Backward Uniqueness for 3D Navier-Stokes Equations with Non-trivial Final Data and Applications

Presented is a backward uniqueness result of bounded mild solutions of 3D Navier-Stokes Equations in the whole space with non-trivial final data. A direct consequence is that a solution must be axi-symmetric in $[0,T]$ if it is so at time $T$. The proof is based on a new weighted estimate which enables to treat terms involving Calderon-Zygmund operators. The new weighted estimate is expected to have certain applications in control theory when classical Carleman-type inequality is not applicable.