Regularity and uniqueness for the 3D incompressible Navier–Stokes equations with damping

This paper studies the Cauchy problem of the 3D incompressible Navier–Stokes equations with damping term ∣ u ∣ β − 1 u ( β ≥ 1 ). We prove that the strong solution exists globally for β ≥ 3 , and establish two regularity criteria as 1 ≤ β < 3 . For any β ≥ 1 , we also prove that the strong solution is unique even among weak solutions.