Small Viscosity Limit of Navier-Stokes Equations in A Quarter Plane with Navier Slip Conditions
Abstract
We study the vanishing-viscosity limit of Leray solutions to the two-dimensional incompressible Navier--Stokes equations in a quarter plane with Navier slip boundary conditions on both edges for a class of compatible well-prepared initial data. By multi-scale analysis, we deduce that a boundary layer of order O(√ε) arises near each edge of the corner and is described by a linearized Prandtl equation, and near the corner, their overlap and nonlinear interaction are captured by a two-dimensional corner profile. Although this profile occurs at order O(√ε), it is governed, in the fast variables, by a nonlinear initial-boundary value problem of Navier--Stokes type. The well-posedness and Sobolev regularity of the outer flow problems, together with the well-posedness and weighted estimates of the boundary-layer and corner-layer problems are obtained on a common time interval [0,T] independent of ε. To ensure the required regularity of the outer Euler flow in the critical right-angle domain, it is assumed that the initial vorticity of the outer leading order flow vanishes in a neighborhood of the corner. We then rigorously justify the resulting composite expansion in the small viscosity limit. More precisely, after retaining the leading outer flow, the two weak boundary layers, the nonlinear corner layer, and the order-ε outer correction, the error between the exact solution and this truncated expansion is O(ε5/4) in L∞(0,T;L2(Q)) and O(ε3/4) in L2(0,T;H1(Q)).
Related
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BibTeX
@misc{guo2026small,
title = {{Small Viscosity Limit of Navier-Stokes Equations in A Quarter Plane with Navier Slip Conditions}},
author = {Wen Guo and Ya-Guang Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28261},
eprint = {2610.28261},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28261},
}
RIS
TY - UNPB
TI - Small Viscosity Limit of Navier-Stokes Equations in A Quarter Plane with Navier Slip Conditions
AU - Wen Guo
AU - Ya-Guang Wang
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28261
UR - https://arxiv.org/abs/2610.28261
AB - We study the vanishing-viscosity limit of Leray solutions to the two-dimensional incompressible Navier--Stokes equations in a quarter plane with Navier slip boundary conditions on both edges for a class of compatible well-prepared initial data. By multi-scale analysis, we deduce that a boundary layer of order O(√ε) arises near each edge of the corner and is described by a linearized Prandtl equation, and near the corner, their overlap and nonlinear interaction are captured by a two-dimensional corner profile. Although this profile occurs at order O(√ε), it is governed, in the fast variables, by a nonlinear initial-boundary value problem of Navier--Stokes type. The well-posedness and Sobolev regularity of the outer flow problems, together with the well-posedness and weighted estimates of the boundary-layer and corner-layer problems are obtained on a common time interval [0,T] independent of ε. To ensure the required regularity of the outer Euler flow in the critical right-angle domain, it is assumed that the initial vorticity of the outer leading order flow vanishes in a neighborhood of the corner. We then rigorously justify the resulting composite expansion in the small viscosity limit. More precisely, after retaining the leading outer flow, the two weak boundary layers, the nonlinear corner layer, and the order-ε outer correction, the error between the exact solution and this truncated expansion is O(ε^5/4) in L^∞(0,T;L^2(Q)) and O(ε^3/4) in L^2(0,T;H^1(Q)).
ER -