arXiv:1502.04197v1 [math.AP] 14 Feb 2015

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arXiv:1502.04197v1 [math.AP] 14 Feb 2015. LONG TIME DECAY OF 3D-NSE IN LEI-LIN-GEVREY SPACES. JAMEL BENAMEUR AND LOTFI JLALI. Abstract.
LONG TIME DECAY OF 3D-NSE IN LEI-LIN-GEVREY SPACES JAMEL BENAMEUR AND LOTFI JLALI

arXiv:1502.04197v1 [math.AP] 14 Feb 2015

Abstract. In this paper, we prove that there exists a unique global solution of 3D Navier-Stokes equation if exp(a|D|1/σ )u0 ∈ X −1 (R3 ) and ku0 kX −1 < ν. Moreover, we will show that k exp(a|D|1/σ )u(t)kX −1 goes to zero if the time t goes to infinity.

Contents 1. Introduction 2. Notations and preliminary results 2.1. Notations 2.2. Preliminary results −1 (R3 ) 3. Well-posedness of (N SE) in Za,σ 4. Global solution 5. Long time decay for the global solution References

1 3 3 3 6 12 14 14

1. Introduction The 3D incompressible Navier-Stokes equations are given by:   ∂t u − ν∆u + u.∇u = −∇p in R+ × R3 div u = 0 in R+ × R3 (N SE)  u(0, x) = u0 (x) in R3 ,

where ν > 0 is the viscosity of fluid, u = u(t, x) = (u1 , u2 , u3 ) and p = p(t, x) denote respectively the unknown velocity and the unknown pressure of the fluid at the point (t, x) ∈ R+ × R3 , and (u.∇u) := u1 ∂1 u + u2 ∂2 u + u3 ∂3 u, while u0 = (uo1 (x), uo2 (x), uo3 (x)) is an initial given velocity. If u0 is quite regular, the divergence free condition determines the pressure p. The study of local existence is studied by serval researchers, Leray [12, 13], Kato [8],etc. The global existence of weak solutions goes back to Leray [13] and Hopf [7]. The global 1 well-posedness of strong solutions for small initial data in the critical Sobolev space H˙ 2 is due to Fujita and Kato [5], also in [3], Chemin has proved the case of H˙ s , s > 21 . In [9], Kato has proved the case of Lebesgue space L3 . In [10], Koch and Tataru have proved the case of the space BMO−1 (see, also [2, 4, 14]). It should be noted, in all these works, that the norms in corresponding spaces of the initial data are assumed to be very small, smaller than the viscosity ν multiplied by tiny positive constant c. For further results and Date: February 17, 2015. 2000 Mathematics Subject Classification. 35-xx, 35Bxx, 35Lxx. Key words and phrases. Navier-Stokes Equations; Critical spaces; Long time decay. 1

2

J. BENAMEUR AND L. JLALI

details the reader can consult the book by Cannone [1]. In [11], the authors consider a new critical space that is contained in BMO−1 , where they show it is sufficient to assumed the norms of initial data are less than exactly the viscosity coefficient ν. Then, the used space in [11] is the following Z |b u(ξ)| dξ < ∞} X −1 (R3 ) = {f ∈ D ′ (R3 ); R3 |ξ|

which is equipped with the norm

kf kX −1 (R3 ) = We will also use the notation, for i = 0, 1, i

3



3

X (R ) = {f ∈ D (R );

Z

R3

Z

R3

|b u(ξ)| dξ. |ξ| |ξ|i |b u(ξ)|dξ < ∞}.

For the small initial data, the global existence is proved in [11]: Theorem 1.1. (See [11]). Let u0 ∈ X −1 (R3 ), such that ku0 kX −1 (R3 ) < ν. Then, there is a unique u ∈ C(R+ , X −1 (R3 )) such that ∆u ∈ L1 (R+ , X −1 (R3 )). Moreover, ∀t ≥ 0   Z t 0 sup ku(t)kX −1 + (ν − ku kX −1 ) k∇ukL∞ dτ ≤ ku0 kX −1 . 0≤t 0, σ > 1 and ρ ∈ R, the following spaces are defined Z 1/σ ρ 3 ′ 3 |ξ|ρ ea|ξ| |fb(ξ)|dξ < ∞} Za,σ (R ) = {f ∈ S (R ); R3

which is equipped with the norm

ρ kf kZa,σ (R3 ) =

Our first result is the following:

Z

R3

|ξ|ρ ea|ξ|

1/σ

|fb(ξ)|dξ.

