This paper is concerned with the large time behavior of the weak solutions for three-dimensional globally modified Navier-Stokes equations. With the aid of energy methods and auxiliary decay estimates together with Lp-Lq estimates of heat semigroup, we derive the optimal upper and lower decay estimates of the weak solutions for the globally modified Navier-Stokes equations as C1(1+t)-3/4≤uL2≤C2(1+t)-3/4, t>1. The decay rate is optimal since it coincides with that of heat equation.

1. Introduction

It is well known that the motion of the viscous incompressible fluids is governed by the following classic Navier-Stokes equations [1]:
(1)∂tu+(u·∇u)-Δu+∇p=0,∇·u=0.
Here u and π denote the unknown velocity and pressure of the fluid motion, respectively. This motion essentially presumes that the derivatives of the components of the velocity are small.

In Leray’s pioneer work [2] in 1930’, for any initial data in L2, Navier-Stokes equations (1) exits a global weak solution u satisfying
(2)u∈L∞(0,T;L2)∩L2(0,T;H1),foranyT>0.
However, the question of global existence for smooth solutions of the 3D Navier-Stokes equations is still a big open problem. In order to overcome this large difficulty, many efforts have been made to study some related modified Navier-Stokes equations (see [3, 4]). Recently, Caraballo et al. [5] (see also Kloeden et al. [6, 7]) introduced an interesting and important mathematical model which is the so-called global modification of the Navier-Stokes equations
(3)∂tu+FN(∥∇u∥L2)(u·∇u)-νΔu+∇p=0,∇·u=0
associated with
(4)u(x,0)=u0,
where FN (for some N∈R+) is defined by
(5)FN(r)=min{1,Nr},r∈R+.

Let us give a profile analysis to this globally defined model. The modifying factor FN(∥∇u∥L2) is a function of ∥∇u∥L2. Essentially, it prevents large gradients dominating flux and leading to explosions. What is the most important is that this model exhibits a unique global weak solution for the system (1) in bounded domain (see [5]).

However, it should be mentioned that although the presence of FN(∥∇u∥L2) actually canceled some singularities of the nonlinear term u·∇u, it cannot increase the effect of low frequency of the solutions of the system (3). Therefore, it is interesting to consider the time decay issue of this model which largely depended on the effect of low frequency of the solutions. In this paper, we are focused on the L2 decay of weak solutions for the modified Navier-Stokes equations (3). To carry out this issue, it is necessary to recall some classic time decay results of the fluid dynamical models. L2 decay of weak solutions for the Navier-Stokes equations was first studied by Schonbek [8] (see also [9]). She first posed interesting methods the so-called Fourier splitting methods and the finite energy weak solutions decay as
(6)c(1+t)-n/4≤∥u(t)∥L2≤c1(1+t)-n/4.
Later on there are large good results to develop the Fourier splitting methods on the incompressible Navier-Stokes equations [10]. One may also refer to some interesting decay issues of the related fluid models [11–13].

Motivated by the upper and lower decay estimates of nonlinear fluid models [14], in this study, we will develop another technique to deal with the time decay problem of the weak solutions for the globally modified Navier-Stokes equations (3). Our trick is mainly based on the energy methods together with the Lp-Lq estimates of heat semigroup in whole space R3. We can get the optimal time decay rate, since it coincides with that of linear equations.

2. Preliminaries and Main Result

In this paper, we denote by C a generic positive constant which may vary from line to line.

Lp(R3) with 1≤p≤∞ is denoted by the Lebesgue space associated with the norm
(7)∥g∥Lq={(∫R3|g(x)|qdx)1/q,1≤q<∞,esssupx∈R3|g(x)|,q=∞.

Hs(R3) with s∈R is denoted by the fractional Sobolev space with the norm
(8)∥g∥Hs=(∫R3|ξ|2s|g^|2dξ)1/2,
where g^ is the Fourier transformation
(9)Fg(ξ)=g^(ξ)=∫R3e-ix·ξg(x)dx.Lq(0,T;X) is the space of all measurable functions u:(0,T)↦X with the norm
(10)∥u∥Lq(0,T;X)=(∫0T∥u∥Xqdt)1/q,1≤q<∞,
and when q=∞,
(11)∥u∥L∞(0,T;X)=esssupt∈[0,T]∥u∥X.

