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Energy methods for fractional Navier–Stokes equations
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Energy methods for fractional Navier–Stokes equationsR Yong Zhou a,b, Li Peng a,∗, Bashir Ahmad b, Ahmed Alsaedi b a
Faculty of Mathematics and Computational Science, Xiangtan University, Hunan 411105, PR China Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
b
a r t i c l e
i n f o
Article history: Received 5 December 2016 Revised 7 March 2017 Accepted 24 March 2017 Available online xxx MSC: 35R11 35Q30 76D03
a b s t r a c t In this paper we make use of energy methods to study the Navier–Stokes equations with time-fractional derivative. Such equations can be used to simulate anomalous diffusion in fractal media. In the first step, we construct a regularized equation by using a smoothing process to transform unbounded differential operators into bounded operators and then obtain the approximate solutions. The second part describes a procedure to take a limit in the approximation program to present a global solution to the objective equation. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Navier-Stokes equations Caputo fractional derivative Energy methods Approximate solutions
1. Introduction Navier–Stokes equations have been investigated by many researchers in view of their crucial role in fluid mechanics and turbulence problems. For more details, we refer the reader to the monographs by Ben-Artzi et al. [1] and Lemarié-Rieusset [14]. The topic of the global existence of weak, mild and strong solutions supplemented with small initial data received considerable attention. For example, Leray [16] carried out a pioneering study on the existence of global weak solutions in the energy space and the uniqueness of such solutions in R2 . Lemarié-Rieusset [15] established the existence of global mild solutions in different types of frameworks in Morrey–Campanato spaces. Later, Iwabuchi and Takada [8] discussed the same problem in function spaces of Besov type. A similar result was established by Lei and Lin [13] in the space X−1 . Similar results were obtained by Kato [9] in Ln (Rn ), Giga and Miyakawa [5], and Taylor [20] in Morrey spaces, Cannone −1+n/p [2] and Planchon [19] in the Besov spaces B p,∞ (Rn ), 1 < p < ∞. On the other hand, fractional calculus gained much popularity during the past decades mainly due to its extensive applications in widespread areas of science and engineering, such as fluid flow, rheology, dynamical processes and porous structures, diffusive
R ∗
Project supported by National Natural Science Foundation of China (11671339). Corresponding author. E-mail addresses: [email protected] (Y. Zhou), [email protected] (L. Peng).
transport akin to diffusion, control theory of dynamical systems, viscoelasticity and so on. For some recent contributions on the topic, we refer the reader to the monographs by Herrmann [6], Hilfer [7], Kilbas et al. [11] and Zhou [22,23], and a series of papers [12,21,24–30] and the references cited therein. Theoretical analysis and experimental data have shown that classical diffusion equation fails to describe diffusion phenomenon in heterogeneous porous media that exhibits fractal characteristics. Fractional calculus tools have been found effective in modelling anomalous diffusion processes as fractional-order operators can characterize the long memory processes. Consequently, it is reasonable and practical to propose the generalized Navier–Stokes equations in terms of Caputo time-fractional derivative operator, which can be used to simulate anomalous diffusion in fractal media. Such models are found to be of great interest for both physicists and pure mathematicians as they exhibit much more complicated phenomena and are more challenging than their corresponding integer-order counterparts. In this paper we consider the following Navier–Stokes equations with time-fractional derivative in R3 :
⎧ α ⎨∂t u + (u · ∇ )u = −∇ p + ν u, ∇ · u = 0, ⎩ u ( 0, x ) = u0 ,
t > 0, (1.1)
where ∂tα is the Caputo fractional derivative of order α ∈ (0, 1), u = (u1 (t, x ), u2 (t, x ), u3 (t, x )) represents the velocity field at a point
http://dx.doi.org/10.1016/j.chaos.2017.03.053 0960-0779/© 2017 Elsevier Ltd. All rights reserved.
Please cite this article as: Y. Zhou et al., Energy methods for fractional Navier–Stokes equations, Chaos, Solitons and Fractals (2017), http://dx.doi.org/10.1016/j.chaos.2017.03.053
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x ∈ R3 and time t > 0, p = p(t, x ) is the pressure, ν is the viscosity and u0 = u0 (x ) is the initial velocity. Recently, Eq. (1.1) attracted significant attention due to its importance in simulating anomalous diffusion in fractal media. For some recent works on analytic and weak solutions of timefractional Navier–Stokes equations, for example, see El-Shahed et al. [4], Momani and Zaid [18] and Zhou and Peng [28]. Another interesting aspect of the study of existence of global mild solutions includes the situation when norms of the initial values are supposed to be small enough, for instance, see Carvalho–Neto and Gabriela [3], Zhou and Peng [27]. It is worth-mentioning that the purpose of this paper is to develop a new global result when we relax the smallness condition on the initial data. The paper is organized as follows. In Section 2 we recall some notations, definitions, and preliminary facts. In Section 3 we construct a regularized equation, and show the local existence of its solutions. Section 4 deals with the global existence and continuation of solutions for a kind of fractional differential equations in a Banach space. Via energy methods, we also obtain the convergence of approximate solutions and the global existence and uniqueness of mild solutions of Eq. (3.1) in different spaces. 2. Preliminaries Here we recall some notations, definitions, and preliminary facts which are used throughout this paper. Denote by · the L2 norm on R3 , where
u =
R3
|u(x )|2 dx
12
12
um =
Dβ u 2
s
R3
It is not difficult to show that if u ∈ H1 , then u ∈ X−1 . Let X be a Banach space and v : [0, ∞ ) → X. The fractional integral of order α ∈ (0, 1] for the function v is defined as
( |x| ) ∈
2 (1 + |ξ |2 ) | u ( ξ )| d ξ
12
( R ), 0 ≤ ≤ 1,
3
.
R3
(x )dx = 1.
We define the mollification Jε u of the function u ∈ Lq (R3 ) (1 ≤ q ≤ ∞ ) as
(Jε u )(x ) = ε −3
R3
x − y ε
u(y )dy.
(α )
t 0
(t − s )α−1 v(s )ds, t > 0.
Further, CDtα v represents the Caputo fractional derivative of order α for the function v and is defined by C α Dt
v(t ) =
d 1 −α I (v(t ) − v(0 ) ) , t > 0. dt t
For u : [0, ∞ ) × R3 → R3 , Caputo time-fractional derivative of the function u can generally be written as
∂tα u(t, x ) =
∂ 1 −α I (u(t, x ) − u(0, x ) ) , t > 0. ∂t t
For more insight into the topic, see Kilbas et al. [11]. Before proceeding further, we present two important results which play a key role in proving the main results. Lemma 2.1. [10] Let T > 0. Then, for any v ∈ L2 ((0, T ] × R3 , R3 ),
R3
v(t, x )∂tα v(t, x )dx ≥ v∂tα v.
Lemma 2.2. Let the function v : (0, T ] × R3 → C3 and ∂tα v exist. Then
v(t )∂tα v(t ) + v(t )∂tα v(t ) ≥ 2|v|∂tα |v|,
v(t )∂tα v(t ) + v(t )∂tα v(t ) = 2(a(t )∂tα a(t ) + b(t )∂tα b(t ) ) = 2 ∂tα a2 (t ) + ∂tα b2 (t ) + t −α a2 (t ) + b2 (t )
V s := {u ∈ H s (R3 ) : ∇ · u = 0} and call the operator P : H s (R3 ) → V s the Leray projector. Clearly, P commutes with Jε . For more details, we refer the reader to Majda and Bertozzi [17]. Let us denote an important space: for m ∈ Z,
Xm = u ∈ D ( R3 ) :
R3
+
t 0
+
t 0
|a(t ) − a(t − s )|2 [α s−α−1 ]ds
|b(t ) − b(t − s )|2 [α s−α−1 ]ds
t 2 −α −1 ≥ 2 ∂tα |v(t )|2 +t −α |v(t )|2 + | v ( t ) | − | v ( t −s ) | [ α s ] ds ( ) 0
= 2|v(t )|∂tα |v(t )|. 3. Local existence
We introduce the space
1
Itα v(t ) =
Here u represents the Fourier transform of u and S (R3 ) denotes the Schwarz space of rapidly decreasing smooth functions. It is clear that the two norms are equivalent for s = m. Let ϱ be the standard mollifier satisfying
C0∞
|ξ |m | u| d ξ .
Proof. Let v(t ) = a(t ) + b(t )i. Then
For s ∈ R, the Sobolev space H s (R3 ) is the completion of S (R3 ) with respect to the norm
R3
where v(t ) denotes the dual of v(t ).
.
0≤|β|≤m
us =
uXm =
The following result is a generalization of Lemma 6.1 in Kemppainen et al. [10].
.
By H m (R3 ), m ∈ Z ∪ {0}, we denote the Sobolev space consisting of functions u ∈ L2 (R3 ) such that Dβ u ∈ L2 (R3 ), 0 ≤ |β | ≤ m with the norm ·m defined by
Here D (R3 ) stands for the space of distributions. The norm of Xm is given by
|ξ |m | u| d ξ < ∞ .
This section is concerned with the construction of an approximate (regularized) equation for the time-fractional Navier–Stokes equations and obtain the existence and some properties of its solution. To do this, we use the mollifier Jε to regularize Eq. (1.1):
⎧ α ε ε ε ε ε ⎨∂t u + Jε [(Jε u ) · ∇ (Jε u )] = −∇ p + ν Jε (Jε u ), ∇ · uε = 0, ⎩ ε u ( 0 ) = Jε u0 .
(3.1)
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By applying the Leray projector P to Eq. (3.1), we get rid of the pressure term in Eq. (3.1), which reduces to an ordinary differential equation (ODE) in Vs :
∂tα uε + PJε [(Jε uε ) · ∇ (Jε uε )] = ν Jε2 uε , uε ( 0 ) = Jε u0 .
(3.2)
3
Since ν ≥ 0, we have
∂tα uε ≤ 0. Then the definition of ∂tα u implies that
sup t∈[0,T ]
u ε ≤ J ε u 0 ≤ u 0 .
We set
Fε (uε ) = ν Jε2 uε − P Jε [(Jε uε ) · ∇ (Jε uε )]. From the argument of Proposition 3.6 in Majda and Bertozzi [17], we know that
Fε (u1 ) − Fε (u2 )m ≤ L(u j , ε , M )u1 − u2 m for u j ∈ BM , j = 1, 2,
(3.3)
where m ∈ Z+ ∪ {0} and
BM = {u ∈ V
m
: um < M }.
∂tα uε m ≤ cm |Jε ∇ uε |L∞ uε m .
C α ([0, T ], V m ) = {u ∈ C ([0, T ], V m ) : ∂tα u ∈ C ([0, T ], V m )}.
