JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 321427 10.1155/2013/321427 321427 Research Article The Asymptotic Stability of the Generalized 3D Navier-Stokes Equations Wang Wen-Juan Jia Yan Guglielmi Nicola 1 School of Mathematical Sciences Anhui University Hefei 230601 China ahu.edu.cn 2013 11 11 2013 2013 08 07 2013 30 09 2013 21 10 2013 2013 Copyright © 2013 Wen-Juan Wang and Yan Jia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the stability issue of the generalized 3D Navier-Stokes equations. It is shown that if the weak solution u of the Navier-Stokes equations lies in the regular class uLp(0,;Bq,0(3)), (2α/p)+(3/q)=2α, 2<q<, 0<α<1, then every weak solution v(x,t) of the perturbed system converges asymptotically to u(x,t) as vt-utL20, t.

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)3×(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 u0L2(3) with ·u0=0 . However, the regularity of Leray weak solutions is still an open problem in mathematical fluid mechanics even if much effort has been made . It is an interesting problem to investigate the stability properties of the Navier-Stokes equations and related fluid models . 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 . 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 ω0L2(3), then every weak solution v(t) of the perturbed system (2) converges asymptotically to u(t) as v(t)-u(t)L20,t.

Now our result reads as follows.

Theorem 1.

Let fL2(0,T;H-α(3)), ω0L2(3); 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)uLp(0,;Bq,0(3)),2αp+3q=2α,2<q<, then v(t)-u(t)L20(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 ). Let 𝒮(3) be Schwartz class of rapidly decreasing functions; supposing f𝒮(3), the Fourier transformation is defined by (5)f(ξ)=3e-ix·ξf(x)dx. Choose two nonnegative radial functions χ,φ𝒮(3), supported in ={ξ3,|ξ|4/3} and 𝒞={ξ3,3/4|ξ|8/3}, respectively, such that (6)χ(ξ)+j0φ(2-jξ)=1,ξ3.

Let h=-1φ and h~=-1χ, we define the dyadic blocks as follows: (7)Δjf=φ(2-jD)f=23j3h(2jy)f(x-y)dy,forj0,Sjf=χ(2-jD)f=-1kj-1Δkf=23j3h~(2jy)f(x-y)dy,Δ-1f=S0f,Δjf=0forj-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 fL2(3), we have the Littlewood-Paley decomposition: (9)f=S0(f)+j0Δjf,f𝒮'(3).

Now we give the definition of the Besov space. Let s and p,q[1,]; the inhomogeneous Besov space Bp,qs(3) (see ) is defined by the full-dyadic decomposition, such as (10)Bp,qs(3)={f𝒮'(3):fBp,qs<}, where (11)fBp,qs={(j=-12jsqΔjfLpq)1/q,1q<,supj-12jsΔjfLp,q=, and 𝒮'(3) is a dual space of 𝒮(3).

The Bony decomposition (see ) 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|1ΔjuΔjv.

The following Bernstein inequality (see ) will be used in the next section.

Lemma 2.

Assume that k,jZ and 1pq, for f𝒮(3), one has (14)sup|α|=kαΔjfLq(3)C2jk+3j((1/p)-(1/q))ΔjfLp(3), 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,wL(0,T;L2)L2(0,T;Hα), for all T>0, vLp(0,;Bq,0), (2α/p)+(3/q)=2α, 2<q<.

Then the trilinear form (15)F(u,v,w)=0T3(u·v)wdxdt is continuous and (16)|F(u,v,w)|CuL(0,T;L2)1/puL2(0,T;Hα)1-(1/p)wL(0,T;L2)1/p×wL2(0,T;Hα)1-(1/p)vLp(0,T;Bq,0). In particular, if u=w, then (17)|F(w,v,w)|120TΛαwL22dt+C0TwL22vBq,0pdt.

Proof of Lemma <xref ref-type="statement" rid="lem2.2">3</xref>.

We borrow the idea of  to prove this lemma. By using of the Littlewood-Paley decomposition and the Bony decomposition, we obtain (18)F(u,v,w)=0T3(uiw)ivdxdt=0T3(Tuiw+Twui+R(ui,w))ffffffff×(jΔjiv)dxdt=|k-j|40T3Sk-1uiΔkwΔjivdxdt+|k-j|40T3ΔkuiSk-1wΔjivdxdt+|k-k|1k,kj-30T3ΔkuiΔkwΔjivdxdt=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|4kk-20TΔkuiL2q/(q-2)ΔkwL2ΔjivLqdtC|k-j|4kk-20T2(3/q)kΔkuiL2ΔkwL2ΔjivLqdtC|k-j|4kk-20T(2(α/p)kΔkuL2)ffffffffffff×(2(α/p)kΔkwL2)ffffffffffff×ΔjvLq2((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(3)B2,2α/p(3)=Hα/p(3), we have the following estimate: (21)|I1|C0TuHα/pwHα/pvBq,0dt.

