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Free boundary problems
Ferková, Terézia ; Schwarzacher, Sebastian (advisor) ; Kampschulte, Malte Laurens (referee)
This thesis deals with the one-phase Bernoulli problem, focusing on the existence and regularity of its solutions. After establishing the necessary preliminary theory on function spaces and convergence in the first chapter, we introduce the one-phase Bernoulli problem in the second chapter, reformulating it as a minimization problem. Then, in the third chapter, we present two illuminating examples of solutions to the problem, which imply that the Lipschitz regularity is optimal. The fourth chapter proves the existence of solutions, employing the direct method of calculus of variations. Finally, the fifth chapter reveals the Lipschitz property of generalized solutions. 1
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Stabilní tekutiny ve vnějších oblastech
Dopita, Jan ; Schwarzacher, Sebastian (advisor) ; Češík, Antonín (referee)
This thesis deals with Stokes and Navier-Stokes descriptions of flow of steady fluids in exterior domain and mainly focuses on presenting Liouville-like results for both cases. Firstly, we introduce the concept of weak derivative and spaces of appropriate functions. Following that, we talk about Stokes flow of incompressible fluids in R2 . We define a notion of a weak solution and we prove the Stokes paradox for generalized solutions in two-dimensional case, which is the main focus of this thesis. In the final chapter we then investigate the Navier- Stokes formulation where we again derive a notion of a weak solution. Last but not least, we present the Liouville property for generalized solution to the Navier-Stokes equations in R3 obeying certain restrictions.
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Convex hull properties for parabolic systems of partial differential equations
Češík, Antonín ; Schwarzacher, Sebastian (advisor) ; Bulíček, Miroslav (referee)
The topic of this thesis is the convex hull property for systems of partial differential equations, which is a natural generalisation of the maximum principle for scalar equations. The main result of this thesis is a theorem asserting the convex hull property for the solutions of a certain class of parabolic systems of nonlinear partial differential equations. It also investigates the coefficients of linear systems. The respective results are sharp which is demonstrated by counterexamples to the convex hull property for solutions of linear elliptic and parabolic systems. The general theme is that the coupling of the system is what breaks the convex hull property, not necessarily the non-linearity.
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Homogenization of flows of non-Newtonian fluids and strongly nonlinear elliptic systems
Kalousek, Martin ; Kaplický, Petr (advisor) ; Diening, Lars (referee) ; Schwarzacher, Sebastian (referee)
The theory of homogenization allows to find for a given system of partial differential equations governing a model with a very complicated internal struc- ture a system governing a model without this structure, whose solution is in a certain sense an approximation of the solution of the original problem. In this thesis, methods of the theory of homogenization are applied to three sys- tems of partial differential equations. The first one governs a flow of a class of non-Newtonian fluid through a porous medium. The second system is utilized for modeling of a flow of a fluid through an electric field wherein the viscosity depends significantly on the intensity of the electric field. For the third system is considered an elliptic operator having growth and coercivity indicated by a general anisotropic inhomogeneous N-function. 1
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