LONG TIME DECAY OF 3D-NSE IN LEI-LIN-GEVREY SPACES

3

−1 (R3 ), such that kuk Theorem 1.4. Let u0 ∈ Za,σ X −1 (R3 ) < ν. Then, there exists a unique + −1 3 1 + 1 global solution u ∈ C(R , Za,σ (R )) ∩ L (R , Za,σ (R3 )) of (N SE).

Our second result is as follows: −1 (R3 )) be the global solution of (N SE). Then Theorem 1.5. Let u ∈ C(R+ , Za,σ

lim sup ku(t)kZa,σ −1 = 0. t→∞

The paper is organized in the following way: In section 2, we give some notations and important preliminary results. Section 3 is devoted to prove that (NSE) is well posed −1 (R3 ). In section 4, we prove the existence under the condition kuk in Za,σ X −1 (R3 ) < ν. −1 (R3 ) goes to zero Finally, in the section 5, we state that the norm of global solution in Za,σ at infinity. 2. Notations and preliminary results 2.1. Notations. In this section, we collect some notations and definitions that will be used later. • The Fourier transformation is normalized as Z b exp(−ix.ξ)f (x)dx, ξ = (ξ1 , ξ2 , ξ3 ) ∈ R3 . F(f )(ξ) = f (ξ) = R3

• The inverse Fourier formula is F

−1

−3

(g)(x) = (2π)

Z

R3

exp(iξ.x)g(ξ)dξ, x = (x1 , x2 , x3 ) ∈ R3 .

• The convolution product of a suitable pair of function f and g on R3 is given by Z f (y)g(x − y)dy. (f ∗ g)(x) := R3

• If f = (f1 , f2 , f3 ) and g = (g1 , g2 , g3 ) are two vector fields, we set f ⊗ g := (g1 f, g2 f, g3 f ),

and

div (f ⊗ g) := (div (g1 f ), div (g2 f ), div (g3 f )). • Let (B, ||.||), be a Banach space, 1 ≤ p ≤ ∞ and T > 0. We define LpT (B) the space of all measurable functions [0, t] ∋ t 7→ f (t) ∈ B such that t 7→ ||f (t)|| ∈ Lp ([0, T ]). 2.2. Preliminary results. In this section, we recall some classical results and we give new technical lemmas. −1 (R3 ) ∩ Z 1 (R3 ). Then Lemma 2.1. Let f, g ∈ Za,σ a,σ

kf gkZa,σ 0 ≤ kf kZa,σ −1 kgkZ 1 + kf kZa,σ 1 kgk −1 . Za,σ a,σ Proof lemma 2.1. We have kf gkZa,σ 0

Z

|fcg(ξ)|dξ  Z Z a|ξ|1/σ b |f (ξ − η)||b g (η)|dη dξ. e ≤

=

ea|ξ|

1/σ

R3

ξ

η

4

J. BENAMEUR AND L. JLALI

Using the inequality ea|ξ| We obtain kf gkZa,σ 0

Put

1/σ

≤ ea|ξ−η|

1/σ

ea|η|

F1 (ξ) = |ξ|e Then

and 1 ≤

|ξ−η| |η|

+

|η| |ξ−η| .

Z Z 1/σ |ea|η| a|ξ−η|1/σ b |f (ξ − η)| ( |ξ − η|e = |b g (η)|dη |η| ξ η Z a|ξ−η|1/σ e 1/σ + g (η)|dη)dξ. |fb(ξ − η)|η||ea|η| |b η |ξ − η| 1/σ

a|ξ|1/σ

1/σ

ea|ξ| |fb(ξ)|, F2 (ξ) = |ξ| kf gkZa,σ 0

1/σ

ea|ξ| 1/σ g (ξ)| and G2 (ξ) = |fb(ξ)|, G1 (ξ) = |ξ|ea|ξ| |b |ξ|

|b g (ξ)|.

≤ kF1 ∗ G2 kL1 + kF2 ∗ G1 kL1

≤ kF1 kL1 kG2 kL1 + kF2 kL1 kG1 kL1 ≤ kf kZa,σ −1 kgkZ 1 + |f kZa,σ 1 kgk −1 . Za,σ a,σ 

3 3 1 1 −1 Lemma 2.2. Let u ∈ L∞ T (Za,σ (R )) ∩ LT (Za,σ (R )). Then Z t eν(t−τ )∆ div(u ⊗ u)dτ kZa,σ k −1 kukL1 (Z 1 ) . −1 ≤ 2kuk ∞ ) L (Za,σ a,σ T

T

0

Proof lemma 2.2. Z t eν(t−τ )∆ div(u ⊗ u)dτ kZa,σ −1 k 0

≤ ≤ ≤

Using the lemma 2.1, we obtain Z t eν(t−τ )∆ div(u ⊗ u)dτ kZa,σ k −1 0

Z

t

0

Z

t

0

Z

0

keν(t−τ )∆ div(u ⊗ u)kZa,σ −1 dτ keν(t−τ )∆ (u ⊗ u)kZa,σ 0 dτ

t

k(u ⊗ u)kZa,σ 0 dτ.

