Differential and Integral Equations

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

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

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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


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