Abstract：

 

Last time, we introduced the concept of ancient bounded mild solutions to the incompressible Navier-Stokes equation. Up to the present time, it is still a long standing open problem to decide whether or not there are any non-trivial, non-constant bounded ancient mild solutions to the Navier-Stokes equation in the setting of 3D-Euclidean space. Since the analog problem in the 2D case was fully resolved already, we will first focus on giving a detailed proof of the Louville's type theorem in the 2D Euclidean setting, which asserts that there is no non-trivial, non-constant bounded ancient mild solution to the Navier-Stokes equation in the setting of 2D Euclidean space.