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We study the high Reynolds number limit of a viscous fluid in the presence ofa rough boundary. We consider the two-dimensional incompressible Navier-Stokesequations with Navier slip boundary condition, in a domain whose boundariesexhibit fast oscillations in the form $x_2 = \varepsilon^{1+\alpha}\eta(x_1/\varepsilon)$, $\alpha > 0$. Under suitable conditions on theoscillating parameter $\varepsilon$ and the viscosity $\nu$, we show thatsolutions of the Navier-Stokes system converge to solutions of the Euler systemin the vanishing limit of both $\nu$ and $\varepsilon$. The main issue is thatthe curvature of the boundary is unbounded as $\varepsilon \rightarrow 0$,which precludes the use of standard methods to obtain the inviscid limit. Ourapproach is to first construct an accurate boundary layer approximation to theEuler solution in the rough domain, and then to derive stability estimates forthis approximation under the Navier-Stokes evolution.

The vanishing viscosity limit for 2D Navier-Stokes in a rough domain