≤ 2

Z

t 0

kukZa,σ −1 kukZ 1 a,σ

−1 kukL1 (Z 1 ) . ≤ 2kukL∞ (Za,σ ) a,σ T

T

 −1 3 1 1 3 Lemma 2.3. Let u ∈ L∞ T (Za,σ (R )) ∩ LT (Za,σ (R )). Then Z T Z t eν(t−τ )∆ div(u ⊗ u)dτ kZa,σ k 1 dt ≤ 2kuk ∞ −1 kukL1 (Z 1 ) . L (Za,σ ) a,σ 0

T

T

0

Proof lemma 2.3. Z T Z tZ Z T Z t 1/σ 2 \ ⊗ u(τ, ξ)|dτ dtdξ e−ν(t−τ )|ξ| |ξ|2 ea|ξ| |u eν(t−τ )∆ div(u ⊗ u)dτ kZa,σ 1 dt ≤ k 3 0 0 0 R 0  Z T Z t Z 2 a|ξ|1/σ −ν(t−τ )|ξ|2 \ |ξ| e e |u ⊗ u(τ, ξ)|dτ dt dξ. ≤ R3

0

0

LONG TIME DECAY OF 3D-NSE IN LEI-LIN-GEVREY SPACES

5

2

Integrating the function e−ν(t−τ )|ξ| twice with respect to τ ∈ [0, t] and t ∈ [0, T ], we get " #  Z T Z TZ t 2 T −ν(t−τ )|ξ| 2 −e  dτ \ \ |u ⊗ u(τ, ξ)|  e−ν(t−τ )|ξ| |u ⊗ u(τ, ξ)|dτ dt = ν|ξ|2 0 0 0 τ

≤ Then Z Z T Z t eν(t−τ )∆ div(u ⊗ u)dτ kZa,σ ≤ k 1 dt 0

Z

T

0

2 a|ξ|1/σ

R3

0

Z



\ |u ⊗ u(τ, ξ)|(

|ξ| e

Z

1−

T

0

2 e−ν(T −τ )|ξ|

ν|ξ|2

ku ⊗ ukZa,σ 0 .

Using the lemma 2.1, we will get the result. The proof of the first main result requires the following lemma. Lemma 2.4.

1

1

ku ⊗ ukZa,σ ≤ kukZ −1 kukZ2 −1 k∆ukZ2 −1 . 0 a a,σ

√ ,σ σ

a,σ

Proof lemma 2.4. It is easy to see that 1

a − √aσ )x σ (σ

Then, for x = |ξ| This implies

≤ ca,σ , ∀x ≥ 0. 1

a (σ − √aσ )|ξ| σ

|ξ|2 e

1

a

|ξ|e σ |ξ| σ ≤ ca,σ Then k∆ukZ −1 a

=

σ ,σ

Z

≤ ca,σ .

1 √aσ |ξ| σ1 . e |ξ| 1

a

R3

Z

u(ξ)|dξ |ξ|e σ |ξ| σ |b

1 √aσ |ξ| σ1 e |b u(ξ)|dξ R3 |ξ| . ≤ ckukZ −1 a

≤ c

2

1 − e−ν(T −τ )|ξ| \ ( )|u ⊗ u(τ, ξ)|dτ ν|ξ|2

T

0

x2 e

)dτ.

√ ,σ σ

Using the previous computations and Cauchy-Schwartz inequality, we get Z Z a|ξ|1/σ u(ξ − η)||b u(η)|dη)dξ e ( |b ku ⊗ ukZa,σ = 0 η

ξ

≤ ckukZ 0a kukZa,σ 0 σ ,σ

≤ ck∆ukZ −1 kukZa,σ 0 a σ ,σ

≤ ckukZ −1a kukZa,σ 0 √ ,σ σ

1

1

≤ ckukZ −1a kukZ2 −1 k∆ukZ2 −1 . √ ,σ σ

a,σ

a,σ



!