Dβ ∂tα uε , Dβ uε = Dβ Jε2 uε , Dβ uε
− D β P J ε [ ( J ε u ε ) · ∇ ( J ε u ε )] , D β u ε
= νJε Dβ ∇ uε 2 − P Jε [(Jε uε ) · ∇ (Dβ Jε uε )], Dβ uε
Theorem 3.1. Assume the initial data u0 ∈ V0 . Then, for any ε > 0, (i) there exists Tε = T (Jε u0 m , ε ) such that Eq. (3.2) has a unique solution uε ∈ Cα ([0, Tε ), Vm ); (ii) uε ∈ Cα ([0, T], V0 ) on any interval [0, T] with
sup
Lemma 3.1. The regularized solution uε of Eq. (3.2) satisfies the inequality:
Proof. We take the derivative Dβ of Eq. (3.2) and L2 inner product with Dβ uε to obtain
It means that Fε is locally Lipschitz continuous on any open set. Let
t∈[0,T ]
In the next lemma, we derive a key estimate which plays an important role in proving the main results.
uε ≤ u0 .
(3.4)
− Dβ P Jε [(Jε uε ) · ∇ (Jε uε )] −P Jε [(Jε uε ) · ∇ (Dβ Jε uε )], Dβ uε . From the discussion of Majda and Bertozzi [17] and Lemma 2.1, we deduce that
uε m ∂tα uε m + νJε ∇ uε 2m ≤ cm |Jε ∇ uε |L∞ uε 2m , which takes the following form for ν > 0:
∂tα uε m ≤ cm |Jε ∇ uε |L∞ uε m .
Proof. (i) If u0 ∈ V0 , then Jε u0 ∈ V m , m ∈ Z+ ∪ {0}. For given r > 0, define
B(r, Tε ) =
uε ∈ C ([0, Tε ), V m ) : sup
t∈[0,Tε )
uε (t ) − Jε u0 m ≤ r .
Notice that
M=
sup
(t,uε )∈[0,Tε )×B(1,Tε )
Fε (uε )m < +∞.
(3.5)
Indeed,
Fε (uε )m ≤ Fε (uε ) − Fε (Jε u0 )m + Fε (Jε u0 )m ≤ Luε − Jε u0 m + Fε (Jε u0 )m ≤ Lr + Fε (Jε u0 )m < +∞. Consider the operator T :
T uε (t ) = Jε u0 +
0
t
(t − s )α−1 Fε (uε (s ))ds.
Obviously T (B(r, Tε )) ⊂ B(r, Tε ). Moreover, for u1 , u2 ∈ B(r, Tε ),
T u1 (t ) − T u2 (t )m ≤ 1
Lt α
α
sup s∈[0,t]
u1 (s ) − u2 (s )m for t ∈ [0, Tε ).
1
Fixing Tε =min{( αMr ) α ,( αL ) α }, it is easy to show that T is a strict contraction mapping on B(r, Tε ). In consequence, we deduce that T has a fixed point. On account of (3.3) and uε ∈ C([0, Tε ), Vm ), it is clear that ∂tα u ∈ C ([0, Tε ), V m ). (ii) Take the L2 inner product of Eq. (3.2) with uε and apply Lemma 2.1 to obtain
uε ∂tα uε ≤ ν
R3
uε Jε2 uε dx −
R3
uε P Jε [(Jε uε ) · ∇ (Jε uε )]dx.
Integrating by parts and using the fact that ∇ · uε = 0, we get
uε ∂tα uε + νJε ∇ uε 2 ≤ 0.
4. Global existence Here we discuss the global existence of solutions by means of mathematical analysis and previous estimate with the initial data being less than the viscosity. Firstly, we consider the following autonomous equation C α Dt z
(t ) = F (z ), z(0 ) = z0 .
(4.1)
Lemma 4.1. Let X be a Banach space and U ⊂ X be an open set. Suppose that the mapping F: U → X satisfies the locally Lipschitz condition, that is, for any z ∈ U there is a constant L > 0 and an open neighborhood Uz ⊂ U of z such that
|F (z1 ) − F (z2 )|X ≤ L|z1 − z2 |X for all z1 , z2 ∈ Uz . Then for any z0 ∈ U, there exists a time T such that Eq. (4.1) has a unique (local) solution z ∈ C([0, T), U). Proof. The method for showing the local existence of solutions for Eq. (4.1) is similar to that of Theorem 3.1 (i), so we omit it. Now we establish the global existence of solutions for a kind of fractional differential equations in Banach space, which furnishes as the key instrument in the proof of the global existence of solutions for the time fractional Navier–Stokes equations. Lemma 4.2. Under the conditions of Lemma 4.1, the unique solution z = z(t ) (t ∈ [0, T )) of the fractional differential equation (4.1) either exists globally in time, or T < ∞ and z(t) preserves the open set U as t →T − . Proof. Suppose that the maximum existing interval for the solution z(t) is [0, T), that is,
T = sup{Tz : z(t ) is defined on [0, Tz ] and z(t ) is a solution of Eq. (4.1 ) on [0, Tz ]}.
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Then T = ∞ or T < ∞. If T = ∞, the conclusion holds. If T < ∞, we show that z(t) preserves the open set U as t → T − . Observe that there exists a sequence {tk } and a positive constant K > 0 such that
tn ≤ tn+1 for n ∈ N,
lim tn = T ,
n→∞
|z(tn )| ≤ K.
z˜(t ) ∈ Br (ϕ¯ , T + h ), that is,
z˜(t ) = ϕ¯ (t ) +
(4.2)
= z0 +
0
with
ε ≤ |z(ηn ) − z∗ | ≤ |z(tn ) − z∗ | + |z(ηn ) − z(tn )| tn ε 1 ≤ + (tn − s )α−1 − (ηn − s )α−1 |F (z(s ))|ds 2
(α ) 0
1 L∞ α ( R+ , X ) =
≤
ε 2
1
ηn
(α )
tn
+
1 + sup |F (z(s ))|(ηn − tn )α
(1 + α ) s∈[tn ,ηn ]
≤
2
Thus limt→T − z(t ) exists. Next we show that z∗ ∈ ∂ U. On the contrary, we assume that z∗ ∈ ∂ U. Since z(t) ∈ U (t ∈ [0, T)), therefore, z∗ ∈ U. Let z(t) be equal to z∗ for t = T and itself for 0 ≤ t < T. Denote
ϕ¯ (t ) = z0 +
0
(t − s )α−1 F (z(s ))ds, t ∈ [T , T1 ],
Sy(t ) = ϕ¯ (t ) +
t T
(t − s )α−1 F (y(s ))ds, t ∈ [T , T1 ],
where y ∈ C([T, T1 ], U). Let
Br =
t
0
Theorem 4.1. For u0 ∈ X−1 satisfying
u0 X−1 < ν,
(4.3)
u(t )X−1 + (ν − u0 X−1 )
t 0
(t − s )α−1 ∇ u(s )L∞ ds
≤ u0 X−1 .
pε =
Ri R j ui u j = (R R )(uε uε ),
i, j=1
where R is the Riesz operator. Firstly we have the estimate
uε (t )X−1 =
(t, y ) : t ∈ [T , T1 ], |y(t )| ≤ sup |ϕ¯ (t )| + r .
|ξ |≤1
|ξ |−1 |u ε (t )|dξ
+
with ϕ¯ ∈ C ([T , T1 ], U ), and define the operator S as follow:
(t − s )α−1 ε ∞ × u (s )X1 ds ∈ L ([0, ∞ ), R ) . u : [0, ∞ ) → X1 :
Proof. For u0 ∈ X−1 , we have Jε u0 X−1 ≤ u0 X−1 by the properties of ϱ. By the arguments similar to the ones employed in Theorem 3.1, there exists a unique locally continuous solution uε (t, x) on the interval [0, Tε ). The associated pressure pε is given by
for sufficiently large n ≥ n0 .
T
Let us introduce
0≤t<∞
(2(ηn − tn )α + tnα − ηnα )
t ∈ [T , T + h].
1 sup |F (z(s ))|(2(ηn − tn )α + tnα − ηnα )
(1 + α ) s∈[0,ηn ] M
t ∈ [0, T ),
z˜(t ),
Evidently z¯ ∈ C ([0, T + h], U ). Thus we find that z¯ (t ) is the solution of (4.1) on the interval [0, T + h], which contradicts the assumption that [0, T) is the maximum existing interval.
sup
which is a contradiction in view of
(1 + α ) ε
z(t ),
Eq. (1.1) has a unique solution u existing globally in time. Moreover, 1 u ∈ C ([0, ∞ ), X−1 ) ∩ L∞ α (R+ , X ) and the following estimate holds:
≤ ε,
≤
(t − s )α−1 F (z˜(s ))ds
(t − s )α−1 F (z¯ (s ))ds, t ∈ [T , T + h],
z¯ (t ) =
(ηn − s )α−1 |F (z(s ))|ds
1 sup |F (z(s ))|( (ηn − tn )α + tnα − ηnα )
(1 + α ) s∈[0,tn ]
t T
t
According to the equicontinuity of z(t) and Lemma 4.1, {z(tn )} has a convergent subsequence. Without loss of generality, let limn→∞ z(tn ) = z∗ . This together with (4.2) implies that for sufficiently small τ > 0, there exists n0 such that T − τ < tn0 < T and for n ≥ n0 , we have |z(tn ) − z∗ | ≤ 2ε . We show that limt→T − z(t ) = z∗ . On the contrary, for n ≥ n0 , there exists ηn ∈ (tn , T) such that |z(ηn ) − z∗ | ≥ ε and |z(t ) − z∗ | < ε, for t ∈ (tn , ηn ). By the continuity of F on [0, T), we denote M = sups∈[0,ηn ] |F (z(s ))|. Thus,
+
|ξ |≥1
|ξ |−2 | −uε (t )|dξ ≤ C uε (t )1 .
(4.4)
This implies that uε ∈ L∞ ([0, Tε ), X−1 ). Next, by applying the Fourier transform to (3.1), we obtain
α ε ε ∂t u − i u (η ) u ε (ξ − η )dη · ξ − iξ p ε + ν|ξ |2 u ε = 0, ξ · u ε = 0. (4.5)
t∈[T,T1 ]
On account of the continuity of F on Br , let M = supy∈Br |F (y(s ))|. Consider
Br (ϕ¯ , T + h ) =
y ∈ C ( [T , T + h], U ) :
sup t∈[T,T +h]
|y(t ) − ϕ¯ (t )| ≤ r,
y(T ) = ϕ¯ (T ) , α )r α1 where h = {T1 − T , ( (1+ ) }. M
As argued in Lemma 4.1, we can derive that S is a strict contraction mapping on Br (ϕ¯ , T + h ). Therefore, S has a fixed point
From (4.5), one infers that
u ε · ∂tα u ε − i and
u ε · ∂tα u ε + i
u ε (ξ ) · u ε (η ) u ε (ξ − η )dη · ξ + ν|ξ |2 |u ε |2 = 0,
u ε (ξ ) · u ε (η ) u ε (ξ − η )dη · ξ + ν|ξ |2 |u ε |2 = 0.
