## Differential and Integral Equations

- Differential Integral Equations
- Volume 19, Number 4 (2006), 449-462.

### On the existence and regularity of solutions for degenerate power-law fluids

J. Málek, D. PraŽák, and M. Steinhauer

#### Abstract

We study time-dependent flows of incompressible *degenerate* power-law fluids
characterized by the power-law index $p-2$ with $p>2$. In this case, the generalized
viscosity vanishes as (the modulus of) the shear rate tends to zero. We prove
global-in-time existence of a weak solution if $p>\max\{\frac{3d-4}{d},2\}$. This improves
the range $p>\frac{3d+2}{d+2}$ for which the existence result was obtained by O.A.\
Ladyzhenskaya and J.L.\ Lions, via standard monotone operator theory. Since we apply
higher differentiability techniques, certain regularity results are also established. The
key step of the proof is an estimate of the velocity gradient in a suitable
Nikol$'$skĭ space. To make the presentation of the method transparent, we restrict
ourselves to the spatially periodic problem. A possible extension of the approach to
no-slip boundary conditions is however discussed as well.

#### Article information

**Source**

Differential Integral Equations, Volume 19, Number 4 (2006), 449-462.

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356050508

**Mathematical Reviews number (MathSciNet)**

MR2215628

**Zentralblatt MATH identifier**

1200.76020

**Subjects**

Primary: 35K65: Degenerate parabolic equations

Secondary: 35B10: Periodic solutions 35B65: Smoothness and regularity of solutions 76A05: Non-Newtonian fluids 76D03: Existence, uniqueness, and regularity theory [See also 35Q30]

#### Citation

Málek, J.; PraŽák, D.; Steinhauer, M. On the existence and regularity of solutions for degenerate power-law fluids. Differential Integral Equations 19 (2006), no. 4, 449--462. https://projecteuclid.org/euclid.die/1356050508