6

J. BENAMEUR AND L. JLALI −1 (R3 ) 3. Well-posedness of (N SE) in Za,σ

In the following theorem, we study the existence and uniqueness of the solution. −1 . Then, there are a time T > 0 and a unique solution Theorem 3.1. Let u0 ∈ Za,σ −1 (R3 )) of (N SE) such that u ∈ L1 ([0, T ], Z 1 (R3 )). u ∈ C([0, T ], Za,σ a,σ

Proof theorem 3.1. (i)Firstly, we wish to prove the existence. The idea of the proof is to write the initial condition as a sum of higher and lower frequencies. For small frequencies, we will give a regular solution of the associated linear system to (N SE). For the higher frequencies, we consider a partial differential equation −1 (R3 ) for which we can solve it by the very small to (N SE) with small initial data in Za,σ Fixed Point Theorem. 1 ). • Let r ∈ (0, 10 • Let N ∈ N, such that Z 1/σ ea|ξ| r |c u0 (ξ)|dξ < . |ξ| 5 |ξ|>N Let’s v 0 = F −1 (1{|ξ|N} c u0 (ξ)). Clearly r kw0 kZa,σ (3.1) . −1 < 5 Let v = eνt∆ v 0 the unique solution to  ∂t v − ν∆v = 0 v(0, x) = v 0 (x), We have

0 kvkZa,σ −1 ≤ ku k −1 , ∀t ≥ 0, Za,σ

and

= kvkL1 (Za,σ 1 ) T

≤ ≤

Z

Z

Z

Z

T 0

Z

T 0

R3

1/σ

|b v (ξ)|dξdt

2

R3 Z T

( R3

|ξ|ea|ξ|

e−νt|ξ| |ξ|ea|ξ| 2

1/σ

|c u0 (ξ)|dξdt

e−νt|ξ| dt)|ξ|ea|ξ|

0

1/σ

|c u0 (ξ)|dξ

Z 1 1/σ 2 u0 (ξ)|dξ. ≤ (1 − e−νT |ξ| )|ξ|−1 ea|ξ| |c ν R3 Using the Dominated Convergence Theorem, we get (3.2)

lim kvkL1 (Za,σ 1 ) = 0.

t→0+

T

Let ε > 0 such that 2εku0 kZa,σ −1
0 such that kvkL1 (Za,σ 1 ) < ε. T

Put w = u − v, clearly w is the solution of the following system  ∂t w − ν∆w + (v + w).∇(v + w) = −∇p w(0, x) = w0 (x) ,

The integral form of w is as follows νt∆

w=e

0

w −

Z

t

eν(t−τ )∆ (v + w).∇(v + w)dτ.

0

To prove the existence of w, we put the following operator Z t νt∆ 0 eν(t−τ )∆ (v + w).∇(v + w)dτ. ψ(w) = e w − 0

Now, we introduce the spaces ZT as follows with the norm

−1 1 ZT = C([0, T ], Za,σ (R3 )) ∩ L1 ([0, T ], Za,σ (R3 )) −1 + kf kL1 (Z 1 ) . kf kZT = kf kL∞ (Za,σ ) a,σ T

T

Using lemmas 2.2 and 2.3, we can prove ψ(ZT ) ⊂ ZT . • Also, denoted by Br the subset of ZT defined by:

Br = {u ∈ ZT ; kukL∞ (Za,σ −1 ≤ r; kukL1 (Z 1 ) ≤ r}. ) a,σ T

T

• For w ∈ Br , we prove that ψ(w) ⊂ Br . In fact, we have kψ(w)(t)kZa,σ −1 ≤ where

4 X

Ik ,

k=0

I0 = keνt∆ w0 kZa,σ −1 Z t keν(t−τ )∆ v∇vkZa,σ I1 = −1 dτ 0 Z t keν(t−τ )∆ v∇wkZa,σ I2 = −1 dτ 0 Z t keν(t−τ )∆ w∇vkZa,σ −1 dτ I3 = 0 Z t keν(t−τ )∆ w∇wkZa,σ −1 dτ. I4 = 0

Using (3.1) the lemma 2.2 and the fact that w ∈ Br , hence we get r I0 ≤ 5

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J. BENAMEUR AND L. JLALI

I1 ≤ kvkL∞ (Za,σ −1 kvkL1 (Z 1 ) ) a,σ T

T

≤ 2εku0 kZa,σ −1 r < 5

I2 , I3 ≤ kvkL1 (Za,σ −1 1 ) kwk ∞ ) L (Za,σ T

T

+ kvkL∞ (Za,σ −1 kwkL1 (Z 1 ) ) a,σ T

T

0

≤ r(ku kZa,σ −1 + ε) r < 5 I4 ≤ kwkL∞ (Za,σ −1 kwkL1 (Z 1 ) ) a,σ T

T

≤ 2r 2 r < . 5

Then (3.3)

kψ(w)(t)kZa,σ −1 ≤ r.

