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

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.