3D Navier-Stokes Equations with Nonvanishing Boundary Condition

TL;DR

This paper studies the existence and regularity of strong solutions to the 3D incompressible Navier-Stokes equations with a specific boundary condition where the normal component of velocity vanishes on the boundary.

Contribution

It establishes the existence and regularity of local-in-time strong solutions for the Navier-Stokes equations under nonvanishing boundary conditions involving the normal component.

Findings

Proves local-in-time existence of strong solutions

Demonstrates regularity properties of solutions

Handles boundary condition $(u ext{·} ) = 0$ on $oundary\

Abstract

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