Thus it follows that
u ε · ∂tα u ε + u ε · ∂tα u ε + 2ν|ξ |2 |u ε |2 =i u ε (ξ ) · u ε (η ) u ε (ξ − η ) − u ε (ξ ) · u ε (η ) u ε (ξ − η ) dη · ξ .
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Using Lemma 2.2, multiplying by |ξ |−1 |u ε |−1 and integrating with respect to ξ , we get
∂tα
|ξ |−1 |u ε |dξ + ν |ξ ||u ε |dξ i ≤ u ε (η ) · |u ε |−1 u ε (ξ ) u ε (ξ − η ) 2 − u ε (η ) · |u ε |−1 u ε (ξ ) u ε (ξ − η ) · |ξ |−1 ξ dηdξ
≤ ≤
1 2
≤
1 ε uε u weakly in L∞ α (R+ , X ), u u weakly in L∞ (R+ , X−1 ), as ε → 0. ∗
uε (0 ) − u0 X−1 → 0, as ε → 0.
|ξ ||u ε (ξ )|dξ .
∂tα
(4.6)
which, according to the definition of ∂tα , yields The above procedure can similarly be applied to each small interval [δ , 2δ ], [2δ , 3δ ], ... . This also ensures that uε (t )X−1 ≤ u0 X−1 < ν for each t ∈ [0, Tε ). Integrating (4.6) from 0 to t, we get t
(t −s )α−1 uε (s )X−1 ds ≤ u0 X−1
for t ∈ [0, Tε ).
∂tα
t 0
(t − s )α−1 ∇ uε (s )L∞ ds ≤
t 0
(t − s )α−1 uε (s )X1 ds
uε (t )m ≤ uε (0 )m +cm ≤ J ε u 0 m + c m
t
(t −s )α−1 |Jε ∇ uε (s )|L∞ uε (s )m ds
0 t
0
uε (t )
m
≤ Jε u0 m exp cm
≤ Jε u0 m exp
0
t
0≤t<∞
1 (uε1 X1 + uε2 X1 )uε1 − uε2 X−1 . 2
(ν − u0 X−1 )
0
t
(t − s )α−1 uε1 − uε2 X1 ds
t
0
× (uε1 X1 + uε2 X1 )ds ≤ uε1 (0 ) − uε2 (0 )X−1 (4.7)
t uε (t )X−1 + (ν − u0 X−1 ) (t − s )α−1 uε (s )X1 ds 0
(4.8)
Estimate (4.8) implies that there exists a subsequence of {uε }, relabeled as {uε }, and 1 u ∈ L∞ (R+ , X−1 ) ∩ L∞ α ( R+ , X )
1 2
(t − s )α−1 (uε1 X1 + uε2 X1 )uε1 − uε2 X−1 ds t 1 ε1 u0 X−1 ε 2 ≤ u (0 ) − u (0 )X−1 exp (t − s )α−1 2 ν − u0 X−1 0 ≤
≤ u0 X−1 .
|ξ ||uε1 − uε2 |dξ
∂tα uε1 − uε2 X−1 + (ν − u0 X−1 )uε1 − uε2 X1
for all t ∈ [0, Tε ). It follows from Lemma 4.2 that Tε = ∞. Moreover, we have the following uniform estimate on uε :
sup
that is,
(t − s )α−1 |∇ uε (s )|L∞ ds
cm u0 X−1 ν − u0 X−1
|ξ |−1 |uε1 − uε2 |dξ + (ν − u0 X−1 )
1 (uε1 X1 + uε2 X1 )uε1 − uε2 X−1 , 2
≤
(t − s )α−1 |∇ uε (s )|L∞ uε (s )m ds.
Using Gronwall’s inequality again, we get
|η|−1 |ξ − η| + |η||ξ − η|−1 |uε1 (η )| + |uε2 (η )|
and
On the other hand, in view of Lemma 3.1, we obtain
uε1 (t ) − uε2 (t )X−1 ≤ uε1 (0 ) − uε2 (0 )X−1 t 1 α −1 ε ε 1 2 exp (t − s ) (u X1 + u X1 )dξ 2 0 u0 X−1 ε ε 1 2 ≤ u (0 ) − u (0 )X−1 exp , ν − u0 X−1
u0 X−1 . ν − u0 X−1
≤
Using Gronwall’s inequality again and noting that ν − u0 X−1 > 0, we get
|ξ ||u ε (ξ )|dξ ,
shows that
≤
∇ uε (t )L∞ = exp(ix · ξ )∇ uε ( ξ )d ξ ≤ |∇ uε ( ξ )|d ξ
1 2
× |u ε1 ( ξ − η ) − u ε2 ( ξ − η )|d η d ξ 1 ≤ (uε1 X−1 + uε2 X−1 )uε1 − uε2 X1 2 1 + (uε1 X1 + uε2 X1 )uε1 − uε2 X−1 . 2
This, together with
=
Combining the above inequality and (4.8), we obtain
uε (t )X−1 ≤ J ε u0 X−1 ≤ u0 X−1 < ν on [0, δ ].
0
≤
∂tα uε (t )X−1 ≤ 0 on [0, δ ],
(4.10)
|ξ |−1 |uε1 − uε2 |dξ + ν |ξ ||uε1 − uε2 |dξ ≤ |u ε1 ( η )| + |u ε2 ( η )| |u ε1 ( ξ − η ) − u ε2 ( ξ − η )|d η d ξ
From (4.3) and (4.4), we see that there exists a small enough δ with δ ∈ (0, Tε ) such that uε (t )X−1 < ν for t ∈ [0, δ ], since uε (t) is local smooth solution on [0, Tε ), that is, uε ∈ C([0, Tε ), Vm ). Therefore, we have
uε (t )X−1 + (ν − u0 X−1 )
(4.9)
In order to prove the strong convergence of uε , we proceed with the same argument as in (4.6) to have the estimate:
|η|−1 |ξ − η| + |η||ξ − η|−1 |u ε (η )||u ε (ξ − η )|dηdξ
|ξ |−1 |u ε (ξ )|dξ
such that
For the relation between uε (0) and u0 , we notice that
|u ε (η )||u ε (ξ − η )|dηdξ
5
u0 X−1 u0 X−1 exp . ν − u0 X−1 ν − u0 X−1
Thus we conclude that {uε } is a Cauchy sequence in L∞ (R+ , X−1 ) ∩ 1 L∞ α (R+ , X ) by (4.10), and that the convergence in (4.9) is a strong one. In practice, the above estimate also ensures that the solution 1 in the space L∞ (R+ , X−1 ) ∩ L∞ α (R+ , X ) is unique under the condition (4.3). To establish continuity of u(t, x), we return to Eq. (4.5). It is obvious that
t 0
=
(t − s )α−1 uε (s )X−1 ds =
t 0
(t − s )α−1
t 0
(t −s )α−1
|ξ ||u ε (ξ )|dξ ds =
0
t
|ξ |−1 | uε ( ξ )|d ξ d s
(t − s )α−1 uε X1 ds.
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In addition, like the estimate (4.6), we can obtain
t 0
(t − s )α−1 ∇ · (uε (s ) uε (s ))X−1 ds ≤
t
t 0
The following results deal with the global existence of approximate solutions and mild solutions, respectively.
(t − s )α−1
Theorem 4.2. Let 0 < T < ∞. If u0 ∈ Vm , m ∈ Z+ ∪ {0}, then there exists a unique solution uε ∈ Cα ([0, T], Vm ) for Eq. (3.2) for any ε > 0.
|u ε (η )||u ε (ξ − η )|dξ dηds
× ≤
(t − s )α−1 uε X−1 uε X1 ds
0
≤ sup
0≤t<∞
uε (t )X−1
t 0
Proof. We show an a priori bound on uε m . To do this, we note that the relation (3.3) with u2 (t, x) ≡ 0 and (3.4) yields
(t − s )α−1 uε X1 ds.
∂tα uε m ≤ L(uε , ε , M )uε m ≤ L(u0 , ε , M )uε m .
The pressures term can be handled in a similar manner, that is, for ξ R uε (ξ ) = −i j · u ε (ξ ), we have |ξ |
j
t 0
(t − s )α−1 ∇ pε X−1 ds ≤
t 0
(t − s )α−1
ε (η )||Ru ε (ξ − η )|dξ dηds ≤ |Ru
×
t 0
≤ sup
0≤t<∞
uε (t )X−1
t 0
Tα uε (t )m ≤ u0 m exp L(u0 , ε , M ) , α which shows a priori bound on uε m . By Lemma 4.2, the regularized Eq. (3.2) has a unique solution uε ∈ Cα ([0, T], Vm ).
(t − s )α−1
Theorem 4.3. Assume that u0 ∈ Vm , m ≥ 3. Then the following results hold:
|u ε (η )||u ε (ξ − η )|dξ dηds
×
Thanks to Gronwall’s inequality, we have
(i) If u0 1 < Cν and 0 < T < ∞, there exists a unique solution
(t − s )α−1 uε X1 ds,
u ∈ C ([0, T ], C 2 (R3 )) ∩ C α ([0, T ], C (R3 ))
Observe that
to the time-fractional Navier–Stokes equations. Moreover, u is the limit of a subsequence of {uε }, where uε are the approximate solutions given by Theorem 4.2; (ii) The following estimates hold true:
uε (t )X−1 − uε (t0 )X−1 t0 ≤ (t0 − s )α−1 − (t − s )α−1 (uε (s )X−1 0
+
∇ · (uε (s ) uε (s ))X−1 + ∇ pε X−1 )ds
+
t
t0
sup t∈[0,T ]
(t − s )α−1 (uε (s )X−1 + ∇ · (uε (s ) uε (s ))X−1
sup t∈[0,T ]
+ ∇ pε X−1 )ds for 0 < t0 ≤ t. Now we estimate each term in the above expression. For I1 (t), notice that t0 0
(t − s )α−1 − (t0 − s )α−1
+
∇
≤2
t0
0
+ ≤2
· (uε (s ) uε (s ))
( u ε ( s )
X−1
X−1
+ ∇ pε
X−1
0
t0
< ∞,
uε (t )
uε (s )X−1
0
t0
lim I1 (t ) =
t−t0 →0
t0
lim
0
+
t−t0 →0
≤ u0 m exp ≤ u0 m exp
(t0 − s )α−1 uε (s )X1 ds
(t0 − s )α−1 − (t − s )
α −1
(uε (s )X−1
∇ · (uε (s ) uε (s ))X−1 + ∇ pε X−1 )ds
= 0.
cm u0 X−1 ν − u0 X−1 cm C u 0 1 ν − C u0 1
.
