Differential and Integral Equations

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

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

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

First available in Project Euclid: 21 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]


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

Export citation