Large-time behavior of the weak solution to 3D Navier-Stokes equations

Abstract

The weak solution to the Navier-Stokes equations in a bounded domain $D\subset {\mathbb{R}}^3$ with a smooth boundary is proved to be unique provided that it satisfies an additional requirement. This solution exists for all $t\ge 0$. In a bounded domain $D$ the solution decays exponentially fast as $t\to \infty$ if the force term decays at a suitable rate.