(4.11)
In consequence, {uε } is a uniformly bounded set of C ([0, T ], H m (R3 )). Next we prove that {uε } constitutes a Cauchy sequence in C ([0, T ], L2 (R3 )). Using Eq. (3.2), we have
uε − uε ∂tα uε − uε ≤ ν Jε2 uε − Jε2 uε , uε − uε − P Jε [(Jε uε ) · ∇ (Jε uε )]
− P J ε [ ( J ε u ε ) · ∇ ( J ε u ε )] , u ε − u ε . Using the argument of [17, Lemma 3.7], we obtain
For I2 (t), by an elementary calculation and the fact that uε ∈ 1 L∞ α (R+ , X ), we obtain the following inequality
I2 (t ) ≤
m
due to the estimate (4.8). Applying the Lebesgue’s dominated convergence theorem, we get
ν −C u0 1 ;
Proof. In the first step, we show that {uε (t)} is a uniformly bounded set of H m (R3 ) independent of ε . By Lemma 3.1, we have
(t0 − s )α−1 uε (s )X1 ds
0≤s
cm C u0 1
Eq. (4.7) and u0 X−1 ≤ C u0 1 ensure that
(t0 − s )α−1 (uε (s )X−1 + ∇ · (uε (s ) uε (s ))X−1
+ 4 sup
ν −C u0 1 ,
∂tα uε m ≤ cm |Jε ∇ uε |L∞ uε m .
)ds
∇ pε X−1 )ds
u(t )m ≤ u0 m exp
cm C u0 1
(iii) uε and u are uniformly bounded in L∞ ([0, T ], H m (R3 )) ∩ Cw ([0, T ], H m (R3 )).
=: I1 (t ) + I2 (t )
uε (t )m ≤ u0 m exp
1 + 2 sup
t0 ≤s
uε (s ))X−1
t
t0
Jε2 − Jε2 uε , uε − uε
− J ε ∇ ( u ε − u ε ) 2
≤ C1 max{ε , ε }uε 3 uε − uε
(t − s )α−1 uε (s )X1 ds and
→ 0, as t − t0 → 0. From the foregoing arguments, it follows that u ∈ 1 L∞ α (R+ , X ). This completes the proof.
Jε2 uε − Jε2 uε , uε − uε ≤
C ([0, ∞ ), X−1 )
∩
P J ε [ ( J ε u ε ) · ∇ ( J ε u ε )] − P J ε [ ( J ε u ε ) · ∇ ( J ε u ε )] , u ε − u ε
= (Jε − Jε )[(Jε uε ) · ∇ (Jε uε )], uε − uε
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Jε [(Jε − Jε )uε · ∇ (Jε uε )], uε − uε
+ J ε [ J ε ( u ε − u ε ) · ∇ ( J ε u ε )] , u ε − u ε
+ Jε {Jε uε · ∇ [(Jε − Jε )uε ]}, uε − uε
+ J ε {J ε u ε · ∇ [ J ε ( u ε − u ε ) ] } , u ε − u ε
Thus
+
≤ C1 max{ε , ε }uε 2m uε − uε
sup t∈[0,T ]
Moreover, for any given t ∈ [0, T], the sequence {uε (t, ·)} is a uniformly bounded set of H m (R3 ); hence there is also a subsequence of {uε (t, ·)}, relabeled as {uε (t, ·)} such that
which imply that
∂tα uε − uε ≤ C1 (M ) max{ε , ε } + uε − uε , where M = sup{uε m } from (4.11). Integrating the above inequalε ity and then using Gronwall’s inequality yields
Tα
uε (t ) − uε (t ) ≤ uε (0 ) − uε (0 ) + C1 (M ) max{ε , ε } α t × exp C1 (M ) (t − s )α−1 ds ≤
0
uε → u in u ∈ C ([0, T ], H m (R3 )),
uε (0 ) − uε (0 ) + C1 (M ) max{ε , ε } α Tα × exp C1 (M ) . α
which implies that
[φ , uε (t, · )] → [φ , u(t, · )] uniformly for t ∈ [0, T ],
Tα
uε (t ) −uε (t ) ≤ uε (0 ) −uε (0 ) +C1 (M ) max{ε , ε } α t∈[0,T ] α T × exp C1 (M ) , α sup
which, in view of (4.10), implies that {uε } is a Cauchy sequence in C ([0, T ], L2 (R3 )). Hence there exists u ∈ C ([0, T ], L2 (R3 )) such that the sequence {uε } converges strongly to u in C ([0, T ], L2 (R3 )), that is,
sup
uε (t ) − u(t ) ≤ C2 ε .
(4.12)
Finally we verify that u ∈ C ([0, T ], C 2 (R3 )) ∩ C α ([0, T ], C (R3 )). By the inequalities (4.11) and (4.12), we get
m
{uε (t ) − uε (t )m } ≤ sup uε (t ) − u(t )1− m t∈[0,T ] t∈[0,T ]
m
ε ε m × u (t ) − u (t )m sup
m
≤ C ( u 0 m ) ε 1− m
for 0 < m < m,
(4.13) m
which implies strong convergence in C ([0, T ], H (R3 )) for all m
< m. With 0 < 72 < m < m, we also have strong convergence in C ([0, T ], C 2 (R3 )). Moreover, from the equation
∂tα uε = ν Jε2 uε − PJε [(Jε uε ) · ∇ (Jε uε )], we deduce that ∂tα uε converges C ([0, T ], C (R3 )). As uε → u, we have
T 0
∂tα uε (t )φ (t )dt =
0
→
T
0
=
T
0
T
to
ν u − P ( u · ∇ u )
Hence we deduce that u(t)m is bounded for each t, which yields
For φ ∈ H −m (R3 ), Denote by [φ , u] the dual pairing of H −m (R3 ) and H m (R3 ). From (4.13), we have
Thus we have
t∈[0,T ]
uε (t, · ) u(t, · ) in H m (R3 ).
u ∈ L∞ ([0, T ], H m (R3 )).
Tα
(4.14)
uε u in L2 ([0, T ], H m (R3 )).
ε
uε (t )m ≤ M.
Accordingly, uε is uniformly bounded in L2 ([0, T ], H m (R3 )) and hence there exists a subsequence, relabeled as {uε }, and u ∈ L2 ([0, T ], H m (R3 )) such that
+ C1 max{ε , ε }uε m uε m × uε − u + C1 uε − u 2 uε m
+ C1 max{ε , ε }uε 2m uε − uε , ε
7
in
(uε (t ) − uε (0 ) )Ct DαT φ (t )dt (u(t ) − u(0 ) )Ct DαT φ (t )dt as ε → 0
∂tα u(t )φ (t )dt for φ ∈ C0∞ ([0, T ], C (R3 ).
Hence the distribution limit of ∂tα uε must be ∂tα u. (iii) Recalling that for any bounded sequence {vε } in H m (R3 ), there exists a subsequence converging weakly to a limit in H m (R3 ), that is, vε v.
for any φ ∈ H −m (R3 ). Applying (4.14) and the conclusion that for
m < m, H −m (R3 ) is dense in H −m (R3 ), we obtain
[φ , uε (t, · )] → [φ , u(t, · )] uniformly for t ∈ [0, T ], for any φ ∈ H −m (R3 ). Thus u ∈ Cw ([0, T ], H m (R3 )). This completes the proof. References [1] Ben-Artzi M, Croisille JP, Fishelov D. Navier-Stokes equations in planar domains. World Scientific; 2013. [2] Cannone M. A generalization of a theorem by kato on navier-stokes equations. Rev Mat Iberoam 1997;13:515–41. [3] De Carvalho-Neto PM, Gabriela P. Mild solutions to the time fractional Navier-Stokes equations in RN . J Differ Equ 2015;259:2948–80. [4] El-Shahed M, Salem A. On the generalized navier-stokes equations. Appl Math Comput 2004;156(1):287–93. [5] Giga Y, Miyakawa T. Navier-stokes flow in R3 with measures as initial vorticity and morrey spaces. Comm Partial Differential Equ 1989;14:577–618. [6] Herrmann R. Fractional calculus: an introduction for physicists. Singapore: World Scientific; 2011. [7] Hilfer R. Applications of fractional calculus in physics. Singapore: World Scientific; 20 0 0. [8] Iwabuchi T, Takada R. Global well-posedness and ill-posedness for the navier-stokes equations with the coriolis force in function spaces of besov type. J Funct Anal 2014;267(5):1321–37. [9] Kato T. Strong lp -solutions of the navier-stokes equation in Rm with applications to weak solutions. Math Z 1984;187:471–80. [10] Kemppainen J, Siljander J, Vergara V, Zacher R. Decay estimates for time-fractional and other non-local in time subdiffusion equations in Rd . Math Ann 2014:1–39. [11] Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications of fractional differential equations. North-Holland math. stud., vol. 204. Elsevier Science B.V., Amsterdam; 2006. [12] Kim I, Kim KH, Lim S. An lq (lp )-theory for the time fractional evolution equations with variable coefficients. Adv Math 2017;306:123–76. [13] Lei Z, Lin FH. Global mild solutions of navier-stokes equations. Comm Pure Appl Math 2011;64(9):1297–304. [14] Lemarié-Rieusset PG. Recent developments in the navier-Stokes problem. Chapman CRC Press; 2002. [15] Lemarié-Rieusset PG. The navier-stokes equations in the critical morrey-campanato space. Rev Mat Iberoam 2007;23(3):897–930. [16] Leray J. Sur le mouvement dun liquide visqueux emplissant l’espace. Acta Math 1934;63:193–248. [17] Majda AJ, Bertozzi AL. Vorticity and incompressible flow, vol. 27. Cambridge University Press; 2002. [18] Momani S, Zaid O. Analytical solution of a time-fractional navier-stokes equation by adomian decomposition method. Appl Math Comput 2006;177:488–94. [19] Planchon F. Global strong solutions in sobolev or lebesgue spaces to the incompressible navier-stokes equations in R3 . Ann Inst Henri Poincare, Anal Non Lineaire 1996;13:319–36.
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[20] Taylor M. Analysis on morrey spaces and applications to navier-stokes equation. Comm Partial Differential Equ 1992;17:1407–56. [21] Zacher R. A de giorgi-nash type theorem for time fractional diffusion equations. Mathematische Annalen 2013;356:99–146. [22] Zhou Y. Basic theory of fractional differential equations. Singapore: World Scientific; 2014. [23] Zhou Y. Fractional evolution equations and inclusions: analysis and control. Academic Press; 2016. [24] Zhou Y, Jiao F, Pecaric J. Abstract cauchy problem for fractional functional differential equations in banach spaces. Topol Methods Nonlinear Anal 2013(42):119–36. [25] Zhou Y, Vijayakumar V, Murugesu R. Controllability for fractional evolution inclusions without compactness. Evol Equ Control Theory 2015;4:507–24.
