1. Introduction and Main Result
In this study, we consider the Cauchy problem of the generalized 3D Navier-stokes equations:
(1)ut+(-Δ)αu+(u·∇)u+∇π=f, (x,t)∈ℝ3×(0,∞),∇·u=0,u(x,0)=u0.
Here, 0<α<1, and u and π denote unknown velocity and pressure, respectively. f is the external force and u0 is a given initial velocity.
It is well known that when α=1, system (1) becomes the classic Navier-Stokes equations. For the Navier-Stokes equations, it is proved that it has a global weak solution
(2)u(x,t)∈L∞(0,T;L2)∩L2(0,T;H1), ∀T>0
for given u0∈L2(ℝ3) with ∇·u0=0 [1]. However, the regularity of Leray weak solutions is still an open problem in mathematical fluid mechanics even if much effort has been made [2–4]. It is an interesting problem to investigate the stability properties of the Navier-Stokes equations and related fluid models [5–11]. As regard to the above system (1), the asymptotic stability of weak solution of the generalized 3D Navier-Stokes equation is described as follows. If u is perturbed initially by ω0 without any smallness assumption, then the perturbed system v is governed by the following equations:
(3)vt+(-Δ)αv+(v·∇)v+∇π=f,∇·v=0,v(x,0)=u0+ω0,
where ω0 is the initial perturbation. There is large literature on the stability issue of the classic Navier-Stokes equations and related fluid models [12–17]. The aim of this paper is to show the stability of weak solution in the framework of the homogeneous Besov space. More precisely, with the use of the Littlewood-Paley decomposition and the classic Fourier splitting technique, we can show that when the initial perturbation ω0∈L2(ℝ3), then every weak solution v(t) of the perturbed system (2) converges asymptotically to u(t) as ∥v(t)-u(t)∥L2→0, t→∞.
Now our result reads as follows.
Theorem 1.
Let f∈L2(0,T;H-α(ℝ3)), ω0∈L2(ℝ3); Suppose that u(x,t) is a weak solution of (1) and that v(x,t) is a weak solution of the perturbed problem (2), respectively. Moreover, if ∇u also lies in the following regular class:
(4)∇u∈Lp(0,∞;Bq,∞0(ℝ3)), 2αp+3q=2α, 2<q<∞,
then ∥v(t)-u(t)∥L2→0 (t→∞).
The remainder of this paper is organized as follows. In the Section 2, we first recall the Littlewood-Paley decomposition and the Bony decomposition; then we give three key lemmas. And we prove asymptotic stability of the weak solution in the Section 3.
2. Some Auxiliary Lemmas
We recall some basic facts about the Littlewood-Paley decomposition (refer to [18]). Let 𝒮(ℝ3) be Schwartz class of rapidly decreasing functions; supposing f∈𝒮(ℝ3), the Fourier transformation ℱ is defined by
(5)ℱf(ξ)=∫ℝ3e-ix·ξf(x)dx.
Choose two nonnegative radial functions χ,φ∈𝒮(ℝ3), supported in ℬ={ξ∈ℝ3,|ξ|≤4/3} and 𝒞={ξ∈ℝ3,3/4≤|ξ|≤8/3}, respectively, such that
(6)χ(ξ)+∑j≥0φ(2-jξ)=1, ξ∈ℝ3.
Let h=ℱ-1φ and h~=ℱ-1χ, we define the dyadic blocks as follows:
(7)Δjf=φ(2-jD)f=23j∫ℝ3h(2jy)f(x-y)dy, for j≥0,Sjf=χ(2-jD)f=∑-1≤k≤j-1Δkf=23j∫ℝ3h~(2jy)f(x-y)dy,Δ-1f=S0f, Δjf=0 for j≤-2.
We can easily verify that
(8)ΔjΔkf=φ(2-jξ)φ(2-kξ)f^=0, if |j-k|≥2,Δj(Sk-1fΔkf)=φ(2-jξ)χ(2-(k-1)ξ)f^×φ(2-kξ)f^=0, if |j-k|≥5.
Especially for any f∈L2(ℝ3), we have the Littlewood-Paley decomposition:
(9)f=S0(f)+∑j≥0Δjf, f∈𝒮'(ℝ3).
Now we give the definition of the Besov space. Let s∈ℝ and p,q∈[1,∞]; the inhomogeneous Besov space Bp,qs(ℝ3) (see [18]) is defined by the full-dyadic decomposition, such as
(10)Bp,qs(ℝ3)={f∈𝒮'(ℝ3):∥f∥Bp,qs<∞},
where
(11)∥f∥Bp,qs={(∑j=-1∞2jsq∥Δjf∥Lpq)1/q,1≤q<∞,supj≥-12js∥Δjf∥Lp,q=∞,
and 𝒮'(ℝ3) is a dual space of 𝒮(ℝ3).
