Besov Regularity of Solutions to NavierStokes Equations
This thesis is concerned with the regularity of solutions to NavierStokes and Stokes equation on domains with point singularities, namely polyhedral domains contained in R3 and general bounded Lipschitz domains in Rd, d ≥ 3 with connected boundary. The NavierStokes equations provide a mathemati...
Saved in:
Main Author:  

Contributors:  
Format:  Doctoral Thesis 
Language:  English 
Published: 
PhilippsUniversität Marburg
2016

Subjects:  
Online Access:  PDF Full Text 
Tags: 
Add Tag
No Tags, Be the first to tag this record!

Summary:  This thesis is concerned with the regularity of solutions to NavierStokes and Stokes
equation on domains with point singularities, namely polyhedral domains contained in
R3 and general bounded Lipschitz domains in Rd, d ≥ 3 with connected boundary. The
NavierStokes equations provide a mathematical model of the motion of a uid. These
NavierStokes equations form the basis for the whole world of computational uid dynamics, and therefore they are considered as maybe the most important PDEs known so far. We consider the stationary (Navier)Stokes equations. The study the Besov regularity of the solution in the scale BsƬ (LƬ (Ω))d, 1/Ƭ = s/d + 1/2 of Besov spaces. This scale is the socalled adaptivity scale. The parameter s determines the approximation order of adaptive numerical wavelet schemes and other nonlinear approximation methods when the error is measured in the L2norm. In contrast to this the convergence order of linear schemes is determined by the classical L2Sobolev regularity.
In many papers the Besov regularity of the solution to various operator equations/partial differential equations was investigated. The proof of Besov regularity in the adaptivity scale was in many contributions performed by combining weighted Sobolev regularity results with characterizations of Besov spaces by wavelet expansions. Choosing a suitable wavelet basis the coeffcients of the wavelet expansion of the solution can be estimated by exploiting the weighted Sobolev regularity of the solution, such that a certain Besov regularity can be established. This technique was applied for the Stokes system in all papers which are part of this thesis. For achieving Besov regularity for NavierStokes equation we used a
fixed point argument. We formulate the NavierStokes equation as a fixed point equation and therefore regularity results for the corresponding Stokes equation can be transferred to the nonlinear case.
In the first paper "Besov regularity for the Stokes and the NavierStokes system in polyhedral domains" we considered the stationary Stokes and the NavierStokes equations in polyhedral domains. Exploiting weighted Sobolev estimates for the solution we proved that the Besov regularity of the solutions to these equations exceed their Sobolev regularity.
In the second paper "Besov Regularity for the Stationary NavierStokes Equation on
Bounded Lipschitz Domains" we have investigated the stationary (Navier)Stokes equations on bounded Lipschitz domain. Based on weighted Sobolev estimates again we could establish a Besov regularity result for the solution to the Stokes system. By applying Banach's fixed point theorem we transferred these results to the nonlinear NavierStokes equation. In order to apply the fixed point theorem we had to require small data and small Reynolds number. 

Physical Description:  79 Pages 
DOI:  10.17192/z2016.0945 