Local Well-Posedness of the Boundary Layer Equations for Dilatant Power-Law Fluids

RESUMO

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 $\partial_y^2(\omega^n)$, whose effective diffusion coefficient vanishes as $\omega=\partial_yu$ 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.