Weak solutions of the Boussinesq equations in domains with rough boundaries
Christian Komo
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Christian Komo, Weak solutions of the Boussinesq equations in domains with rough boundaries (2013), Logos Verlag, Berlin, ISBN: 9783832591342
Beschreibung / Abstract
In mathematical fluid dynamics the Boussinesq equations constitute a widely used model of motion of a viscous, incompressible buoyancy-driven fluid flow coupled with heat convection. This thesis deals with existence and uniqueness of weak solutions of the Boussinesq equations with no slip boundary condition for the velocity field u and Robin boundary condition for the temperature θ in domains satisfying a uniform Lipschitz condition. We investigate the influence of surface roughness to weak solutions of the Boussinesq equations. The main tool of our approach is the theory of Young measures. Optimizing the heat transfer is an important application of fluid mechanics. We address the problem of optimizing the surface roughness such that the heat energy transferred through the boundary becomes maximal/minimal.
Furthermore, we deal with existence and uniqueness of strong solutions of the Boussinesq equations with Dirichlet boundary conditions for u, θ in arbitrary domains Omega subseteq mathbbR³.
These results will be used to formulate regularity criteria for weak solutions of the Boussinesq equations.
Furthermore, we deal with existence and uniqueness of strong solutions of the Boussinesq equations with Dirichlet boundary conditions for u, θ in arbitrary domains Omega subseteq mathbbR³.
These results will be used to formulate regularity criteria for weak solutions of the Boussinesq equations.
Inhaltsverzeichnis
- BEGINN
- 1 Introduction
- 1.1 Weak solutions
- 1.2 Strong solutions
- 1.3 The influence of surface roughness to the boundary behaviour to weak solutions
- 1.4 Construction of some special domains with rough boundaries
- 1.5 Optimizing the heat energy transferred to the exterior
- 2 Preliminaries
- 2.1 Notation, function spaces and basic inequalities
- 2.2 The Bochner integral and spaces of Bochner integrable functions
- 3 Modelisation
- 3.1 Conservation equations
- 3.2 The Boussinesq approximation
- 4 Weak solutions of the Boussinesq equations
- 4.1 Domains satisfying the uniform Lipschitz condition and the surface measure
- 4.2 Definition of weak solutions
- 4.3 Formulation of the Boussinesq system as a nonlinear operator equation
- 4.4 Existence of weak solutions
- 4.5 Further properties of weak solutions.
- 5 Strong solutions of the Boussinesq equations in general domains
- 5.1 Weak solutions and strong solutions
- 5.2 Construction of strong solutions in general domains
- 5.3 Regularity of strong solutions of the Boussinesq equations
- 5.4 Regularity criteria for the Boussinesq equations in general domains .
- 6 Young Measures
- 6.1 Motivation and interpretation of Young Measures
- 6.2 Preliminaries
- 6.3 Existence theorems
- 6.4 Generating some special gradient Young measures
- 7 Influence of boundary rugosity to weak solutions of the Boussinesq equations
- 7.1 Sequential stability of the Boussinesq equations on a fixed domain
- 7.2 The Young measure approach to determine the boundary behaviour
- 7.3 Main results
- 8 Construction of some special domains with rough boundaries
- 9 Optimization of the heat energy transferred to the exterior
- 9.1 Mathematical formulation of the optimization problems
- 9.2 Preliminary lemmata
- 9.3 Existence of solutions of the optimization problems
- 10 Appendix