Global long time uniform well-posedness of 3D incompressible Navier-Stokes equations under time-independent uniqueness condition

Abstract

In this work, inspired by the uniqueness condition of the 3D steady incompressible Navier-Stokes equations, we present a time-independent uniqueness condition depending on $(\nu ,u_0,f_{\infty },\Omega )$ with $f_{\infty }={\sup }_{t\ge 0}{\Vert f\left(t\right)\Vert }_{0,\Omega }$ and consider the fully discrete Galerkin method for the 3D time-dependent incompressible Navier-Stokes equations in the infinite time interval $[0,\infty )$. Furthermore, we provide the long time uniform stability and convergence of the fully discrete Galerkin solution and obtain the global uniform well-posedness (or the existence, uniqueness and long time stability of the solution) of the 3D time-dependent incompressible Navier-Stokes equations under the time-independent uniqueness condition by use of the compact theorem and a new a priori estimate of the fully discrete Galerkin solution.