[26] Zhou Y, Peng L. Topological properties of solutions sets for partial functional evolution inclusions. CR Math 2017;355:45–64. [27] Zhou Y, Peng L. On the time-fractional navier-stokes equations. Comput Math Appl 2017;73(6):874–91. [28] Zhou Y, Peng L. Weak solutions of the time-fractional navier-stokes equations and optimal control. Comput Math Appl 2017;73(6):1016–27. [29] Zhou Y, Zhang L, Shen XH. Existence of mild solutions for fractional evolution equations. J Int Equs Appl 2013;25:557–86. [30] Zhou Y, Zhang L. Existence and multiplicity results of homoclinic solutions for fractional hamiltonian systems. Comput Math Appl 2017;73(6):1325–45.
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Chaos, Solitons and Fractals 0 0 0 (2017) 1–8
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Energy methods for fractional Navier–Stokes equationsR Yong Zhou a,b, Li Peng a,∗, Bashir Ahmad b, Ahmed Alsaedi b a
Faculty of Mathematics and Computational Science, Xiangtan University, Hunan 411105, PR China Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
b
a r t i c l e
i n f o
Article history: Received 5 December 2016 Revised 7 March 2017 Accepted 24 March 2017 Available online xxx MSC: 35R11 35Q30 76D03
a b s t r a c t In this paper we make use of energy methods to study the Navier–Stokes equations with time-fractional derivative. Such equations can be used to simulate anomalous diffusion in fractal media. In the first step, we construct a regularized equation by using a smoothing process to transform unbounded differential operators into bounded operators and then obtain the approximate solutions. The second part describes a procedure to take a limit in the approximation program to present a global solution to the objective equation. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Navier-Stokes equations Caputo fractional derivative Energy methods Approximate solutions
1. Introduction Navier–Stokes equations have been investigated by many researchers in view of their crucial role in fluid mechanics and turbulence problems. For more details, we refer the reader to the monographs by Ben-Artzi et al. [1] and Lemarié-Rieusset [14]. The topic of the global existence of weak, mild and strong solutions supplemented with small initial data received considerable attention. For example, Leray [16] carried out a pioneering study on the existence of global weak solutions in the energy space and the uniqueness of such solutions in R2 . Lemarié-Rieusset [15] established the existence of global mild solutions in different types of frameworks in Morrey–Campanato spaces. Later, Iwabuchi and Takada [8] discussed the same problem in function spaces of Besov type. A similar result was established by Lei and Lin [13] in the space X−1 . Similar results were obtained by Kato [9] in Ln (Rn ), Giga and Miyakawa [5], and Taylor [20] in Morrey spaces, Cannone −1+n/p [2] and Planchon [19] in the Besov spaces B p,∞ (Rn ), 1 < p < ∞. On the other hand, fractional calculus gained much popularity during the past decades mainly due to its extensive applications in widespread areas of science and engineering, such as fluid flow, rheology, dynamical processes and porous structures, diffusive
R ∗
Project supported by National Natural Science Foundation of China (11671339). Corresponding author. E-mail addresses: [email protected] (Y. Zhou), [email protected] (L. Peng).
transport akin to diffusion, control theory of dynamical systems, viscoelasticity and so on. For some recent contributions on the topic, we refer the reader to the monographs by Herrmann [6], Hilfer [7], Kilbas et al. [11] and Zhou [22,23], and a series of papers [12,21,24–30] and the references cited therein. Theoretical analysis and experimental data have shown that classical diffusion equation fails to describe diffusion phenomenon in heterogeneous porous media that exhibits fractal characteristics. Fractional calculus tools have been found effective in modelling anomalous diffusion processes as fractional-order operators can characterize the long memory processes. Consequently, it is reasonable and practical to propose the generalized Navier–Stokes equations in terms of Caputo time-fractional derivative operator, which can be used to simulate anomalous diffusion in fractal media. Such models are found to be of great interest for both physicists and pure mathematicians as they exhibit much more complicated phenomena and are more challenging than their corresponding integer-order counterparts. In this paper we consider the following Navier–Stokes equations with time-fractional derivative in R3 :
⎧ α ⎨∂t u + (u · ∇ )u = −∇ p + ν u, ∇ · u = 0, ⎩ u ( 0, x ) = u0 ,
t > 0, (1.1)
where ∂tα is the Caputo fractional derivative of order α ∈ (0, 1), u = (u1 (t, x ), u2 (t, x ), u3 (t, x )) represents the velocity field at a point
http://dx.doi.org/10.1016/j.chaos.2017.03.053 0960-0779/© 2017 Elsevier Ltd. All rights reserved.
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x ∈ R3 and time t > 0, p = p(t, x ) is the pressure, ν is the viscosity and u0 = u0 (x ) is the initial velocity. Recently, Eq. (1.1) attracted significant attention due to its importance in simulating anomalous diffusion in fractal media. For some recent works on analytic and weak solutions of timefractional Navier–Stokes equations, for example, see El-Shahed et al. [4], Momani and Zaid [18] and Zhou and Peng [28]. Another interesting aspect of the study of existence of global mild solutions includes the situation when norms of the initial values are supposed to be small enough, for instance, see Carvalho–Neto and Gabriela [3], Zhou and Peng [27]. It is worth-mentioning that the purpose of this paper is to develop a new global result when we relax the smallness condition on the initial data. The paper is organized as follows. In Section 2 we recall some notations, definitions, and preliminary facts. In Section 3 we construct a regularized equation, and show the local existence of its solutions. Section 4 deals with the global existence and continuation of solutions for a kind of fractional differential equations in a Banach space. Via energy methods, we also obtain the convergence of approximate solutions and the global existence and uniqueness of mild solutions of Eq. (3.1) in different spaces. 2. Preliminaries Here we recall some notations, definitions, and preliminary facts which are used throughout this paper. Denote by · the L2 norm on R3 , where
u =
R3
|u(x )|2 dx
12
12
um =
Dβ u 2
s
R3
It is not difficult to show that if u ∈ H1 , then u ∈ X−1 . Let X be a Banach space and v : [0, ∞ ) → X. The fractional integral of order α ∈ (0, 1] for the function v is defined as
( |x| ) ∈
2 (1 + |ξ |2 ) | u ( ξ )| d ξ
12
( R ), 0 ≤ ≤ 1,
3
.
R3
(x )dx = 1.
We define the mollification Jε u of the function u ∈ Lq (R3 ) (1 ≤ q ≤ ∞ ) as
(Jε u )(x ) = ε −3
R3
x − y ε
u(y )dy.
(α )
t 0
(t − s )α−1 v(s )ds, t > 0.
Further, CDtα v represents the Caputo fractional derivative of order α for the function v and is defined by C α Dt
v(t ) =
d 1 −α I (v(t ) − v(0 ) ) , t > 0. dt t
For u : [0, ∞ ) × R3 → R3 , Caputo time-fractional derivative of the function u can generally be written as
∂tα u(t, x ) =
∂ 1 −α I (u(t, x ) − u(0, x ) ) , t > 0. ∂t t
For more insight into the topic, see Kilbas et al. [11]. Before proceeding further, we present two important results which play a key role in proving the main results. Lemma 2.1. [10] Let T > 0. Then, for any v ∈ L2 ((0, T ] × R3 , R3 ),
R3
v(t, x )∂tα v(t, x )dx ≥ v∂tα v.
Lemma 2.2. Let the function v : (0, T ] × R3 → C3 and ∂tα v exist. Then
v(t )∂tα v(t ) + v(t )∂tα v(t ) ≥ 2|v|∂tα |v|,
v(t )∂tα v(t ) + v(t )∂tα v(t ) = 2(a(t )∂tα a(t ) + b(t )∂tα b(t ) ) = 2 ∂tα a2 (t ) + ∂tα b2 (t ) + t −α a2 (t ) + b2 (t )
V s := {u ∈ H s (R3 ) : ∇ · u = 0} and call the operator P : H s (R3 ) → V s the Leray projector. Clearly, P commutes with Jε . For more details, we refer the reader to Majda and Bertozzi [17]. Let us denote an important space: for m ∈ Z,
Xm = u ∈ D ( R3 ) :
R3
+
t 0
+
t 0
|a(t ) − a(t − s )|2 [α s−α−1 ]ds
|b(t ) − b(t − s )|2 [α s−α−1 ]ds
t 2 −α −1 ≥ 2 ∂tα |v(t )|2 +t −α |v(t )|2 + | v ( t ) | − | v ( t −s ) | [ α s ] ds ( ) 0
= 2|v(t )|∂tα |v(t )|. 3. Local existence
We introduce the space
1
Itα v(t ) =
Here u represents the Fourier transform of u and S (R3 ) denotes the Schwarz space of rapidly decreasing smooth functions. It is clear that the two norms are equivalent for s = m. Let ϱ be the standard mollifier satisfying
C0∞
|ξ |m | u| d ξ .
Proof. Let v(t ) = a(t ) + b(t )i. Then
For s ∈ R, the Sobolev space H s (R3 ) is the completion of S (R3 ) with respect to the norm
R3
where v(t ) denotes the dual of v(t ).
.
0≤|β|≤m
us =
uXm =
The following result is a generalization of Lemma 6.1 in Kemppainen et al. [10].
.
By H m (R3 ), m ∈ Z ∪ {0}, we denote the Sobolev space consisting of functions u ∈ L2 (R3 ) such that Dβ u ∈ L2 (R3 ), 0 ≤ |β | ≤ m with the norm ·m defined by
Here D (R3 ) stands for the space of distributions. The norm of Xm is given by
|ξ |m | u| d ξ < ∞ .
This section is concerned with the construction of an approximate (regularized) equation for the time-fractional Navier–Stokes equations and obtain the existence and some properties of its solution. To do this, we use the mollifier Jε to regularize Eq. (1.1):
⎧ α ε ε ε ε ε ⎨∂t u + Jε [(Jε u ) · ∇ (Jε u )] = −∇ p + ν Jε (Jε u ), ∇ · uε = 0, ⎩ ε u ( 0 ) = Jε u0 .
(3.1)
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By applying the Leray projector P to Eq. (3.1), we get rid of the pressure term in Eq. (3.1), which reduces to an ordinary differential equation (ODE) in Vs :
∂tα uε + PJε [(Jε uε ) · ∇ (Jε uε )] = ν Jε2 uε , uε ( 0 ) = Jε u0 .
(3.2)
3
Since ν ≥ 0, we have
∂tα uε ≤ 0. Then the definition of ∂tα u implies that
sup t∈[0,T ]
u ε ≤ J ε u 0 ≤ u 0 .
