of the balance of entropy. This in- equality is paramount in the analysis and as its consequence, we obtain sufficient a priori estimates, positivity of temperature and also regularity of the elastic deformation. The second part of the thesis deals with the existence analysis for the isothermal case, however using a completely different method, which is of independent interest. 1
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Konjugovaná funkce
Bathory, Michal ; Opic, Bohumír (advisor) ; Bulíček, Miroslav (referee)
Using interpolation methods, new results on the boundedness of quasilinear joint weak type operators on Lorentz-Karamata (LK) spaces are established. LK spaces generalize many function spaces introduced before in literature, for example, the generalized Lorentz- Zygmund spaces, the Zygmund spaces, the Lorentz spaces and, of course, the Lebesgue spaces. The focus is mainly on the limiting cases of interpolation, where the spaces involved are, in certain sense, very close to the endpoint spaces. The results contain both necessary and sufficient conditions for the boundedness of the given operator on LK spaces. The complete characterization of embeddings of LK spaces is also included and the optimality of achieved results is then discussed. Finally, we apply our results to the conjugate function operator, which is known to be bounded on $L_p$ only if $1<p<\infty.$ Powered by TCPDF (www.tcpdf.org)
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Conjugate series to Fourier's ones
Bathory, Michal ; Opic, Bohumír (advisor) ; Zelený, Miroslav (referee)
This thesis focuses entirely on conjugate series to Fourier's ones. It provides a quick and intuitive introduction to this topic for the reader who is familiar with classical Fourier's series. The thesis contains simple tests for the convergence of conjugate series and for the existence of related conjugate functions. These concepts are illustrated with examples. Extensive and comprehensive works of Antoni Zygmund are dedicated to the conjugate series (among other topics) but the corresponding proofs are far from detailed. Thus, this thesis summarizes the basic assertions systematically, gives proofs in detail and offers author's own solutions to selected examples. Powered by TCPDF (www.tcpdf.org)
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