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Preprint (Vorabdruck) zugänglich unter
Steady states of anisotropic generalized Newtonian fluids
URN: urn:nbn:de:bsz:291-scidok-44345
URL: http://scidok.sulb.uni-saarland.de/volltexte/2012/4434/
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Dokument 1.pdf (471 KB)
Institut:
Fachrichtung 6.1 - Mathematik
DDC-Sachgruppe:
Mathematik
Dokumentart:
Preprint (Vorabdruck)
Schriftenreihe:
Preprint / Fachrichtung Mathematik, Universität des Saarlandes
Bandnummer:
88
Sprache:
Englisch
Erstellungsjahr:
2003
Publikationsdatum:
10.02.2012
Kurzfassung auf Englisch:
We consider the stationary flow of a generalized Newtonian fluid which is modelled by an anisotopic dissipative potential f. More precisely, we are looking for a solution u:\Omega\rightarrow\mathbb{R}^{n}, \Omega\subset\mathbb{R}^{n},n=2,3, of the following system of nonlinear partial differential equations
\left.\begin{array}{c}
-\mbox{div}\{T(\varepsilon(u))\}+u^{k}\frac{\partial u}{\partial x_{k}}+\nabla\pi=g\mbox{ in}\Omega,\
\mbox{div}u=0\mbox{ in}\Omega,\mbox{ }u=0\mbox{ on}\partial\Omega.\mbox{ }
\end{array}\right\} (*)
Here \pi:\Omega\rightarrow\mathbb{R} denotes the pressure, g is a system of volume forces, and the tensor T is the gradient of the potential f. Our main hypothesis imposed on f is the existence of exponents 1<p\leq q_{0}<\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_{0}-2}{2}}\left|\sigma\right|^{2}
holds with constants \lambda,\Lambda>0. Under natural assumptions on p and q_{0} we prove the existence of a weak solution u to the problem (*), moreover we prove interior C^{1,\alpha}-regularity of u in the two-dimensional case. If n=3, then interior partial regularity is established.
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