The Bony decomposition (see [19]) will be frequently used; it is followed by
(12)uv=Tuv+Tvu+R(u,v),
where
(13)Tuv=∑jSj-1uΔjv, R(u,v)=∑|j′-j|≤1ΔjuΔj′v.
The following Bernstein inequality (see [18]) will be used in the next section.
Lemma 2.
Assume that k,j∈Z and 1≤p≤q≤∞, for f∈𝒮(ℝ3), one has
(14)sup|α|=k∥∂αΔjf∥Lq(ℝ3)≤C2jk+3j((1/p)-(1/q))∥Δjf∥Lp(ℝ3),
and the constant C is independent of j and k.
In the following, we will introduce two lemmas, which will be employed in the proof of our theorem.
Lemma 3.
Suppose that u,w∈L∞(0,T;L2) ∩ L2(0,T;Hα), for all T>0, ∇v∈Lp(0,∞;Bq,∞0), (2α/p)+(3/q)=2α, 2<q<∞.
Then the trilinear form
(15)F(u,v,w)=∫0T∫ℝ3(u·∇v)w dx dt
is continuous and
(16)|F(u,v,w)|≤C∥u∥L∞(0,T;L2)1/p∥u∥L2(0,T;Hα)1-(1/p)∥w∥L∞(0,T;L2)1/p×∥w∥L2(0,T;Hα)1-(1/p)∥∇v∥Lp(0,T;Bq,∞0).
In particular, if u=w, then
(17)|F(w,v,w)|≤12∫0T∥Λαw∥L22dt+C∫0T∥w∥L22∥∇v∥Bq,∞0pdt.
Proof of Lemma 3.
We borrow the idea of [20] to prove this lemma. By using of the Littlewood-Paley decomposition and the Bony decomposition, we obtain
(18)F(u,v,w)=∫0T∫ℝ3(uiw)∂iv dx dt=∫0T∫ℝ3(Tuiw+Twui+R(ui,w))ffffffff×(∑jΔj∂iv)dx dt=∑|k-j|≤4∫0T∫ℝ3Sk-1uiΔkwΔj∂iv dx dt +∑|k-j|≤4∫0T∫ℝ3ΔkuiSk-1wΔj∂iv dx dt +∑|k-k′|≤1 ∑k,k′≥j-3∫0T∫ℝ3ΔkuiΔk′wΔj∂iv dx dt=I1+I2+I3.
Then we estimate I1, I2, and I3 one by one. Applying the Hölder inequality and the Bernstein inequality (40), we derive
(19)|I1|≤C∑|k-j|≤4 ∑k′≤k-2∫0T∥Δk′ui∥L2q/(q-2)∥Δkw∥L2∥Δj∂iv∥Lqdt≤C∑|k-j|≤4 ∑k′≤k-2∫0T2(3/q)k′∥Δk′ui∥L2∥Δkw∥L2∥Δj∂iv∥Lqdt≤C∑|k-j|≤4 ∑k′≤k-2∫0T(2(α/p′)k′∥Δk′u∥L2)ffffffffffff ×(2(α/p′)k∥Δkw∥L2)ffffffffffff ×∥Δj∇v∥Lq2((3/q)-(α/p′))k′-(α/p′)kdt,
where (1/p)+(1/p′)=1.
Since |k-j|≤4, k′<k and (2α/p)+(3/q)=2α with 2<q<∞, then
(20)2((3/q)-(α/p′))k′-(α/p′)k=2((3/q)-α+(α/p))k′-(α-(α/p))k=2(3/2q)(k′-k)≤C.
Thanks to the Sobolev embedding B2,∞α/p′(ℝ3)↪B2,2α/p′(ℝ3)=Hα/p′(ℝ3), we have the following estimate:
(21)|I1|≤C∫0T∥u∥Hα/p′∥w∥Hα/p′∥∇v∥Bq,∞0dt.
Similarly, for I2, we also have
(22)|I2|≤C∫0T∥u∥Hα/p′∥w∥Hα/p′∥∇v∥Bq,∞0dt.
To estimate the last term I3, by using the Hölder inequality and the Bernstein inequality we obtain
(23)|I3|≤C∑|k-k′|≤1 ∑k,k′≥j-3∫0T∥Δkui∥L2∥Δk′w∥L2∥Δj∂iv∥L∞dt≤C∑|k-k′|≤1 ∑k,k′≥j-3∫0T∥Δkui∥L2∥Δk′w∥L2∑|k-k′|≤1fff∑k,k′≥j-3∫0T×(2(3/q)j∥Δj∂iv∥Lq)dt≤C∑|k-k′|≤1 ∑k,k′≥j-3∫0T(2(α/p′)k∥Δku∥L2)fffffffffffffff ×(2(α/p′)k′∥Δk′w∥L2)fffffffffffffff ×∥Δj∇v∥Lq2-(3/q)j -(α/p′)(k+k′)dt.
