# Steady States of Anisotropic Generalized Newtonian Fluids

## Abstract.

We consider the stationary flow of a generalized Newtonian fluid which is modelled by an anisotropic dissipative potential f. More precisely, we are looking for a solution $$u:\Omega \to \mathbb{R}^n ,\Omega \subset \mathbb{R}^n ,\,n = 2,3,$$ of the following system of nonlinear partial differential equations

\left. \begin{aligned} - {\text{div}}\{ T(\varepsilon (u))\} + u^k \tfrac{{\partial u}} {{\partial x_k }} + \nabla \pi & = g{\text{ in}}\,\Omega , \\ {\text{div }}u = 0{\text{ in}}\,\Omega ,\quad u & = 0{\text{ on}}\,\partial \Omega . \\ \end{aligned} \right\}
((*))

Here $$\pi :\Omega \to \mathbb{R}$$ 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 ≤ q0 < ∞ such that

$$\lambda (1 + |\varepsilon |^2 )^{\tfrac{{p - 2}} {2}} |\sigma |^2 \leq D^2 f(\varepsilon )(\sigma ,\sigma ) \leq \Lambda (1 + |\varepsilon |^2 )^{\tfrac{{q_0 - 2}} {2}} |\sigma |^2$$

holds with constants λ, Λ > 0. Under natural assumptions on p and q0 we prove the existence of a weak solution u to the problem (*), moreover we prove interior C1,α-regularity of u in the two-dimensional case. If n = 3, then interior partial regularity is established.

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Authors

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Correspondence to Darya Apushkinskaya.

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Apushkinskaya, D., Bildhauer, M. & Fuchs, M. Steady States of Anisotropic Generalized Newtonian Fluids. J. math. fluid mech. 7, 261–297 (2005). https://doi.org/10.1007/s00021-004-0118-6

• Accepted:

• Issue Date:

• 76M30
• 49N60
• 35J50
• 35Q30

### Keywords.

• Generalized Newtonian fluids
• anisotropic dissipative potentials
• regularity