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−1u∣u∣β−1u (β≥1β≥1). We prove that the strong solution exists globally for β≥3β≥3, and establish two regularity criteria as 1≤β<31≤β<3. For any β≥1β≥1, we also prove that the strong solution is unique even among weak solutions.