3D Navier-Stokes Equations with Nonvanishing Boundary Condition

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 $(u\cdot \vec{n})|_{\partial \Omega}=0$.