Convergence Rates for the Stationary and Non-stationary Navier-Stokes Equations over Non-Lipschitz Boundaries

TL;DR

This paper establishes higher-order convergence rates for the Navier-Stokes equations over complex oscillating domains, extending previous results to non-stationary cases with explicit error bounds.

Contribution

It provides new convergence rate estimates for both stationary and non-stationary Navier-Stokes equations over non-Lipschitz, highly oscillating domains, generalizing prior wall law results.

Findings

Achieved $O( ext{small power of } ext{epsilon})$ convergence in stationary case.

Derived exponential decay convergence rate in non-stationary case.

Extended Navier's wall laws to complex oscillating boundary domains.

Abstract

In this paper, we consider the higher-order convergence rates for the 2D stationary and non-stationary Navier-Stokes Equations over highly oscillating periodic bumpy John domains with $C^{2}$ regularity in some neighborhood of the boundary point (0,0). For the stationary case and any $\gamma\in (0,1/2)$, using the variational equation satisfied by the solution and the correctors for the bumpy John domains obtained by Higaki, Prange and Zhuge \cite{higaki2021large,MR4619004} after correcting the values on the inflow/outflow boundaries $(\{0\}\cup\{1\})\times(0,1)$, we can obtain an $O(\varepsilon^{2-\gamma})$ approximation in $L^2$ for the velocity and an $O(\varepsilon^{2-\gamma})$ convergence rates in $L^1$ approximated by the so called Navier's wall laws, which generalized the results obtained by J\"{a}ger and Mikeli\'{c} \cite{MR1813101}. Moreover, for the non-stationary case, using the energy method, we can obtain an $O(\varepsilon^{2-\gamma}+\exp(-Ct))$ convergence rate for the velocity in $L_x^1$.