An inviscid limit problem for Navier-Stokes equations in 3D domains with oscillatory boundaries
Tuoc Phan, Dario A. Valdebenito
TL;DR
The paper addresses the inviscid limit for 3D Navier–Stokes flows with a measurable, anisotropic viscosity in oscillatory domains that converge to the flat half-space. The authors flatten the oscillatory domain to a fixed half-space and construct boundary-layer corrections alongside a perturbation around the Euler solution, enabling a Grönwall-based energy analysis for the remainder. Under precise anisotropic rates linking the viscosity eigenvalues $\eta,\nu$ and the oscillation parameter $\delta$, they prove convergence of Leray-Hopf weak solutions to the Euler solution in the upper half-space with explicit rates $O(\beta(\eta,\nu)+\delta^{\alpha-5/2})$. This advances the understanding of inviscid limits in geometrically complex domains with non-smooth viscosity data, with potential implications for geophysical and multiphysics flows.
Abstract
We study an inviscid limit problem for a class of Navier-Stokes equations with vanishing measurable viscous coefficients in 3-dimensional spatial domains whose boundaries are oscillatory, depending on a small parameter, and become flat when the parameter converges to zero. Under some sufficient conditions on the anisotropic vanishing rates of the eigenvalues of the matrices of the viscous coefficients and the oscillatory parameter, we show that Leray-Hopf weak solutions of the Navier-Stokes equations with no slip boundary condition converge to solutions of the Euler equations in the upper half space. To prove the result, we apply a change of variables to flatten the boundaries of the spatial domains for the Navier-Stokes equations, and then construct the boundary layer terms. As the Navier-Stokes equations and the Euler equations are originally written in two different domains, additional boundary layer terms are constructed and their estimates are obtained.