Similarly, kψ(w)(t)kL1 (Za,σ 1 ) ≤ where J0 =

Z

T

0

J1 =

Z

T

0

J2 = J3 = J4 =

Z

Z

Z

k

T

k

0 T 0

k

T

0

k

Z

Z

Z

Z

t 0 t

0 t 0 t

0

4 X

Jk ,

k=0

keνt∆ w0 kZa,σ 1 dt

eν(t−τ )∆ v∇vdτ kZa,σ 1 dt eν(t−τ )∆ v∇wdτ kZa,σ 1 dt eν(t−τ )∆ w∇vdτ kZa,σ 1 dt

eν(t−τ )∆ w∇wdτ kZa,σ 1 dt.

Using lemmas 2.3 and the fact that w ∈ Br , we get r J0 ≤ 5 J1 ≤ 2kvkL∞ (Za,σ −1 kvkL1 (Z 1 ) ) a,σ T

≤ 2εku0 kZa,σ −1 r < 5

T

LONG TIME DECAY OF 3D-NSE IN LEI-LIN-GEVREY SPACES

J2 , J3 ≤ kvkL1 (Za,σ −1 1 ) kwk ∞ ) L (Za,σ T

T

−1 kwkL1 (Z 1 ) + kvkL∞ (Za,σ ) a,σ T

T

≤ r(ku0 kZa,σ −1 + ε) r < 5

−1 kwkL1 (Z 1 ) J4 ≤ 2kwkL∞ (Za,σ ) a,σ T

T

≤ 2r 2 r . < 5

Then (3.4)

kψ(w)(t)kL1 (Za,σ 1 ) ≤ r.

Combining (3.3) and (3.4), we get ψ(w) ⊂ Br and we can deduce (3.5)

• Proof of the following estimate

ψ(Br ) ⊂ Br .

kψ(w2 ) − ψ(w1 )kZT ≤ In fact, we have ψ(w2 ) − ψ(w1 ) = − = − and

Z

Z

t

1 kw2 − w1 k|ZT , w1 , w2 ∈ Br . 2

eν(t−τ )∆ ((v + w2 )∇(v + w2 ) − (v + w1 )∇(v + w1 ))dτ

0 t

eν(t−τ )∆ ((v + w2 )∇(w2 − w1 ) + (w2 − w1 )∇(v + w1 ))dτ

0

kψ(w2 ) − ψ(w1 )kZa,σ −1 ≤ K1 + K2 ,

with

K1 = k K2 = k

Z

t

Z0 t 0

eν(t−τ )∆ (v + w2 )∇(w2 − w1 )dτ kZa,σ −1 , eν(t−τ )∆ (w2 − w1 )∇(v + w1 )dτ kZa,σ −1 .

Using lemma 2.2, we can deduce K1 ≤ kv + w2 kZa,σ + kv + w2 kZa,σ −1 kw2 − w1 k|Z 1 1 kw2 − w1 k| −1 Za,σ a,σ ≤ (kvkZa,σ −1 + kw2 k −1 )kw2 − w1 k|Z 1 Za,σ a,σ

+ (kvkZa,σ 1 + kw2 kZa,σ 1 )kw2 − w1 k| −1 Za,σ ≤ (ε + 2r + ku0 kZa,σ −1 )kw2 − w1 k|ZT .

Similarly, we get Then (3.6)

K2 ≤ (ε + 2r + ku0 kZa,σ −1 )kw2 − w1 k|Z . T

0 kψ(w2 ) − ψ(w1 )kL∞ (Za,σ −1 ≤ 2(ε + 2r + ku k −1 )kw2 − w1 k|ZT . ) Za,σ T

Therefore, we have kψ(w2 ) − ψ(w1 )kL1 (Za,σ 1 ) ≤ K3 + K4 ,

9

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J. BENAMEUR AND L. JLALI

with K3 = K4 =

Z

Z

T

k

0 T 0

k

Z

Z

t 0 t 0

eν(t−τ )∆ (v + w2 )∇(w2 − w1 )dτ kZa,σ 1 dt, eν(t−τ )∆ (w2 − w1 )∇(v + w1 )dτ kZa,σ 1 dt.

Using lemma 2.3, then we can deduce −1 K3 ≤ kv + w2 kL1 (Za,σ 1 ) kw2 − w1 k| ∞ ) L (Za,σ T

T

+ kv + w2 kL∞ (Za,σ −1 kw2 − w1 k| 1 −1 ) L (Za,σ ) T

0

≤ (ε + 2r + ku kZa,σ −1 )kw2 − w1 k|ZT . Similarly, we get K4 ≤ (ε + 2r + ku0 kZa,σ −1 )kw2 − w1 k|ZT .

