Weak and Strong Solutions for the Incompressible Navier-Stokes Equations with Damping Term

Abstract: In this paper we intend to understand the influences of the damping term |u|β−1u on the well-posedness of the classical incompressible Navier-Stokes equations. Our results show that the Cauchy problem of the damped Navier-Stokes equations will have global weak solutions for any β ≥ 1, global strong solutions for any β ≥ 7/2 and will have unique strong solution for any 7/2 ≤ β ≤ 5. Note that when 1 ≤ β ≤ 4, the solutions of the damped Navier-Stokes equations do not belong to the Serrin’s class, which is the regularity class of the classical Navier-Stokes equations.