We study time-dependent flows of incompressible degenerate power-law fluids characterized by the power-law index p — 1 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 so?lution if p > max{ ^—, 2}. This improves the range p > ^f for which the existence result was obtained by O.A. Ladyzhenskaya and J.L. Lions, via standard monotone operator theory. Since we apply higher differen?tiability techniques, certain regularity results are also established. The key step of the proof is an estimate of the velocity gradient in a suitable Nikol'skii space. To make the presentation of the method transparent, we restrict ourselves to the spatially periodic problem, A possible exten?sion of the approach to no-slip boundary conditions is however discussed as well.
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机译：我们研究了随时间变化的不可压缩简并幂律流体，其特征在于幂律指数p_1为p>2。在这种情况下，随着剪切速率（的模数）趋于零，广义粘度消失了。如果p> max {^ —，2}，我们证明了弱解在时间上的整体存在。这提高了通过O.A获得存在结果的范围p> ^ f。 Ladyzhenskaya和J.L. Lions，通过标准单调算子理论。由于我们应用了更高的微分技术，因此还建立了某些规律性结果。证明的关键步骤是在合适的Nikol'skii空间中估计速度梯度。为了使该方法的表示透明，我们将自己限制在空间周期性问题上。但是，也讨论了防滑边界条件方法的可能扩展。
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