Global well-posedness 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). It is shown that the strong solution exists globally for any β ≥ 1.


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.
This model 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 problem (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 [8], 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 [9], Gala and Ragusa [3,4], improved the results in X. Zhong [1,8].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].However, the global existence of strong solution to the problem (1.1) for 1 ≤ β < 3 is still unknown.In fact, this is the main motivation of this paper.
Before stating our main result, we first explain the notations and conventions used throughout this paper.We denote by For 1 ≤ p ≤ ∞ and integer k ≥ 0, the standard Sobolev spaces are denoted by: Our main result reads as follows.
Remark 1.3.Due to 1 ≤ β < 3, we get from Hölder's and Sobolev's inequalities that Consequently, for the given initial velocity u 0 ∈ H 1 (R 3 ) with div u 0 = 0, it follows from (1.4) that the problem (1.1) has a unique global strong solution when the viscosity constant µ is sufficiently large or u 0 L 2 ∇u 0 L 2 is small enough.
The rest of this paper is organized as follows.In Section 2, we collect some elementary facts and inequalities that will be used later.In Section 3, we show the local existence and uniqueness of solutions of the Cauchy problems (1.1).Finally, we give the proof of Theorem 1.2 in Section 4.

Preliminaries
In this section, we will recall some known facts and elementary inequalities that will be used frequently later.
We begin with the following Gronwall's inequality, which plays a central role in proving a priori estimates on strong solutions (u, P).
Next, the following Gagliardo-Nirenberg inequality will be used later.
Lemma 2.2.Let 1 ≤ p, q, r ≤ ∞, and j, m are arbitrary integers satisfying The constant C depends only on n, m, j, q, r, a.
Finally, we need the following lemma to obtain the uniform bounds in the next section.
where F is bounded on bounded sets from R into R. Then for every ε > 0, there exists T ε independent of g such that g(t) Proof.See [6, Lemma 6].

Local existence and uniqueness of solutions
In this section, we shall prove the following local existence and uniqueness of strong solutions to the Cauchy problem (1.1).

A priori estimates
The main goal of this subsection is to derive the following key a priori estimates on Φ(t) defined by Φ(t) u(t) 2 H 1 + 1, which are needed for the proof of Theorem 3.1.Proposition 3.2.Assume that u 0 ∈ H 1 (R 3 ) with div u 0 = 0. Let (u, P) be a solution to the problem (1.1) on R 3 × (0, T].Then there exist a small time T 0 ∈ (0, T] and a positive constant C depending only on µ, α, β, and E 0 Proof.Multiplying (1.1) 1 by u and integrating (by parts) the resulting equation over R 3 , we obtain that 1 2 By the divergence theorem and (1.1) 2 , we have 2) and integrating with respect to t, we get Multiplying (1.1) 1 by u t and integrating (by parts) the resulting equation over R 3 , we obtain from Cauchy-Schwartz inequality that µ 2 Thus Similarly, multiplying (1.1) 1 by −∆u and integrating the resulting equations over R 3 , we have 1 2 Hence, we get which combined with (3.5) yields Applying the Gagliardo-Nirenberg inequality and Sobolev's inequality, the right hand side of (3.7) can be bounded as Then integrating (3.9) with respect to t, we have ∇u(t) Hence, the desired (3.1) follows from this inequality and Lemma 2.3.This completes the proof of Proposition 3.2.

Proof of Theorem 3.1
Since a priori estimates in higher norms have been derived, the local existence of strong solutions can be established by a standard Galerkin method (see for example [2]), and we omit the details.Thus we complete the proof of the existence part of Theorem 3.1.
The uniqueness part of Theorem 3.1 is an immediate consequence of the weak-strong unique result in [9,Theorem 3.1].This finishes the proof of Theorem 3.1.

Proof of Theorem 1.2
Throughout this section, we denote C 0 u 0 2 L 2 .Sometimes we use C( f ) to emphasize the dependence on f .Let (u, P) be the strong solution to the problem (1.1) on R 3 × (0, T), then one has the following results.Lemma 4.1.For any t ∈ (0, T), there holds Proof.This follows from (3.4).

Lemma 4.2.
There exists an absolute constant C independent of T, u 0 , µ, α, and β, such that for any t ∈ (0, T), there holds Proof.We infer from (3.7) that Then we obtain after integrating (4.3) with respect to t that 2µ sup 0≤s≤t By virtue of the Gagliardo-Nirenberg and Sobolev's inequalities, one finds that Substituting (4.5) into (4.4) and employing (4.1), we derive that This implies the desired (4.2) and finishes the proof of Lemma 4.2.

Lemma 4.3.
There exists a positive constant ε 0 independent of T, u 0 , µ, α, and β, such that provided that Proof.Define function E(t) as follows In view of the regularity of u, one can easily check that E(t) is a continuous function on [0, T].By (4.2), there is an absolute constant M such that Now suppose that and set We claim that T * = T.
Choosing ε 0 = 1 8M , by virtue of the claim we showed in the above, we derive that provided that (4.7) holds true.This gives the desired (4.6) and consequently completes the proof of Lemma 4.3.

X. Zhong
Now, we can give the proof of Theorem 1.2.
Proof of Theorem 1.2.Let ε 0 be the constant stated in Lemma 4.3 and suppose that the initial velocity u 0 ∈ H 1 (R 3 ) with div u 0 = 0, and According to Theorem 3.1, there is a unique local strong solution (u, P) to the system (1.1).Let T * be the maximal existence time to the solution.We will show that T * = ∞.Suppose, by contradiction, that T * < ∞, then by (1.2), we deduce that for any (s, γ) with 2 s + 3 γ ≤ 1, 3 < γ < ∞,

Lemma 2 . 1 .
Suppose that h and r are integrable on (a, b) and nonnegative a.e. in (a, b).Further assume that y ∈ C[a, b], y ∈ L 1 (a, b), and