We set
Fε (uε ) = ν Jε2 uε − P Jε [(Jε uε ) · ∇ (Jε uε )]. From the argument of Proposition 3.6 in Majda and Bertozzi [17], we know that
Fε (u1 ) − Fε (u2 )m ≤ L(u j , ε , M )u1 − u2 m for u j ∈ BM , j = 1, 2,
(3.3)
where m ∈ Z+ ∪ {0} and
BM = {u ∈ V
m
: um < M }.
∂tα uε m ≤ cm |Jε ∇ uε |L∞ uε m .
C α ([0, T ], V m ) = {u ∈ C ([0, T ], V m ) : ∂tα u ∈ C ([0, T ], V m )}.
Dβ ∂tα uε , Dβ uε = Dβ Jε2 uε , Dβ uε
− D β P J ε [ ( J ε u ε ) · ∇ ( J ε u ε )] , D β u ε
= νJε Dβ ∇ uε 2 − P Jε [(Jε uε ) · ∇ (Dβ Jε uε )], Dβ uε
Theorem 3.1. Assume the initial data u0 ∈ V0 . Then, for any ε > 0, (i) there exists Tε = T (Jε u0 m , ε ) such that Eq. (3.2) has a unique solution uε ∈ Cα ([0, Tε ), Vm ); (ii) uε ∈ Cα ([0, T], V0 ) on any interval [0, T] with
sup
Lemma 3.1. The regularized solution uε of Eq. (3.2) satisfies the inequality:
Proof. We take the derivative Dβ of Eq. (3.2) and L2 inner product with Dβ uε to obtain
It means that Fε is locally Lipschitz continuous on any open set. Let
t∈[0,T ]
In the next lemma, we derive a key estimate which plays an important role in proving the main results.
uε ≤ u0 .
(3.4)
− Dβ P Jε [(Jε uε ) · ∇ (Jε uε )] −P Jε [(Jε uε ) · ∇ (Dβ Jε uε )], Dβ uε . From the discussion of Majda and Bertozzi [17] and Lemma 2.1, we deduce that
uε m ∂tα uε m + νJε ∇ uε 2m ≤ cm |Jε ∇ uε |L∞ uε 2m , which takes the following form for ν > 0:
∂tα uε m ≤ cm |Jε ∇ uε |L∞ uε m .
Proof. (i) If u0 ∈ V0 , then Jε u0 ∈ V m , m ∈ Z+ ∪ {0}. For given r > 0, define
B(r, Tε ) =
uε ∈ C ([0, Tε ), V m ) : sup
t∈[0,Tε )
uε (t ) − Jε u0 m ≤ r .
Notice that
M=
sup
(t,uε )∈[0,Tε )×B(1,Tε )
Fε (uε )m < +∞.
(3.5)
Indeed,
Fε (uε )m ≤ Fε (uε ) − Fε (Jε u0 )m + Fε (Jε u0 )m ≤ Luε − Jε u0 m + Fε (Jε u0 )m ≤ Lr + Fε (Jε u0 )m < +∞. Consider the operator T :
T uε (t ) = Jε u0 +
0
t
(t − s )α−1 Fε (uε (s ))ds.
Obviously T (B(r, Tε )) ⊂ B(r, Tε ). Moreover, for u1 , u2 ∈ B(r, Tε ),
T u1 (t ) − T u2 (t )m ≤ 1
Lt α
α
sup s∈[0,t]
u1 (s ) − u2 (s )m for t ∈ [0, Tε ).
1
Fixing Tε =min{( αMr ) α ,( αL ) α }, it is easy to show that T is a strict contraction mapping on B(r, Tε ). In consequence, we deduce that T has a fixed point. On account of (3.3) and uε ∈ C([0, Tε ), Vm ), it is clear that ∂tα u ∈ C ([0, Tε ), V m ). (ii) Take the L2 inner product of Eq. (3.2) with uε and apply Lemma 2.1 to obtain
uε ∂tα uε ≤ ν
R3
uε Jε2 uε dx −
R3
uε P Jε [(Jε uε ) · ∇ (Jε uε )]dx.
Integrating by parts and using the fact that ∇ · uε = 0, we get
uε ∂tα uε + νJε ∇ uε 2 ≤ 0.
4. Global existence Here we discuss the global existence of solutions by means of mathematical analysis and previous estimate with the initial data being less than the viscosity. Firstly, we consider the following autonomous equation C α Dt z
(t ) = F (z ), z(0 ) = z0 .
(4.1)
Lemma 4.1. Let X be a Banach space and U ⊂ X be an open set. Suppose that the mapping F: U → X satisfies the locally Lipschitz condition, that is, for any z ∈ U there is a constant L > 0 and an open neighborhood Uz ⊂ U of z such that
|F (z1 ) − F (z2 )|X ≤ L|z1 − z2 |X for all z1 , z2 ∈ Uz . Then for any z0 ∈ U, there exists a time T such that Eq. (4.1) has a unique (local) solution z ∈ C([0, T), U). Proof. The method for showing the local existence of solutions for Eq. (4.1) is similar to that of Theorem 3.1 (i), so we omit it. Now we establish the global existence of solutions for a kind of fractional differential equations in Banach space, which furnishes as the key instrument in the proof of the global existence of solutions for the time fractional Navier–Stokes equations. Lemma 4.2. Under the conditions of Lemma 4.1, the unique solution z = z(t ) (t ∈ [0, T )) of the fractional differential equation (4.1) either exists globally in time, or T < ∞ and z(t) preserves the open set U as t →T − . Proof. Suppose that the maximum existing interval for the solution z(t) is [0, T), that is,
T = sup{Tz : z(t ) is defined on [0, Tz ] and z(t ) is a solution of Eq. (4.1 ) on [0, Tz ]}.
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Then T = ∞ or T < ∞. If T = ∞, the conclusion holds. If T < ∞, we show that z(t) preserves the open set U as t → T − . Observe that there exists a sequence {tk } and a positive constant K > 0 such that
tn ≤ tn+1 for n ∈ N,
lim tn = T ,
n→∞
|z(tn )| ≤ K.
z˜(t ) ∈ Br (ϕ¯ , T + h ), that is,
z˜(t ) = ϕ¯ (t ) +
(4.2)
= z0 +
0
with
ε ≤ |z(ηn ) − z∗ | ≤ |z(tn ) − z∗ | + |z(ηn ) − z(tn )| tn ε 1 ≤ + (tn − s )α−1 − (ηn − s )α−1 |F (z(s ))|ds 2
(α ) 0
1 L∞ α ( R+ , X ) =
≤
ε 2
1
ηn
(α )
tn
+
1 + sup |F (z(s ))|(ηn − tn )α
(1 + α ) s∈[tn ,ηn ]
≤
2
Thus limt→T − z(t ) exists. Next we show that z∗ ∈ ∂ U. On the contrary, we assume that z∗ ∈ ∂ U. Since z(t) ∈ U (t ∈ [0, T)), therefore, z∗ ∈ U. Let z(t) be equal to z∗ for t = T and itself for 0 ≤ t < T. Denote
ϕ¯ (t ) = z0 +
0
(t − s )α−1 F (z(s ))ds, t ∈ [T , T1 ],
Sy(t ) = ϕ¯ (t ) +
t T
(t − s )α−1 F (y(s ))ds, t ∈ [T , T1 ],
where y ∈ C([T, T1 ], U). Let
Br =
t
0
Theorem 4.1. For u0 ∈ X−1 satisfying
u0 X−1 < ν,
(4.3)
u(t )X−1 + (ν − u0 X−1 )
t 0
(t − s )α−1 ∇ u(s )L∞ ds
≤ u0 X−1 .
pε =
Ri R j ui u j = (R R )(uε uε ),
i, j=1
where R is the Riesz operator. Firstly we have the estimate
uε (t )X−1 =
(t, y ) : t ∈ [T , T1 ], |y(t )| ≤ sup |ϕ¯ (t )| + r .
|ξ |≤1
|ξ |−1 |u ε (t )|dξ
+
with ϕ¯ ∈ C ([T , T1 ], U ), and define the operator S as follow:
(t − s )α−1 ε ∞ × u (s )X1 ds ∈ L ([0, ∞ ), R ) . u : [0, ∞ ) → X1 :
Proof. For u0 ∈ X−1 , we have Jε u0 X−1 ≤ u0 X−1 by the properties of ϱ. By the arguments similar to the ones employed in Theorem 3.1, there exists a unique locally continuous solution uε (t, x) on the interval [0, Tε ). The associated pressure pε is given by
for sufficiently large n ≥ n0 .
T
Let us introduce
0≤t<∞
(2(ηn − tn )α + tnα − ηnα )
t ∈ [T , T + h].
1 sup |F (z(s ))|(2(ηn − tn )α + tnα − ηnα )
(1 + α ) s∈[0,ηn ] M
t ∈ [0, T ),
z˜(t ),
Evidently z¯ ∈ C ([0, T + h], U ). Thus we find that z¯ (t ) is the solution of (4.1) on the interval [0, T + h], which contradicts the assumption that [0, T) is the maximum existing interval.
sup
which is a contradiction in view of
(1 + α ) ε
z(t ),
Eq. (1.1) has a unique solution u existing globally in time. Moreover, 1 u ∈ C ([0, ∞ ), X−1 ) ∩ L∞ α (R+ , X ) and the following estimate holds:
≤ ε,
≤
(t − s )α−1 F (z˜(s ))ds
(t − s )α−1 F (z¯ (s ))ds, t ∈ [T , T + h],
z¯ (t ) =
(ηn − s )α−1 |F (z(s ))|ds
1 sup |F (z(s ))|( (ηn − tn )α + tnα − ηnα )
(1 + α ) s∈[0,tn ]
t T
t
According to the equicontinuity of z(t) and Lemma 4.1, {z(tn )} has a convergent subsequence. Without loss of generality, let limn→∞ z(tn ) = z∗ . This together with (4.2) implies that for sufficiently small τ > 0, there exists n0 such that T − τ < tn0 < T and for n ≥ n0 , we have |z(tn ) − z∗ | ≤ 2ε . We show that limt→T − z(t ) = z∗ . On the contrary, for n ≥ n0 , there exists ηn ∈ (tn , T) such that |z(ηn ) − z∗ | ≥ ε and |z(t ) − z∗ | < ε, for t ∈ (tn , ηn ). By the continuity of F on [0, T), we denote M = sups∈[0,ηn ] |F (z(s ))|. Thus,
+
|ξ |≥1
|ξ |−2 | −uε (t )|dξ ≤ C uε (t )1 .
