发表机构
Wuhan University; Shantou University（武汉大学; 汕头大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对膨胀型幂律流体边界层方程，在 $1<n<7/3$ 范围内通过切向正则化和加权能量估计证明了单调解的局部存在唯一性，部分解决了Oleinik-Samokhin第十一个开放问题。
AI 中文摘要
我们建立了在周期半空间中，对于 $1<n<\frac{7}{3}$ 的膨胀型幂律流体，二维非定常边界层方程单调解的局部时间存在性和唯一性。在 Oleinik 单调性条件以及对初始数据和外部流动的适当加权 Sobolev 假设下，我们通过切向正则化构造解，并推导出一致先验估计。一个合适的良好未知量补偿了由法向速度引起的切向导数损失。通过结合加权能量估计、Faà di Bruno 公式以及最大值和最小值原理，我们控制了非线性退化扩散项 $\partial_y^2(\omega^n)$，其有效扩散系数在 $\omega=\partial_yu$ 于无穷远处衰减时消失，并传播涡量的加权单调性。在该指数范围内，我们的结果部分解决了 Oleinik 和 Samokhin 提出的第十一个开放问题，即膨胀型流体非定常边界层系统解的存在性和唯一性。
英文摘要
We establish the local-in-time existence and uniqueness of monotone solutions to the two-dimensional nonstationary boundary-layer equations for a dilatant power-law fluid in a periodic half-space for $1<n<\frac{7}{3}$. Under Oleinik's monotonicity condition and suitable weighted Sobolev assumptions on the initial data and outer flow, we construct solutions through tangential regularization and derive uniform a priori estimates. A suitable good unknown compensates for the loss of one tangential derivative caused by the normal velocity. By combining weighted energy estimates, the Faà di Bruno formula, and the maximum and minimum principles, we control the nonlinear degenerate diffusion term $\partial_y^2(ω^n)$, whose effective diffusion coefficient vanishes as $ω=\partial_yu$ decays at infinity, and propagate the weighted monotonicity of the vorticity. Within this exponent range, our result partially resolves the eleventh open problem posed by Oleinik and Samokhin \cite{OAO} on the existence and uniqueness of solutions to nonstationary boundary-layer systems for dilatant fluids.