To state the main results of this paper, we first give the definition of the weak solutions of the three-dimensional globally modified Navier-Stokes equations (3) [5].

Definition 1.

u(x,t) is called a weak solution for three-dimensional globally modified Navier-Stokes equations (3) associated with u0∈L2(R3) if the following properties

u∈L∞(0,T;L2(R3))∩L2(0,T;H1(R3));

for any ϕ∈C0∞(R3×[0,T)) with ∇·ϕ=0,
(12)∫0T∫R3{u·∂tϕ-∇u·∇ϕ+FN(∥∇u∥L2)∇ϕ:u⊗u}dxdt=-∫R3u0ϕ(0)dx;

the energy inequality
(13)∥u(t)∥L22+2∫0t∫R3|∇u(x,s)|2dxds≤∥u0∥L22,mmmmmmmmmmmmmmimfor0≤t≤T
hold true.

Our results read as follows.

Theorem 2.

Suppose that u(x,t) is a weak solution for three-dimensional globally modified Navier-Stokes equations (3). Moreover, if the solution eΔtu0 of the heat equation ∂tu-Δu=0 satisfies
(14)C1(1+t)-3/4≤∥eΔtu0∥L2≤C2(1+t)-3/4,
then the weak solution u(x,t) of (1) possesses the following optimal upper and lower decay rate:
(15)C1(1+t)-3/4≤∥u∥L2≤C2(1+t)-3/4,t>1.

Remark 3.

The decay rate is optimal since it coincides with that of heat equation. The finding is mainly based on energy methods and auxiliary decay estimates together with Lp-Lq estimates of heat semigroup.

In this section, we will first study auxiliary L2 decay of weak solutions for three-dimensional globally modified Navier-Stokes equations (3).

Lemma 4.

Suppose that u(t) is a weak solution of three-dimensional globally modified Navier-Stokes equations (3); then one has
(16)t1/2∥∇u(t)∥L2⟶0,t⟶∞.

Proof of Lemma <xref ref-type="statement" rid="lem3.1">4</xref>.

Taking the inner product of (3) with -Δu gives
(17)ddt∥∇u(t)∥L22+2∥Δu(t)∥L22≤|2∫R3FN(∥∇u∥L2)(u·∇)uΔudx|,
where we have used the following properties:
(18)∫R3∇pΔudx=∫R3pΔ(divu)dx=0
due to the divergence free of the velocity fields.

Since
(19)FN(∥∇u∥L2)=min{1,N∥∇u∥L2}≤1,
then the right hand side of inequality (17) can be estimated after by applying Hölder inequality
(20)|2∫R3FN(∥∇u∥L2)(u·∇)uΔudx|≤C∥Δu∥L2∥u∥L∞∥∇u∥L2.
With the aid of the Gagliardo-Nirenberg inequality,
(21)∥u∥L∞≤C∥Δu∥L23/4∥u∥L21/4,∥∇u∥L2≤C∥u∥L21/2∥Δu∥L21/2.

Plugging (21) into (20), one shows that
(22)|2∫R3FN(∥∇u∥L2)(u·∇)uΔudx|≤C∥Δu∥L27/4∥u∥L21/4∥∇u∥L2≤C∥Δu∥L22(∥u∥L21/4∥∇u∥L2∥Δu∥L2-1/4)≤C∥Δu∥2(∥u∥L21/2∥∇u∥L21/2).
Then inserting the above inequality into (17), one gets
(23)ddt∥∇u(t)∥L22+2∥Δu(t)∥L22≤C∥Δu(t)∥L22∥u∥L21/2∥∇u∥L21/2.

Thus we rewrite inequality (23) as
(24)ddt∥∇u(t)∥L22≤2∥Δu(t)∥L22(C∥u∥L21/2∥∇u∥L21/2-1).
Now for any small ε>0, there exists a large M>0, such that, for t≥M,
(25)∥u∥L2∥∇u∥L2≤ε.
Otherwise, there exists a positive constant ε0, such that, for all t≥0(26)∥u∥L2∥∇u∥L2>ε0
from which and together with energy inequality we have
(27)∥u0∥L2∥∇u∥L2>ε0
which implies that
(28)∥∇u∥L2>Cε0.

On the other hand, from energy inequality we have
(29)∫0∞∥∇u∥L2dt<∞,
which contradicts (28).

Hence, we have
(30)∥u∥L2∥∇u∥L2≤ε,forlarget.