(4.4)
This implies that uε ∈ L∞ ([0, Tε ), X−1 ). Next, by applying the Fourier transform to (3.1), we obtain
α ε ε ∂t u − i u (η ) u ε (ξ − η )dη · ξ − iξ p ε + ν|ξ |2 u ε = 0, ξ · u ε = 0. (4.5)
t∈[T,T1 ]
On account of the continuity of F on Br , let M = supy∈Br |F (y(s ))|. Consider
Br (ϕ¯ , T + h ) =
y ∈ C ( [T , T + h], U ) :
sup t∈[T,T +h]
|y(t ) − ϕ¯ (t )| ≤ r,
y(T ) = ϕ¯ (T ) , α )r α1 where h = {T1 − T , ( (1+ ) }. M
As argued in Lemma 4.1, we can derive that S is a strict contraction mapping on Br (ϕ¯ , T + h ). Therefore, S has a fixed point
From (4.5), one infers that
u ε · ∂tα u ε − i and
u ε · ∂tα u ε + i
u ε (ξ ) · u ε (η ) u ε (ξ − η )dη · ξ + ν|ξ |2 |u ε |2 = 0,
u ε (ξ ) · u ε (η ) u ε (ξ − η )dη · ξ + ν|ξ |2 |u ε |2 = 0.
Thus it follows that
u ε · ∂tα u ε + u ε · ∂tα u ε + 2ν|ξ |2 |u ε |2 =i u ε (ξ ) · u ε (η ) u ε (ξ − η ) − u ε (ξ ) · u ε (η ) u ε (ξ − η ) dη · ξ .
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Using Lemma 2.2, multiplying by |ξ |−1 |u ε |−1 and integrating with respect to ξ , we get
∂tα
|ξ |−1 |u ε |dξ + ν |ξ ||u ε |dξ i ≤ u ε (η ) · |u ε |−1 u ε (ξ ) u ε (ξ − η ) 2 − u ε (η ) · |u ε |−1 u ε (ξ ) u ε (ξ − η ) · |ξ |−1 ξ dηdξ
≤ ≤
1 2
≤
1 ε uε u weakly in L∞ α (R+ , X ), u u weakly in L∞ (R+ , X−1 ), as ε → 0. ∗
uε (0 ) − u0 X−1 → 0, as ε → 0.
|ξ ||u ε (ξ )|dξ .
∂tα
(4.6)
which, according to the definition of ∂tα , yields The above procedure can similarly be applied to each small interval [δ , 2δ ], [2δ , 3δ ], ... . This also ensures that uε (t )X−1 ≤ u0 X−1 < ν for each t ∈ [0, Tε ). Integrating (4.6) from 0 to t, we get t
(t −s )α−1 uε (s )X−1 ds ≤ u0 X−1
for t ∈ [0, Tε ).
∂tα
t 0
(t − s )α−1 ∇ uε (s )L∞ ds ≤
t 0
(t − s )α−1 uε (s )X1 ds
uε (t )m ≤ uε (0 )m +cm ≤ J ε u 0 m + c m
t
(t −s )α−1 |Jε ∇ uε (s )|L∞ uε (s )m ds
0 t
0
uε (t )
m
≤ Jε u0 m exp cm
≤ Jε u0 m exp
0
t
0≤t<∞
1 (uε1 X1 + uε2 X1 )uε1 − uε2 X−1 . 2
(ν − u0 X−1 )
0
t
(t − s )α−1 uε1 − uε2 X1 ds
t
0
× (uε1 X1 + uε2 X1 )ds ≤ uε1 (0 ) − uε2 (0 )X−1 (4.7)
t uε (t )X−1 + (ν − u0 X−1 ) (t − s )α−1 uε (s )X1 ds 0
(4.8)
Estimate (4.8) implies that there exists a subsequence of {uε }, relabeled as {uε }, and 1 u ∈ L∞ (R+ , X−1 ) ∩ L∞ α ( R+ , X )
1 2
(t − s )α−1 (uε1 X1 + uε2 X1 )uε1 − uε2 X−1 ds t 1 ε1 u0 X−1 ε 2 ≤ u (0 ) − u (0 )X−1 exp (t − s )α−1 2 ν − u0 X−1 0 ≤
≤ u0 X−1 .
|ξ ||uε1 − uε2 |dξ
∂tα uε1 − uε2 X−1 + (ν − u0 X−1 )uε1 − uε2 X1
for all t ∈ [0, Tε ). It follows from Lemma 4.2 that Tε = ∞. Moreover, we have the following uniform estimate on uε :
sup
that is,
(t − s )α−1 |∇ uε (s )|L∞ ds
cm u0 X−1 ν − u0 X−1
|ξ |−1 |uε1 − uε2 |dξ + (ν − u0 X−1 )
1 (uε1 X1 + uε2 X1 )uε1 − uε2 X−1 , 2
≤
(t − s )α−1 |∇ uε (s )|L∞ uε (s )m ds.
Using Gronwall’s inequality again, we get
|η|−1 |ξ − η| + |η||ξ − η|−1 |uε1 (η )| + |uε2 (η )|
and
On the other hand, in view of Lemma 3.1, we obtain
uε1 (t ) − uε2 (t )X−1 ≤ uε1 (0 ) − uε2 (0 )X−1 t 1 α −1 ε ε 1 2 exp (t − s ) (u X1 + u X1 )dξ 2 0 u0 X−1 ε ε 1 2 ≤ u (0 ) − u (0 )X−1 exp , ν − u0 X−1
u0 X−1 . ν − u0 X−1
≤
Using Gronwall’s inequality again and noting that ν − u0 X−1 > 0, we get
|ξ ||u ε (ξ )|dξ ,
shows that
≤
∇ uε (t )L∞ = exp(ix · ξ )∇ uε ( ξ )d ξ ≤ |∇ uε ( ξ )|d ξ
1 2
× |u ε1 ( ξ − η ) − u ε2 ( ξ − η )|d η d ξ 1 ≤ (uε1 X−1 + uε2 X−1 )uε1 − uε2 X1 2 1 + (uε1 X1 + uε2 X1 )uε1 − uε2 X−1 . 2
This, together with
=
Combining the above inequality and (4.8), we obtain
uε (t )X−1 ≤ J ε u0 X−1 ≤ u0 X−1 < ν on [0, δ ].
0
≤
∂tα uε (t )X−1 ≤ 0 on [0, δ ],
(4.10)
|ξ |−1 |uε1 − uε2 |dξ + ν |ξ ||uε1 − uε2 |dξ ≤ |u ε1 ( η )| + |u ε2 ( η )| |u ε1 ( ξ − η ) − u ε2 ( ξ − η )|d η d ξ
From (4.3) and (4.4), we see that there exists a small enough δ with δ ∈ (0, Tε ) such that uε (t )X−1 < ν for t ∈ [0, δ ], since uε (t) is local smooth solution on [0, Tε ), that is, uε ∈ C([0, Tε ), Vm ). Therefore, we have
uε (t )X−1 + (ν − u0 X−1 )
(4.9)
In order to prove the strong convergence of uε , we proceed with the same argument as in (4.6) to have the estimate:
|η|−1 |ξ − η| + |η||ξ − η|−1 |u ε (η )||u ε (ξ − η )|dηdξ
|ξ |−1 |u ε (ξ )|dξ
such that
For the relation between uε (0) and u0 , we notice that
|u ε (η )||u ε (ξ − η )|dηdξ
5
u0 X−1 u0 X−1 exp . ν − u0 X−1 ν − u0 X−1
Thus we conclude that {uε } is a Cauchy sequence in L∞ (R+ , X−1 ) ∩ 1 L∞ α (R+ , X ) by (4.10), and that the convergence in (4.9) is a strong one. In practice, the above estimate also ensures that the solution 1 in the space L∞ (R+ , X−1 ) ∩ L∞ α (R+ , X ) is unique under the condition (4.3). To establish continuity of u(t, x), we return to Eq. (4.5). It is obvious that
t 0
=
(t − s )α−1 uε (s )X−1 ds =
t 0
(t − s )α−1
t 0
(t −s )α−1
|ξ ||u ε (ξ )|dξ ds =
0
t
|ξ |−1 | uε ( ξ )|d ξ d s
(t − s )α−1 uε X1 ds.
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In addition, like the estimate (4.6), we can obtain
t 0
(t − s )α−1 ∇ · (uε (s ) uε (s ))X−1 ds ≤
t
t 0
The following results deal with the global existence of approximate solutions and mild solutions, respectively.
(t − s )α−1
Theorem 4.2. Let 0 < T < ∞. If u0 ∈ Vm , m ∈ Z+ ∪ {0}, then there exists a unique solution uε ∈ Cα ([0, T], Vm ) for Eq. (3.2) for any ε > 0.
|u ε (η )||u ε (ξ − η )|dξ dηds
× ≤
(t − s )α−1 uε X−1 uε X1 ds
0
≤ sup
0≤t<∞
uε (t )X−1
t 0
Proof. We show an a priori bound on uε m . To do this, we note that the relation (3.3) with u2 (t, x) ≡ 0 and (3.4) yields
(t − s )α−1 uε X1 ds.
∂tα uε m ≤ L(uε , ε , M )uε m ≤ L(u0 , ε , M )uε m .
The pressures term can be handled in a similar manner, that is, for ξ R uε (ξ ) = −i j · u ε (ξ ), we have |ξ |
j
t 0
(t − s )α−1 ∇ pε X−1 ds ≤
t 0
(t − s )α−1
ε (η )||Ru ε (ξ − η )|dξ dηds ≤ |Ru
×
t 0
≤ sup
0≤t<∞
uε (t )X−1
t 0
Tα uε (t )m ≤ u0 m exp L(u0 , ε , M ) , α which shows a priori bound on uε m . By Lemma 4.2, the regularized Eq. (3.2) has a unique solution uε ∈ Cα ([0, T], Vm ).
(t − s )α−1
Theorem 4.3. Assume that u0 ∈ Vm , m ≥ 3. Then the following results hold:
|u ε (η )||u ε (ξ − η )|dξ dηds
×
Thanks to Gronwall’s inequality, we have
(i) If u0 1 < Cν and 0 < T < ∞, there exists a unique solution
(t − s )α−1 uε X1 ds,
u ∈ C ([0, T ], C 2 (R3 )) ∩ C α ([0, T ], C (R3 ))
Observe that
to the time-fractional Navier–Stokes equations. Moreover, u is the limit of a subsequence of {uε }, where uε are the approximate solutions given by Theorem 4.2; (ii) The following estimates hold true:
uε (t )X−1 − uε (t0 )X−1 t0 ≤ (t0 − s )α−1 − (t − s )α−1 (uε (s )X−1 0
+
∇ · (uε (s ) uε (s ))X−1 + ∇ pε X−1 )ds
+
t
t0
sup t∈[0,T ]
(t − s )α−1 (uε (s )X−1 + ∇ · (uε (s ) uε (s ))X−1
sup t∈[0,T ]
+ ∇ pε X−1 )ds for 0 < t0 ≤ t. Now we estimate each term in the above expression. For I1 (t), notice that t0 0
(t − s )α−1 − (t0 − s )α−1
+
∇
≤2
t0
0
+ ≤2
· (uε (s ) uε (s ))
( u ε ( s )
X−1
X−1
+ ∇ pε
X−1
0
t0
< ∞,
uε (t )
uε (s )X−1
0
t0
lim I1 (t ) =
t−t0 →0
t0
lim
0
+
t−t0 →0
≤ u0 m exp ≤ u0 m exp
(t0 − s )α−1 uε (s )X1 ds
(t0 − s )α−1 − (t − s )
α −1
(uε (s )X−1
∇ · (uε (s ) uε (s ))X−1 + ∇ pε X−1 )ds
= 0.
cm u0 X−1 ν − u0 X−1 cm C u 0 1 ν − C u0 1
.
