  This paper investigates the existence and regularity of strong solutions to
the incompressible Navier-Stokes equations within a bounded domain $\Omega
\subset \mathbb{R}^3$, subject to the boundary condition $(u\cdot
\vec{n})|_{\partial \Omega}=0$. Here, $\vec{n}$ represents the normal vector to
the boundary $\partial\Omega$, and the equation is given by $\partial_t u = \nu
\Delta u - (u \cdot \nabla) u - \nabla p + f$, with initial condition
$u|_{t=0}=u_o\in H$ and the divergence constraint $div\,u = 0$. This paper aims
to establish the existence and the regularity of local-in-time strong solutions
when the boundary condition is $(u\cdot \vec{n})|_{\partial \Omega}=0$.
