A note on the uniqueness of strong solution to the incompressible Navier – Stokes equations with damping

We study the Cauchy problem of the 3D incompressible Navier–Stokes equations with nonlinear damping term α|u|β−1u (α > 0 and β ≥ 1). In [J. Math. Anal. Appl. 377(2011), 414–419], Zhang et al. obtained global strong solution for β > 3 and the solution is unique provided that 3 < β ≤ 5. In this note, we aim at deriving the uniqueness of global strong solution for any β > 3.


Introduction
We are concerned with the following incompressible Navier-Stokes equations with damping in R 3 : where u = (u 1 (t, x), u 2 (t, x), u 3 (t, x)) is the velocity field, P(t, x) is a scalar pressure.t ≥ 0 is the time, x ∈ R 3 is the spatial coordinate.In the damping term, α > 0 and β ≥ 1 are two constants.The prescribed function u 0 (x) is the initial velocity field with div u 0 = 0, while the constant µ > 0 represents the viscosity coefficient of the flow.
When there is no damping term α|u| β−1 u, the system (1.1) is reduced to the classical incompressible Navier-Stokes equations, which has been attracted quite a lot of attention, refer to [2][3][4][5][6]8] and references therein.The model (1.1) comes from porous media flow, friction effects, or some dissipative mechanisms, mainly as a limiting system from compressible flows (see [1] for the physical background).The system (1.1) was studied firstly by Cai and Jiu [1], they showed the existence of a global weak solution for any β ≥ 1 and global strong solutions for β ≥ 7  2 .Moreover, the uniqueness is shown for any 7  2 ≤ β ≤ 5.In [7], Zhang et al. proved for β > 3 and u 0 ∈ H 1 ∩ L β+1 that the system (1.1) has a global strong solution and the strong solution is unique when 3 < β ≤ 5. Later, Zhou [10] improved the results in [1,7].He obtained that the strong solution exists globally for β ≥ 3 and u 0 ∈ H 1 .Moreover, regularity criteria for (1.1) is also established for 1 ≤ β < 3 as follows: if u(t, x) satisfies then the solution remains smooth on [0, T].Recently, Zhong [9] showed the global unique strong solution for any β ≥ 1 provided that the viscosity constant µ is sufficiently large or u 0 L 2 ∇u 0 L 2 is small enough.Now we define precisely what we mean by strong solutions to the system (1.1).
The aim of this paper is to show the uniqueness of global strong solution.Our main result reads as follows.

Remark 1.3.
It should be noted that the uniqueness of global strong solutions was shown in [1] for 7  2 ≤ β ≤ 5, while the authors [7] extended the uniqueness of global strong solutions for 3 < β ≤ 5. Thus, our theorem improves the uniqueness results in [1,7].

Proof of Theorem 1.2
Throughout this section, we denote Since the global existence of strong solutions for β > 3 has been obtained in [7,Theorem 3.1], we only need to show the uniqueness for β > 3. To this end, let (u, P) and ( ū, P) be two strong solutions to the system (1.1) on R 3 × (0, T) with the same initial data, and denote U u − ū, π P − P.