Please use this identifier to cite or link to this item: doi:10.22028/D291-26296
Title: Steady states of anisotropic generalized Newtonian fluids
Author(s): Apushkinskaya, Darya
Bildhauer, Michael
Fuchs, Martin
Language: English
Year of Publication: 2003
DDC notations: 510 Mathematics
Publikation type: Other
Abstract: 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.
Link to this record: urn:nbn:de:bsz:291-scidok-44345
Series name: Preprint / Fachrichtung Mathematik, Universität des Saarlandes
Series volume: 88
Date of registration: 10-Feb-2012
Faculty: MI - Fakultät für Mathematik und Informatik
Department: MI - Mathematik
Collections:SciDok - Der Wissenschaftsserver der Universität des Saarlandes

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