  We prove the inviscid limit of the incompressible Navier-Stokes equations in
the same topology of Besov spaces as the initial data. The proof is based on
proving the continuous dependence of the Navier-Stokes equations uniformly with
respect to the viscosity. To show the latter, we rely on some Bona-Smith type
method in the $L^p$ setting. Our obtained result implies a new result that the
Cauchy problem of the Euler equations is locally well-posed in the borderline
Besov space $B^{\frac dp+1}_{p,1}(\mathbb{R}^d)$, $1\leq p\leq \infty$, $d\geq
2$, in the sense of Hadmard, which is an open problem left in recent works by
Bourgain and Li in \cite{BL,BL1} and by Misio{\l}ek and Yoneda in \cite{MY,MY2,
MY3}.