Since |k-k′|≤1, k,k′≥j-3 and (2α/p)+(3/q)=2α, 2<q<∞, we have
(24)2-(3/q)j-(α/p′)(k+k′)=2-(3/q)j-(3/2)(k+k′)(1/q)≤29/q≤C,|I3|≤C∫0T∥u∥Hα/p′∥w∥Hα/p′∥∇v∥Bq,∞0dt.
So, we can derive
(25)|F(u,v,w)| ≤C∫0T∥u∥Hα/p′∥w∥Hα/p′∥∇v∥Bq,∞0dt ≤C(∫0T∥u∥Hα/p′2p′dt)1/2p′(∫0T∥w∥Hα/p′2p′dt)1/2p′ ×(∫0T∥∇v∥Bq,∞0pdt)1/p ≤C∥u∥L2p′(0,T;Hα/p′)∥w∥L2p′(0,T;Hα/p′)∥∇v∥Lp(0,T;Bq,∞0).
Applying the interpolation inequality, we have
(26)∥u∥L2p′(0,T;Hα/p′)≤C∥u∥L∞(0,T;L2)1-(1/p′)·∥u∥L2(0,T;Hα)1/p′≤C∥u∥L∞(0,T;L2)1/p·∥u∥L2(0,T;Hα)1-(1/p).
Then
(27)|F(u,v,w)|≤C∥u∥L∞(0,T;L2)1/p∥u∥L2(0,T;Hα)1-(1/p)∥w∥L∞(0,T;L2)1/p×∥w∥L2(0,T;Hα)1-(1/p)∥∇v∥Lp(0,T;Bq,∞0).
Especially if u=w, by using the interpolation inequality, we get
(28)|F(u,v,w)|≤C∫0T∥w∥Hα/p′2∥∇v∥Bq,∞0dt≤C∫0T∥w∥L22(1-(1/p′))∥Λαw∥L22/p′∥∇v∥Bq,∞0dt≤12∫0T∥Λαw∥L22dt+C∫0T∥w∥L22∥∇v∥Bq,∞0pdt.
Hence, the proof of the lemma is complete.
Let w(x,t)=v(x,t)-u(x,t) denote the difference of v(x,t) and u(x,t), where u(x,t) is a weak solution of (1) and v(x,t) is a weak solution of the perturbed problem (2). Thus w(x,t) satisfies the following equations:(29)wt+(-Δ)αw+(v·∇)w+(w·∇)u+∇π=0, f(x,t)∈ℝ3×(0,∞), ∇·w=0,w(x,0)=w0.
Lemma 4.
Let w(x,t) be the solution of the above problem. Then
(30)|w^(ξ,t)|≤e-|ξ|2αt|w^0(ξ)|+C|ξ|t.
Proof of Lemma 4.
Taking the Fourier transformation of the first equation of (38), we get
(31)w^t+|ξ|2αw^=F[-(v·∇)w-(w·∇)u-∇π]=:G(ξ,t).
We can easily obtain
(32)|F[-(v·∇)w]|≤∑i,j∫ℝ3|viwj||ξj|dx≤|ξ|∥v∥L2∥w∥L2,|F[-(w·∇)u]|≤∑i,j∫ℝ3|wiuj||ξj|dx≤|ξ|∥w∥L2∥u∥L2.
Applying the operator ∇div to the first equation of (38), we have
(33)Δπ=∑i,j∂2∂xi∂xj(-viwj-wiuj),
and taking the Fourier transformation, we get
(34)|ξ|2F[π]=∑i,jξiξjF[-viwj-wiuj];
thus
(35)|F[∇π]|≤|ξ||F[π]|≤|ξ|∥w∥L2(∥u∥L2+∥v∥L2).
Then we have
(36)|G(ξ,t)|≤|ξ|∥w∥L2(∥u∥L2+∥v∥L2).
Thus solving the ordinary differential equation (31) and using (36) gives
(37)|w^(ξ,t)|=|w^0(ξ)e-|ξ|2αt+∫0te-|ξ|2α(t-s)G(ξ,s)ds|≤|w^0(ξ)|e-|ξ|2αt+C|ξ|∫0t∥w∥L2(∥u∥L2+∥v∥L2)ds≤e-|ξ|2αt|w^0(ξ)|+C|ξ|t,
which is the desired assertion of Lemma 4.
3. Proof of Theorem 1
The following argument is follows the classic Fourier splitting methods which is first used by Schonbek [21] (see also [22]).
