Dokument

Summary:
This thesis is concerned with the regularity of solutions to Navier-Stokes 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 Navier-Stokes equations provide a mathematical model of the motion of a uid. These Navier-Stokes 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 so-called 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 L2-norm. In contrast to this the convergence order of linear schemes is determined by the classical L2-Sobolev 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 Navier-Stokes equation we used a fixed point argument. We formulate the Navier-Stokes equation as a fixed point equation and therefore regularity results for the corresponding Stokes equation can be transferred to the non-linear case. In the first paper "Besov regularity for the Stokes and the Navier-Stokes system in polyhedral domains" we considered the stationary Stokes- and the Navier-Stokes 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 Navier-Stokes 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 non-linear Navier-Stokes equation. In order to apply the fixed point theorem we had to require small data and small Reynolds number.

Bibliographie / References

1. 2 Besov Regularity for the Stokes and the Navier-Stokes System in Polyhedral Domains 19 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Besov regularity for the Navier-Stokes System in polyhedral domains . . . 22 2.3 Besov regularity for the Stokes system in polyhedral domains . . . . . . . . 31 2.4 Norm estimates for Navier-Stokes and Stokes equations on polyhedral domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.5 Appendix A - Sobolev and weighted Sobolev regularity of solutions of the Stokes and the Navier-Stokes system . . . . . . . . . . . . . . . . . . . . . 35 2.6 Appendix B - Function spaces and wavelets . . . . . . . . . . . . . . . . . 37 (x) jr 2u(x)j2 dx + (x)j2 dx Theorem 3.3.5. Let be a bounded Lipschitz domain in Rd, d and 2 N with < 2 . Then,
2. [14] Albert Cohen, Ingrid Daubechies and Jean-Christophe Feauveau, Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math. 45 (1992), no. 5, 485{560.
3. [12] Albert Cohen, Wolfgang Dahmen and Ronald DeVore, Adaptive wavelet methods II|Beyond the elliptic case, Found. Comput. Math. 2 (2002), no. 3, 203{245.
4. [13] Albert Cohen, Wolfgang Dahmen and Ronald DeVore, Adaptive wavelet schemes for nonlinear variational problems, SIAM J. Numer. Anal. 41 (2003), no. 5, 1785{1823.

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