We now choose ε=1/C in (24) and apply (30) to yield
(31)ddt∥∇u(t)∥L22≤0,
from which and together with the energy inequality we have
(32)(t-M)∥∇u(t)∥L22≤∫Mt∥∇u(s)∥L22ds≤12∥u(M)∥2≤12∥u0∥2=C
which implies that
(33)t∥∇u(t)∥L22⟶0,t⟶∞.

4. Optimal Upper and Lower Decay Estimates4.1. Upper Decay Estimate

Consider the integral equations of (3)
(34)u(t)=eΔtu0-∫0teΔ(t-s)PFN(∥∇u∥L2)(u·∇u)ds,
where
(35)Pg=g-∇Δ-1∇·g.

Taking the L2 norm of the integral equation and applying the Lp-Lq estimates of heat equation, it follows that together with Hölder inequality
(36)∥u(t)∥L2≤∥eΔtu0∥L2+C∫0t∥eΔ(t-s)PFN(∥∇u∥L2)(u·∇u)∥L2ds≤C(1+t)-3/4+C∫0t(t-s)-1/2∥u∥L2∥∇u∥L2ds,
where we have used the properties
(37)∥P∥≤1,FN(∥∇u∥L2)≤1.

According to Lemma 4, we let
(38)η(t)≜t1/2∥∇u(t)∥L2
and it is obvious that
(39)η(t)⟶0,t⟶∞.

Thus we rewrite (36) as
(40)∥u(t)∥L2≤∥eΔtu0∥L2+c∫0t∥eΔ(t-s)PFN(∥∇u∥L2)(u·∇u)∥L2ds≤C(1+t)-3/4+C∫0t(t-s)-1/2(1+s)-1/2∥u∥L2η(s)ds.
That is to say,
(41)(1+t)3/4∥u(t)∥L2≤C+Cθ(t)sup0≤s≤t(1+s)3/4∥u(s)∥L2
with
(42)θ(t)=(1+t)3/4∫0t(t-s)-1/2(1+s)-5/4η(s)ds.

It is easy to check that
(43)θ(t)⟶0,ast⟶∞,
or for large t>0,
(44)θ(t)≤12C.

Thus we obtain the optimal upper decay estimates of the weak solution for three-dimensional globally modified Navier-Stokes equations (3) as
(45)∥u(t)∥L2≤C(1+t)-3/4.

4.2. Lower Decay Estimate

From the integral equations (34), we will investigate the error estimates of solutions between three-dimensional globally modified Navier-Stokes equations (3) and the heat equation:
(46)∥u(t)-eΔtu0∥L2≤∫0t∥eΔ(t-s)PFN(∥∇u∥L2)(u·∇u)∥L2ds≤∫0t∥eΔ(t-s)(u·∇u)∥L2ds≤∫0t/2∥∇eΔ(t-s)(u⊗u)∥L2ds+∫t/2t∥∇eΔ(t-s)(u⊗u)∥L2ds=I+J.
For I, employing the Lp-Lq estimates of heat semigroup and upper decay estimates gives
(47)I≤C∫0t/2(t-s)-5/4∥u∥L22ds≤C∫0t/2(t-s)-5/4(1+s)-3/2ds≤C(1+t)-3/2.

For J, similarly,
(48)J≤∫t/2t∥∇eΔ(t-s)(u⊗u)∥L2ds≤∫t/2t(t-s)-7/8∥u⊗u∥L4/3ds≤∫t/2t(t-s)-7/8∥u∥L2∥u∥L4ds≤∫t/2t(t-s)-7/8∥u∥L25/4∥∇u∥L23/4ds≤∫t/2t(t-s)-7/8(1+s)-21/16η3/4(t)ds=o((1+t)-19/16),t⟶∞.
Thus we have from the estimates I and J(49)∥u(t)-eΔtu0∥L2=o((1+t)-19/16),t⟶∞.

Hence by the triangle inequality, one shows that
(50)∥u(t)∥L2≥∥e-Δtu0∥L2-∥u(t)-eΔtu0∥L2≥C(1+t)-3/4,forlarget.

Combination of the upper and lower decay estimates for weak solutions of three-dimensional globally modified Navier-Stokes equations (3) completes the proof of Theorem 2.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is partially supported by the NSF of China nos. 11361026, 11161022, and 71161013.

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