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

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 $r\geq 2$. 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.