Local Well-Posedness of the Boundary Layer Equations for Dilatant Power-Law Fluids
Abstract
We establish local-in-time existence and uniqueness of monotone classical solutions to the two-dimensional nonstationary boundary layer equations for dilatant power-law fluids in a periodic half-space, for every fixed exponent n>1. The result holds under Oleinik's monotonicity condition and suitable weighted Sobolev, algebraic decay, and compatibility assumptions. The main difficulties are the tangential derivative loss caused by the normal velocity and the degeneracy at infinity of the principal coefficient nωn−1 in the nonlinear vorticity diffusion ∂y2(ωn), where ω=∂y u. We adapt the Masmoudi-Wong energy method and use a good unknown to control the tangential derivative loss. For the nonlinear diffusion, we separate the estimates near the boundary from the treatment of degeneracy in the far field. Nonlinear boundary reduction and localized trace estimates control the boundary terms, while weighted estimates exploit the compensating decay of vorticity derivatives to control the diffusion commutators. Together with tangential regularization and weighted maximum and minimum principles, these estimates yield bounds uniform in the regularization parameter and preserve weighted monotonicity. Under these assumptions, the result provides an affirmative local-in-time answer to the existence and uniqueness question in Open Problem~11 of Oleinik and Samokhin.
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BibTeX
@misc{zhang2026local,
title = {{Local Well-Posedness of the Boundary Layer Equations for Dilatant Power-Law Fluids}},
author = {Mingxue Zhang and Zhonger Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06944},
eprint = {2609.06944},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06944},
}
RIS
TY - UNPB
TI - Local Well-Posedness of the Boundary Layer Equations for Dilatant Power-Law Fluids
AU - Mingxue Zhang
AU - Zhonger Wu
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.06944
UR - https://arxiv.org/abs/2609.06944
AB - We establish local-in-time existence and uniqueness of monotone classical solutions to the two-dimensional nonstationary boundary layer equations for dilatant power-law fluids in a periodic half-space, for every fixed exponent n>1. The result holds under Oleinik's monotonicity condition and suitable weighted Sobolev, algebraic decay, and compatibility assumptions. The main difficulties are the tangential derivative loss caused by the normal velocity and the degeneracy at infinity of the principal coefficient nω^n−1 in the nonlinear vorticity diffusion ∂_y^2(ω^n), where ω=∂_y u. We adapt the Masmoudi-Wong energy method and use a good unknown to control the tangential derivative loss. For the nonlinear diffusion, we separate the estimates near the boundary from the treatment of degeneracy in the far field. Nonlinear boundary reduction and localized trace estimates control the boundary terms, while weighted estimates exploit the compensating decay of vorticity derivatives to control the diffusion commutators. Together with tangential regularization and weighted maximum and minimum principles, these estimates yield bounds uniform in the regularization parameter and preserve weighted monotonicity. Under these assumptions, the result provides an affirmative local-in-time answer to the existence and uniqueness question in Open Problem~11 of Oleinik and Samokhin.
ER -