Non-uniqueness of solutions for 3D Navier-Stokes equations in bounded domains

Abstract

This paper examines the uniqueness/non-uniqueness of local-in-time strong solutions for the incompressible 3D Navier-Stokes equations in bounded domains, which are $\partial_t u=\nu \Delta u- u\cdot \nabla u-\nabla p+ f$ and $div~u=0$. The focus of this study is on the case where the boundary condition is defined as $u\cdot \vec{n}|_{\partial\Omega}=0$. This paper demonstrates the existence of two distinct strong solutions to the Navier-Stokes equations under this boundary condition.