Then (3.7)

0 kψ(w2 ) − ψ(w1 )kL1 (Za,σ 1 ) ≤ 2(ε + 2r + ku k −1 )kw2 − w1 k|ZT . Za,σ T

By (3.6) and (3.7), we obtain kψ(w2 ) − ψ(w1 )kZT ≤ 4(ε + 2r + ku0 kZa,σ −1 )kw2 − w1 k|Z . T This implies 1 kψ(w2 ) − ψ(w1 )kZT ≤ kw2 − w1 k|ZT . 2 So, combining (3.5) and (3.8) and the Fixed Point Theorem, there is a unique w ∈ Br such that u = v + w is the solution of (N SE) with u ∈ ZT (R3 ). (3.8)

(ii) Secondly, we want to prove the uniqueness. −1 (R3 )) ∩ L1 ([0, T ], Z 1 (R3 )) of (N SE) such that u (0) = u (0). Let u1 , u2 ∈ C([0, T ], Za,σ 1 2 a,σ Put δ = U1 − U2 . We have

(3.9)

Then

∂t δ − ν∆δ + u1 .∇δ + δ.∇u2 = −∇(p1 − p2 ). \ ∂t δb + ν|ξ|2 δb + (u\ 1 .∇δ) + (δ.∇u2 ) = 0.

b we get Multiplying the previous equation by δ, (3.10)

\ b b δb + ν|ξ|2 δ. b b δb + (u\ ∂t δ. 1 .∇δ).δ + (δ.∇u2 ).δ = 0.

From Eq (3.9) we have

\ ∂t δb + ν|ξ|2 δb + (u\ 1 .∇δ) + (δ.∇u2 ) = 0.

b we get Multiplying this equation by δ, (3.11)

\ b b b δb + ν|ξ|2 δ. b δb + (u\ ∂t δ. 1 .∇δ).δ + (δ.∇u2 ).δ = 0.

By summing (3.10) and (3.11), we get

and

\ b b 2 + 2ν|ξ|2 |δ| b 2 + 2Re((u\ b ∂t |δ| 1 .∇δ).δ) + 2Re((δ.∇u2 ).δ) = 0, \ b b b 2 + 2ν|ξ|2 |δ| b 2 ≤ 2|(u\ ∂t |δ| 1 .∇δ)||δ| + 2|(δ.∇u2 )||δ|.

LONG TIME DECAY OF 3D-NSE IN LEI-LIN-GEVREY SPACES

11

Let ε > 0, thereby we have

q q 2 2 2 2 2 b b b b 2 + ε2 ∂t |δ| = ∂t (|δ| + ε ) = 2 |δ| + ε .∂t |δ|

then 2∂t

q

b 2 + ε2 + 2ν|ξ|2 q |δ|

b2 |δ|

≤ 2|(u\ 1 .∇δ)| q

b 2 + ε2 |δ|

b |δ|

\2 )| q + 2|(δ.∇u

b 2 + ε2 |δ|

\ ≤ 2|(u\ 1 .∇δ)| + 2|(δ.∇u2 )|.

b |δ|

b 2 + ε2 |δ|

By integrating with respect to time Z t Z t Z t q b2 |δ| \2 )|dτ. b 2 + ε2 + ν |(δ.∇u |(u\ .∇δ)|dτ + ≤ |ξ|2 q |δ| 1 0 0 0 2 2 b |δ| + ε

Letting ε → 0, we get

b +ν |δ|

Z

1

Multiplying by

ea|ξ| σ |ξ|

kδkZa,σ −1

t 0

b |ξ| |δ|dτ ≤ 2

Z

0

t

|(u\ 1 .∇δ)|dτ +

Z

0

t

\2 )|dτ. |(δ.∇u

and integrating with respect to ξ , thereafter we get Z t Z t Z t kδ.∇u2 kZa,σ ku1 .∇δkZa,σ k∆δkZa,σ ≤ +ν −1 dτ −1 dτ + −1 dτ 0 0 0 Z t Z t ku2 δkZa,σ kδu1 kZa,σ ≤ 0 dτ. 0 dτ + 0

0

Using the elementary inequality xy ≤ kδu1 kZa,σ 0

x2 2

+

y2 2

, we get

≤ kδkZa,σ 0 ku1 kZ 0 a,σ 1

1

1

1

≤ kδkZ2 −1 k∆δkZ2 −1 ku1 kZ2 −1 k∆u1 kZ2 −1 a,σ



a,σ

a,σ

a,σ

2 ν kδkZa,σ −1 ku1 k −1 k∆u1 k −1 + k∆δkZa,σ −1 . Za,σ Za,σ ν 2

Similarly, ku2 δkZa,σ ≤ 0

2 ν kδkZa,σ k∆δkZa,σ −1 ku2 k −1 k∆u2 k −1 + −1 . Z Z a,σ a,σ ν 2

Then kδkZa,σ −1

≤ +

Z 2 t kδkZa,σ −1 ku1 k −1 k∆u1 k −1 dτ Za,σ Za,σ ν 0 Z t 2 kδkZa,σ −1 ku2 k −1 k∆u2 k −1 dτ. Za,σ Za,σ ν 0

