The Inviscid Limit of the Navier-Stokes Equations with Kinematic and Navier Boundary Conditions

Abstract

We are concerned with the inviscid limit of the Navier-Stokes equations on bounded regular domains in ${\mathbb{R}}^3$ with the kinematic and Navier boundary conditions. We first establish the existence and uniqueness of strong solutions in the class $C([0,T_{\star });H^r(\Omega ;{\mathbb{R}}^3))\cap C^1([0,T_{\star });H^{r-2}(\Omega ;{\mathbb{R}}^3))$ with some $T_{\star }>0$ for the initial-boundary value problem with the kinematic and Navier boundary conditions on $\partial \Omega$ and divergence-free initial data in the Sobolev space $H^r(\Omega ;{\mathbb{R}}^3)$ for $r\ge 2$. Then, for the strong solution with $H^{r+1}$--regularity in the spatial variables, we establish the inviscid limit in $H^r(\Omega ;{\mathbb{R}}^3)$ uniformly on $[0,T_{\star })$ for $r>\frac{5}{2}$. This shows that the boundary layers do not develop up to the highest order Sobolev norm in $H^r(\Omega ;{\mathbb{R}}^3)$ in the inviscid limit. Furthermore, we present an intrinsic geometric proof for the failure of the strong inviscid limit under a non-Navier slip-type boundary condition.

30 pages

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