## 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

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* ≤ *q*_{0} < ∞ such that

holds with constants λ, Λ > 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,α}-regularity of *u* in the two-dimensional case. If *n* = 3, then interior partial regularity is established.

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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

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### Mathematics Subject Classification (2000).

- 76M30
- 49N60
- 35J50
- 35Q30

### Keywords.

- Generalized Newtonian fluids
- anisotropic dissipative potentials
- regularity