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Preprint (Vorabdruck) zugänglich unter
URN: urn:nbn:de:bsz:291-scidok-43820
URL: http://scidok.sulb.uni-saarland.de/volltexte/2011/4382/


Variants of the Stokes problem : the case of anisotropic potentials

Bildhauer, Michael ; Fuchs, Martin

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Dokument 1.pdf (453 KB)

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Institut: Fachrichtung 6.1 - Mathematik
DDC-Sachgruppe: Mathematik
Dokumentart: Preprint (Vorabdruck)
Schriftenreihe: Preprint / Fachrichtung Mathematik, Universit├Ąt des Saarlandes
Bandnummer: 57
Sprache: Englisch
Erstellungsjahr: 2002
Publikationsdatum: 02.12.2011
Kurzfassung auf Englisch: We investigate the smoothness properties of local solutions of the nonlinear Stokes problem
-\mbox{div}\{T(\varepsilon(v))\}+\nabla\pi=g\mbox{ on }\Omega,
\mbox{ div }v\equiv0\mbox{ on }\Omega,
where v:\Omega\rightarrow\mathbb{R}^{n} is the velocity field, \pi:\Omega\rightarrow\mathbb{R} denotes the pressure function, and g:\Omega\rightarrow\mathbb{R}^{n} represents a system of volume forces, \Omega denoting an open subset of \mathbb{R}^{n}. The tensor t is assumed to be the gradient of some potential f acting on symmetric matrices. Our main hypothesis imposed on f is the existence of exponents 1<p\leq q<\infty such that
\lambda(1+\left|\varepsilon\right|^{2})^{\frac{p-2}{2}}\left|\sigma\right|^{2}\leq D^{2}f(\varepsilon)(\sigma,\sigma)\leq\Lambda(1+\left|\varepsilon\right|^{2})^{\frac{q-2}{2}}\left|\sigma\right|^{2}
holds with suitable constants \lambda,\Lambda>0, i.e. the potential f is of anisotropic power growth. Under natural assumptions on p and q we prove that velocity fields from the space W_{p,loc}^{1}(\Omega;\mathbb{R}^{n}) are of class C^{1,\alpha} on an open subset of \Omega with full measure. If n=2, then the set of interior singularities is empty.
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