1 Using Gronwall lemma and the fact (t 7→ ku1 kZa,σ −1 k∆u1 k −1 ) ∈ L ([0, T ]), (t 7→ ku2 k −1 k∆u2 k −1 ) ∈ Za,σ Za,σ Za,σ L1 ([0, T ]), we can deduce that δ = 0 in [0, T ] which gives the uniqueness. 

In the following, we prove a global existence if the initial condition is small in the Lei-Lin-Gevrey spaces.

12

J. BENAMEUR AND L. JLALI

−1 (R3 ) such that ku0 k Theorem 3.2. Let u0 ∈ Za,σ −1 < ν. Then, there exists a unique Za,σ 3 3 1 + 1 + −1 global solution u ∈ C(R , Za,σ (R )) ∩ L (R , Za,σ (R )) of (N SE) such that

ku(t)kZa,σ −1 + (

Z t ν − ku0 kZa,σ −1 0 k∆ukZa,σ ) −1 dτ ≤ ku k −1 . Za,σ 2 0

Proof theorem 3.2. −1 (R3 ), we have a local existence From theorem 1.4, if u0 ∈ Za,σ −1 3 1 1 3 u ∈ L∞ T (Za,σ (R )) ∩ LT (Za,σ (R )). ∗ −1 3 1 ([0, T ∗ ), Z 1 (R3 )) is the Assume that ku0 kZa,σ −1 < ν and u ∈ C([0, T ), Za,σ (R )) ∩ L a,σ loc maximal solution of (N SE). We have

∂t ku(t)kZa,σ −1 + νk∆uk −1 ≤ kdiv(u ⊗ u)k −1 . Za,σ Za,σ Integrating over (0, t) we get Z t ku(t)kZa,σ k∆ukZa,σ −1 + ν −1 dτ

0

≤ ku kZa,σ −1 +

0

≤ ku0 kZa,σ −1 +

(3.12)

Z

Z

t

ku ⊗ ukZa,σ 0 dτ

0 t 0

kukZa,σ −1 k∆uk −1 dτ. Za,σ ν+ku0 k

Z −1

a,σ Therefore, for T∗ = sup{t ∈ [0, T ∗ ) / ku(t)kZa,σ −1 < α}, where α = . 2 Take t ∈ [0, T∗ ). Then we have Z t Z t 0 k∆ukZa,σ k∆ukZa,σ ku(t)kZa,σ −1 dτ. −1 dτ ≤ ku k −1 + α −1 + ν Za,σ

0

0

This implies

ku(t)kZa,σ −1 + (ν − α)

Z

t

k∆ukZa,σ −1 dτ

0

≤ ku0 kZa,σ −1 < α.

Then T∗ = T ∗ . Particularly if T < T ∗ , we have Z T 0 k∆ukZa,σ ku(T )kZa,σ −1 dτ ≤ ku k −1 . −1 + (ν − α) Za,σ 0

Therefore,

T∗

=∞

 4. Global solution

In this section, we prove the first main theorem 1.4. ∗ ), Z −1 (R3 )) ∩ L1 ([0, T ∗ ), Z 1 (R3 )) be the maximal solution of (N SE), Let u ∈ C([0, Ta,σ a,σ a,σ a,σ loc such that ku0 kX −1 < ν. Therefore, we have Z t Z t 0 ku(t)kZa,σ k∆ukZa,σ ≤ ku kZa,σ kdiv(u ⊗ u)kZa,σ −1 + ν −1 dτ −1 + −1 dτ 0 0 Z t ≤ ku0 kZa,σ ku ⊗ ukZa,σ −1 + 0 dτ. 0

LONG TIME DECAY OF 3D-NSE IN LEI-LIN-GEVREY SPACES

13

2

2

Using the lemma 2.4 and the inequality xy ≤ x2 + y2 , thus we get Z t Z t 1 1 0 kukZ −1a kukZ2 −1 k∆ukZ2 −1 dτ k∆uk dτ ≤ ku k + c ku(t)kZa,σ + ν −1 −1 −1 Za,σ Za,σ 0

≤ ku0 kZa,σ −1 + c This implies that ν + 2

ku(t)kZa,σ −1

Z

t 0

Z

ku(t)kZa,σ −1 ≤ ku k −1 exp(c Za,σ

ku(t)kZa,σ −1 + ν

Z

t

0

√a ,σ σ

0

Z

t

Z

t 0

√a ,σ σ

kuk2Z −1 dτ ). √a ,σ σ

0

√a ,σ σ

0

≤ ku0 kZa,σ −1 (1 + c ≤ ku0 kZa,σ −1 exp(c

ν k∆ukZa,σ −1 )dτ. 2

kuk2Z −1 kukZa,σ −1 dτ.

