Blow-up and uniqueness of Leray-Hopf solutions to forced Navier-Stokes equations

We prove the existence of forces and smooth initial data such that the associated Leray-Hopf solution of the 3D Navier-Stokes equations is unique, global-in-time, satisfies the energy equality, and has infinitely many (countable) blow-up instants. Different classes of forces and blow-up strengths are considered. All of them are sharp compared to the boundedness statements in literature. Our method also enables us to construct examples of blow-up for the forced 3D Euler equations.