(4.11)
In consequence, {uε } is a uniformly bounded set of C ([0, T ], H m (R3 )). Next we prove that {uε } constitutes a Cauchy sequence in C ([0, T ], L2 (R3 )). Using Eq. (3.2), we have
uε − uε ∂tα uε − uε ≤ ν Jε2 uε − Jε2 uε , uε − uε − P Jε [(Jε uε ) · ∇ (Jε uε )]
− P J ε [ ( J ε u ε ) · ∇ ( J ε u ε )] , u ε − u ε . Using the argument of [17, Lemma 3.7], we obtain
For I2 (t), by an elementary calculation and the fact that uε ∈ 1 L∞ α (R+ , X ), we obtain the following inequality
I2 (t ) ≤
m
due to the estimate (4.8). Applying the Lebesgue’s dominated convergence theorem, we get
ν −C u0 1 ;
Proof. In the first step, we show that {uε (t)} is a uniformly bounded set of H m (R3 ) independent of ε . By Lemma 3.1, we have
(t0 − s )α−1 uε (s )X1 ds
0≤s
cm C u0 1
Eq. (4.7) and u0 X−1 ≤ C u0 1 ensure that
(t0 − s )α−1 (uε (s )X−1 + ∇ · (uε (s ) uε (s ))X−1
+ 4 sup
ν −C u0 1 ,
∂tα uε m ≤ cm |Jε ∇ uε |L∞ uε m .
)ds
∇ pε X−1 )ds
u(t )m ≤ u0 m exp
cm C u0 1
(iii) uε and u are uniformly bounded in L∞ ([0, T ], H m (R3 )) ∩ Cw ([0, T ], H m (R3 )).
=: I1 (t ) + I2 (t )
uε (t )m ≤ u0 m exp
1 + 2 sup
t0 ≤s
uε (s ))X−1
t
t0
Jε2 − Jε2 uε , uε − uε
− J ε ∇ ( u ε − u ε ) 2
≤ C1 max{ε , ε }uε 3 uε − uε
(t − s )α−1 uε (s )X1 ds and
→ 0, as t − t0 → 0. From the foregoing arguments, it follows that u ∈ 1 L∞ α (R+ , X ). This completes the proof.
Jε2 uε − Jε2 uε , uε − uε ≤
C ([0, ∞ ), X−1 )
∩
P J ε [ ( J ε u ε ) · ∇ ( J ε u ε )] − P J ε [ ( J ε u ε ) · ∇ ( J ε u ε )] , u ε − u ε
= (Jε − Jε )[(Jε uε ) · ∇ (Jε uε )], uε − uε
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Jε [(Jε − Jε )uε · ∇ (Jε uε )], uε − uε
+ J ε [ J ε ( u ε − u ε ) · ∇ ( J ε u ε )] , u ε − u ε
+ Jε {Jε uε · ∇ [(Jε − Jε )uε ]}, uε − uε
+ J ε {J ε u ε · ∇ [ J ε ( u ε − u ε ) ] } , u ε − u ε
Thus
+
≤ C1 max{ε , ε }uε 2m uε − uε
sup t∈[0,T ]
Moreover, for any given t ∈ [0, T], the sequence {uε (t, ·)} is a uniformly bounded set of H m (R3 ); hence there is also a subsequence of {uε (t, ·)}, relabeled as {uε (t, ·)} such that
which imply that
∂tα uε − uε ≤ C1 (M ) max{ε , ε } + uε − uε , where M = sup{uε m } from (4.11). Integrating the above inequalε ity and then using Gronwall’s inequality yields
Tα
uε (t ) − uε (t ) ≤ uε (0 ) − uε (0 ) + C1 (M ) max{ε , ε } α t × exp C1 (M ) (t − s )α−1 ds ≤
0
uε → u in u ∈ C ([0, T ], H m (R3 )),
uε (0 ) − uε (0 ) + C1 (M ) max{ε , ε } α Tα × exp C1 (M ) . α
which implies that
[φ , uε (t, · )] → [φ , u(t, · )] uniformly for t ∈ [0, T ],
Tα
uε (t ) −uε (t ) ≤ uε (0 ) −uε (0 ) +C1 (M ) max{ε , ε } α t∈[0,T ] α T × exp C1 (M ) , α sup
which, in view of (4.10), implies that {uε } is a Cauchy sequence in C ([0, T ], L2 (R3 )). Hence there exists u ∈ C ([0, T ], L2 (R3 )) such that the sequence {uε } converges strongly to u in C ([0, T ], L2 (R3 )), that is,
sup
uε (t ) − u(t ) ≤ C2 ε .
(4.12)
Finally we verify that u ∈ C ([0, T ], C 2 (R3 )) ∩ C α ([0, T ], C (R3 )). By the inequalities (4.11) and (4.12), we get
m
{uε (t ) − uε (t )m } ≤ sup uε (t ) − u(t )1− m t∈[0,T ] t∈[0,T ]
m
ε ε m × u (t ) − u (t )m sup
m
≤ C ( u 0 m ) ε 1− m
for 0 < m < m,
(4.13) m
which implies strong convergence in C ([0, T ], H (R3 )) for all m
< m. With 0 < 72 < m < m, we also have strong convergence in C ([0, T ], C 2 (R3 )). Moreover, from the equation
∂tα uε = ν Jε2 uε − PJε [(Jε uε ) · ∇ (Jε uε )], we deduce that ∂tα uε converges C ([0, T ], C (R3 )). As uε → u, we have
T 0
∂tα uε (t )φ (t )dt =
0
→
T
0
=
T
0
T
to
ν u − P ( u · ∇ u )
Hence we deduce that u(t)m is bounded for each t, which yields
For φ ∈ H −m (R3 ), Denote by [φ , u] the dual pairing of H −m (R3 ) and H m (R3 ). From (4.13), we have
Thus we have
t∈[0,T ]
uε (t, · ) u(t, · ) in H m (R3 ).
u ∈ L∞ ([0, T ], H m (R3 )).
Tα
(4.14)
uε u in L2 ([0, T ], H m (R3 )).
ε
uε (t )m ≤ M.
Accordingly, uε is uniformly bounded in L2 ([0, T ], H m (R3 )) and hence there exists a subsequence, relabeled as {uε }, and u ∈ L2 ([0, T ], H m (R3 )) such that
+ C1 max{ε , ε }uε m uε m × uε − u + C1 uε − u 2 uε m
+ C1 max{ε , ε }uε 2m uε − uε , ε
7
in
(uε (t ) − uε (0 ) )Ct DαT φ (t )dt (u(t ) − u(0 ) )Ct DαT φ (t )dt as ε → 0
∂tα u(t )φ (t )dt for φ ∈ C0∞ ([0, T ], C (R3 ).
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for any φ ∈ H −m (R3 ). Applying (4.14) and the conclusion that for
m < m, H −m (R3 ) is dense in H −m (R3 ), we obtain
[φ , uε (t, · )] → [φ , u(t, · )] uniformly for t ∈ [0, T ], for any φ ∈ H −m (R3 ). Thus u ∈ Cw ([0, T ], H m (R3 )). This completes the proof. References [1] Ben-Artzi M, Croisille JP, Fishelov D. Navier-Stokes equations in planar domains. World Scientific; 2013. [2] Cannone M. A generalization of a theorem by kato on navier-stokes equations. Rev Mat Iberoam 1997;13:515–41. [3] De Carvalho-Neto PM, Gabriela P. Mild solutions to the time fractional Navier-Stokes equations in RN . J Differ Equ 2015;259:2948–80. [4] El-Shahed M, Salem A. On the generalized navier-stokes equations. Appl Math Comput 2004;156(1):287–93. [5] Giga Y, Miyakawa T. Navier-stokes flow in R3 with measures as initial vorticity and morrey spaces. Comm Partial Differential Equ 1989;14:577–618. [6] Herrmann R. Fractional calculus: an introduction for physicists. Singapore: World Scientific; 2011. [7] Hilfer R. Applications of fractional calculus in physics. Singapore: World Scientific; 20 0 0. [8] Iwabuchi T, Takada R. Global well-posedness and ill-posedness for the navier-stokes equations with the coriolis force in function spaces of besov type. J Funct Anal 2014;267(5):1321–37. [9] Kato T. Strong lp -solutions of the navier-stokes equation in Rm with applications to weak solutions. Math Z 1984;187:471–80. [10] Kemppainen J, Siljander J, Vergara V, Zacher R. Decay estimates for time-fractional and other non-local in time subdiffusion equations in Rd . Math Ann 2014:1–39. [11] Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications of fractional differential equations. North-Holland math. stud., vol. 204. Elsevier Science B.V., Amsterdam; 2006. [12] Kim I, Kim KH, Lim S. An lq (lp )-theory for the time fractional evolution equations with variable coefficients. Adv Math 2017;306:123–76. [13] Lei Z, Lin FH. Global mild solutions of navier-stokes equations. Comm Pure Appl Math 2011;64(9):1297–304. [14] Lemarié-Rieusset PG. Recent developments in the navier-Stokes problem. Chapman CRC Press; 2002. [15] Lemarié-Rieusset PG. The navier-stokes equations in the critical morrey-campanato space. Rev Mat Iberoam 2007;23(3):897–930. [16] Leray J. Sur le mouvement dun liquide visqueux emplissant l’espace. Acta Math 1934;63:193–248. [17] Majda AJ, Bertozzi AL. Vorticity and incompressible flow, vol. 27. Cambridge University Press; 2002. [18] Momani S, Zaid O. Analytical solution of a time-fractional navier-stokes equation by adomian decomposition method. Appl Math Comput 2006;177:488–94. [19] Planchon F. Global strong solutions in sobolev or lebesgue spaces to the incompressible navier-stokes equations in R3 . Ann Inst Henri Poincare, Anal Non Lineaire 1996;13:319–36.
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Please cite this article as: Y. Zhou et al., Energy methods for fractional Navier–Stokes equations, Chaos, Solitons and Fractals (2017), http://dx.doi.org/10.1016/j.chaos.2017.03.053