≤ ku0 kZa,σ −1 Z t Z kuk2Z −1 ku0 kZa,σ + c −1 exp(c

k∆ukZa,σ −1 dτ

a,σ

(kuk2Z −1 kukZa,σ −1 +

k∆ukZa,σ −1 dτ ≤ ku k −1 + c Za,σ

0

Then

t

0

By the Gronwall lemma, we get

a,σ

√ ,σ σ

0

Z

Z

t

√a ,σ σ

0

0

kuk2Z −1

t

s 0

kuk2Z −1 )

exp(c

Z

√a ,σ σ

s 0

kuk2Z −1 )) √a ,σ σ

kuk2Z −1 )dτ. √a ,σ σ

R ∗ ∗ < ∞, by the previous inequality Ta,σ k∆uk Assumed that Ta,σ −1 dτ = ∞. This implies 0 Za,σ that Z Ta,σ ∗ kuk2Z −1 dτ = ∞. √a ,σ σ

0

−1 (R3 ) ֒→ Z −1 (R3 ). Then T ∗ = T ∗ As Za,σ a,σ √a √a ,σ

σ

σ

(4.1)

∗ ∗ a Ta,σ = T√

σ



,σ .

Thus

= ... = T ∗an ,σ , ∀n ∈ N. σ2

Therefore, from the dominated convergence theorem lim ku0 kZ −1a

n→∞

n ,σ σ2

Then, there exists n0 ∈ N such that

ku0 kZ −1a

n ,σ σ2

= ku0 kX −1 < ν.

< ν, ∀n ≥ n0 .

Applying theorem 1.5, so we have ∀n ≥ n0

(4.2)

u ∈ C(R+ , Z −1na

0 σ 2

,σ ).

∗ = T∗ = ∞. This is Using the inequalities (4.1)-(4.2) and for n = n0 , we obtain Ta,σ a n ,σ ∗ = ∞. absurd, so Ta,σ

σ2



14

J. BENAMEUR AND L. JLALI

5. Long time decay for the global solution In this section, we prove the second main theorem 1.5. −1 (R3 )). As Z −1 (R3 ) ֒→ X −1 (R3 ). Then u ∈ C(R+ , X −1 (R3 )). Let u ∈ C(R+ , Za,σ a,σ For the results of Hantaek Bae (see [15]). There exist t0 > 0 and α > 0 such that (5.1)

keα|D| u(t)kX −1 (R3 ) ≤ c0 , ∀t ≥ t0 ,

√ where t0 = ϕ(t) = t − t0 . Therefore, let a > 0 and β > 0. Then, there exists c1 > 0 such that (5.2)

1

ax σ ≤ c1 + βx, x ≥ 0.

Take β = α2 and using the inequalities (5.1)-(5.2) and the Cauchy-Schwartz inequality, so we obtain Z 1/σ ea|ξ| ku(t)kZa,σ = |b u(ξ)|dξ −1 |ξ| R3 Z ec1 +β|ξ| |b u(ξ)|dξ ≤ |ξ| R3 Z eβ|ξ| ≤ ec1 |b u(ξ)|dξ R3 |ξ| 1

≤ ec1 keα|D| u(t)kX −1 kukX2 −1 1

≤ c0 ec1 kukX2 −1 . Using theorem 1.3. So, limt→∞ ku(t)kZa,σ −1 = 0.



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15

[14] F.Planchon. Global strong solutions in Sobolev or Lebesgue spaces to the incompressible Navier Stokes equations in R3 ,Ann.Inst.H.Poincare Anal.Non Lineaire 13 (3)(1996) 319 336. [15] Hanteak. Bae. Existence and Analyticity of Lie-Lin Solution to the Navier Stokes equations. [16] Z.Zhang,Z.Yin. Global well-posedness for the generalized Navier Stokes system. arXiv:1306.3735v1 [Math.Ap] 17 june 2013. Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Kingdom of Saudi Arabia E-mail address: [email protected] Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Kingdom of Saudi Arabia E-mail address: [email protected]