Similarly, for I2, we also have (22)|I2|C0TuHα/pwHα/pvBq,0dt.

To estimate the last term I3, by using the Hölder inequality and the Bernstein inequality we obtain (23)|I3|C|k-k|1k,kj-30TΔkuiL2ΔkwL2ΔjivLdtC|k-k|1k,kj-30TΔkuiL2ΔkwL2|k-k|1fffk,kj-30T×(2(3/q)jΔjivLq)dtC|k-k|1k,kj-30T(2(α/p)kΔkuL2)fffffffffffffff×(2(α/p)kΔkwL2)fffffffffffffff×ΔjvLq2-(3/q)j  -(α/p)(k+k)dt.   Since |k-k|1,  k,kj-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/qC,|I3|C0TuHα/pwHα/pvBq,0dt.

So, we can derive (25)|F(u,v,w)|C0TuHα/pwHα/pvBq,0dtC(0TuHα/p2pdt)1/2p(0TwHα/p2pdt)1/2p×(0TvBq,0pdt)1/pCuL2p(0,T;Hα/p)wL2p(0,T;Hα/p)vLp(0,T;Bq,0).

Applying the interpolation inequality, we have (26)uL2p(0,T;Hα/p)CuL(0,T;L2)1-(1/p)·uL2(0,T;Hα)1/pCuL(0,T;L2)1/p·uL2(0,T;Hα)1-(1/p).

Then (27)|F(u,v,w)|CuL(0,T;L2)1/puL2(0,T;Hα)1-(1/p)wL(0,T;L2)1/p×wL2(0,T;Hα)1-(1/p)vLp(0,T;Bq,0).

Especially if u=w, by using the interpolation inequality, we get (28)|F(u,v,w)|C0TwHα/p2vBq,0dtC0TwL22(1-(1/p))ΛαwL22/pvBq,0dt120TΛαwL22dt+C0TwL22vBq,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)3×(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 <xref ref-type="statement" rid="lem2.3">4</xref>.

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,j3|viwj||ξj|dx|ξ|vL2wL2,|F[-(w·)u]|i,j3|wiuj||ξj|dx|ξ|wL2uL2. Applying the operator div to the first equation of (38), we have (33)Δπ=i,j2xixj(-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|wL2(uL2+vL2).

Then we have (36)|G(ξ,t)||ξ|wL2(uL2+vL2). 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|0twL2(uL2+vL2)dse-|ξ|2αt|w^0(ξ)|+C|ξ|t, which is the desired assertion of Lemma 4.

3. Proof of Theorem <xref ref-type="statement" rid="thm1.1">1</xref>

The following argument is follows the classic Fourier splitting methods which is first used by Schonbek  (see also ).

Taking the inner product of the first equation in (38) with w together with the divergence-free condition of v,w we have (38)12ddtwL22+3|Λαw|2dx=-3(w·)u·wdx.

Applying Plancherel’s theorem to (38) yields (39)12ddt3|w^(ξ,t)|2dξ+3|ξ|2α|w^(ξ,t)|2dξ=-3(w·)u·wdx.

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)3|w^(ξ,t)|2dξ)+2f(t)3|ξ|2α|w^(ξ,t)|2dξ=-2f(t)3(w·)u·wdx+f(t)3|w^(ξ,t)|2dξ.

By integrating in time from 0 to t for (40), we have (41)f(t)3|w^(ξ,t)|2dξ+20tf(s)3|ξ|2α|w^(ξ,s)|2dξds=3|w^0|2dξ-20tf(s)3(w·)u·wdxds+0tf(s)3|w^(ξ,s)|2dξds.

Noting that f(t) is a scalar function and applying Lemma 3, we get (42)|0tf(s)3(w·)u·wdxds|120tf(s)ΛαwL22ds+C0tf(s)wL22uBq,0pds120tf(s)3|ξ|2α|w^(ξ,s)|2dξdt+C0tf(s)wL22uBq,0pds.

Then, (43)f(t)3|w^(ξ,t)|2dξ+0tf(s)3|ξ|2α|w^(ξ,s)|2dξds3|w^0|2dξ+0tf(s)3|w^(ξ,s)|2dξds+C0tf(s)wL22uBq,0pds.

Let B(t)={ξ3:f(t)|ξ|2α<f(t)}, we have (44)f(s)3|ξ|2α|w^(ξ,s)|2dξf(s)3|w^(ξ,s)|2dξ-f(s)B(s)|w^(ξ,s)|2dξ.

Then, (45)f(t)3|w^(ξ,t)|2dξ3|w^0(ξ)|2dξ+C0tf(s)wL22uBq,0pds+0tf(s)B(s)|w^(ξ,s)|2dξds.