Taking the inner product of the first equation in (38) with w together with the divergence-free condition of v,w we have
(38)12ddt∥w∥L22+∫ℝ3|Λαw|2dx=-∫ℝ3(w·∇)u·w dx.
Applying Plancherel’s theorem to (38) yields
(39)12ddt∫ℝ3|w^(ξ,t)|2dξ+∫ℝ3|ξ|2α|w^(ξ,t)|2dξ =-∫ℝ3(w·∇)u·w dx.
Let f(t) be a continuous function of t with f(0)=1, f(t)>0 and f′(t)>0, we can derive the following:
(40)ddt(f(t)∫ℝ3|w^(ξ,t)|2dξ) +2f(t)∫ℝ3|ξ|2α|w^(ξ,t)|2dξ =-2f(t)∫ℝ3(w·∇)u·w dx +f′(t)∫ℝ3|w^(ξ,t)|2dξ.
By integrating in time from 0 to t for (40), we have
(41)f(t)∫ℝ3|w^(ξ,t)|2dξ +2∫0tf(s)∫ℝ3|ξ|2α|w^(ξ,s)|2dξ ds =∫ℝ3|w^0|2dξ-2∫0tf(s)∫ℝ3(w·∇)u·w dx ds +∫0tf′(s)∫ℝ3|w^(ξ,s)|2dξ ds.
Noting that f(t) is a scalar function and applying Lemma 3, we get
(42)|∫0tf(s)∫ℝ3(w·∇)u·w dx ds| ≤12∫0tf(s)∥Λαw∥L22ds+C∫0tf(s)∥w∥L22∥∇u∥Bq,∞0pds ≤12∫0tf(s)∫ℝ3|ξ|2α|w^(ξ,s)|2dξ dt +C∫0tf(s)∥w∥L22∥∇u∥Bq,∞0pds.
Then,
(43)f(t)∫ℝ3|w^(ξ,t)|2dξ +∫0tf(s)∫ℝ3|ξ|2α|w^(ξ,s)|2dξ ds ≤∫ℝ3|w^0|2dξ+∫0tf′(s)∫ℝ3|w^(ξ,s)|2dξ ds +C∫0tf(s)∥w∥L22∥∇u∥Bq,∞0pds.
Let B(t)={ξ∈ℝ3:f(t)|ξ|2α<f′(t)}, we have
(44)f(s)∫ℝ3|ξ|2α|w^(ξ,s)|2dξ≥f′(s)∫ℝ3|w^(ξ,s)|2dξ-f′(s)∫B(s)|w^(ξ,s)|2dξ.
Then,
(45)f(t)∫ℝ3|w^(ξ,t)|2dξ ≤∫ℝ3|w^0(ξ)|2dξ+C∫0tf(s)∥w∥L22∥∇u∥Bq,∞0pds +∫0tf′(s)∫B(s)|w^(ξ,s)|2dξ ds.
In addition,
(46)∫0tf′(s)∫B(s)|w^(ξ,s)|2dξ ds ≤C∫0tf′(s)∫B(s)(e-2|ξ|2αs|w^0(ξ)|2+|ξ|2s2)dξ ds ≤C∫0tf′(s)(∫ℝ3e-2|ξ|2αs|w^0(ξ)|2dξ)ds +C∫0tf′(s)s2(f′(s)f(s))5/2αds.
Choose f(t)=(1+t)2, then
(47)(1+t)2∫ℝ3|w^(ξ,t)|2dξ ≤C+C∫0t(1+s)2∥w∥L22∥∇u∥Bq,∞0pds +C∫0t(1+s)∫ℝ3e-2|ξ|2αs|w^0(ξ,s)|2dξ ds +C(1+t)4-(5/2α),(1+t)2∥w∥L22 ≤C∫0t(1+s)∫ℝ3e-2|ξ|2αs|w^0(ξ)|2dξ ds +C∫0t(1+s)2∥w∥L22∥∇u∥Bq,∞0pds +C(1+t)4-(5/2α).
By using the Gronwall inequality, it follows that
(48)(1+t)2∥w∥L22≤{C∫0t(1+s)∫ℝ3e-2|ξ|2αs|w^0(ξ)|2dξ ds+C(1+t)4-(5/2α)} ×exp(∫0t∥∇u∥Bq,∞0pds).
Since
(49)∫ℝ3e-2|ξ|2αt|w^0(ξ)|2dξ≤C(1+t)-3/2α⟶0, t⟶∞,
we derive
(50)∥w∥L2≤C(1+t)-2∫0t(1+s)∫ℝ3e-2|ξ|2αs|w^0(ξ)|2dξ ds+C(1+t)2-(5/2α)⟶0, t⟶∞,
which completes the proof of Theorem 1.