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On the time-fractional Navier–Stokes equations
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On the time-fractional Navier–Stokes equations✩ Yong Zhou a,b,∗ , Li Peng a a
Faculty of Mathematics and Computational Science, Xiangtan University, Hunan 411105, PR China
b
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
article
info
Article history: Available online xxxx Keywords: Navier–Stokes equations Caputo fractional derivative Mittag-Leffler functions Mild solutions Regularity
abstract This paper is concerned with the Navier–Stokes equations with time-fractional derivative of order α ∈ (0, 1). This type of equations can be used to simulate anomalous diffusion in fractal media. We establish the existence and uniqueness of local and global mild solutions in H β,q . Meanwhile, we also give local mild solutions in Jq . Moreover, we prove the existence and regularity of classical solutions for such equations in Jq . © 2016 Elsevier Ltd. All rights reserved.
1. Introduction The Navier–Stokes equations describe the motion of the incompressible Newtonian fluid flows ranging from large scale atmospheric motions to the lubrication of ball bearings, and express the conservation of mass and momentum. For more details we refer to the monographs of Cannone [1] and Varnhorn [2]. We find this system which is so rich in phenomena that the whole power of mathematical theory is needed to discuss the existence, regularity and boundary conditions; see, e.g., Lemarié-Rieusset [3] and Von Wahl [4]. It is worth mentioning that Leray carried out an initial study that a boundary-value problem for the time-dependent Navier–Stokes equations possesses a unique smooth solution on some intervals of time provided the data are sufficiently smooth. Since then many results on the existence for weak, mild and strong solutions for the Navier–Stokes equations have been investigated intensively by many authors; see, e.g., Almeida and Ferreira [5], Heck et al. [6], Iwabuchi and Takada [7], Koch et al. [8], Masmoudi and Wong [9], and Weissler [10]. Moreover, one can find results on regularity of weak and strong solution from Amrouche and Rejaiba [11], Chemin and Gallagher [12], Chemin et al. [13], Choe [14], Danchin [15], Giga [16], Kozono [17], Raugel and Sell [18] and the references therein. On the other hand, fractional calculus has gained considerable popularity during the past decades due mainly to its demonstrated applications in numerous seemingly diverse and wide-spread fields of science and engineering, including fluid flow, rheology, dynamical processes and porous structures, diffusive transport akin to diffusion, control theory of dynamical systems, viscoelasticity and so on; see, e.g., Herrmann [19], Hilfer [20] and Zhou et al. [21–23]. The most important among such models are those described by partial differential equations with fractional derivatives. Such models are interesting for not only physicists but also pure mathematicians. Recent theoretical analysis and experimental data have shown that classical diffusion equation fails to describe diffusion phenomenon in heterogeneous porous media that exhibits fractal characteristics. How is the classical diffusion equation
✩ Project supported by National Natural Science Foundation of China (11271309).
∗
Corresponding author at: Faculty of Mathematics and Computational Science, Xiangtan University, Hunan 411105, PR China. E-mail address: [email protected] (Y. Zhou).
http://dx.doi.org/10.1016/j.camwa.2016.03.026 0898-1221/© 2016 Elsevier Ltd. All rights reserved.
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Y. Zhou, L. Peng / Computers and Mathematics with Applications (
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modified to make it appropriate to depict anomalous diffusion phenomena? This problem is interesting for researchers. Fractional calculus have been found effective in modelling anomalous diffusion processes since it has been recognized as one of the best tools to characterize the long memory processes. Consequently, it is reasonable and significative to propose the generalized Navier–Stokes equations with Caputo fractional derivative operator, which can be used to simulate anomalous diffusion in fractal media. Its evolutions behave in a much more complex way than in classical inter-order case and the corresponding investigation becomes more challenging. The main effort on time-fractional Navier–Stokes equations has been put into attempts to derive numerical solutions and analytical solutions; see Ganji et al. [24], El-Shahed et al. [25], and Momani and Zaid [26]. However, to the best of our knowledge, there are very few results on the existence and regularity of mild solutions for time-fractional Navier–Stokes equations. Recently, Carvalho-Neto [27] dealt with the existence and uniqueness of global and local mild solutions for the time-fractional Navier–Stokes equations. Motivated by above discussion, in this paper we study the following time-fractional Navier–Stokes equations in an open set Ω ⊂ Rn (n ≥ 3):
α ∂ u − ν 1u + (u · ∇)u t ∇ ·u u|∂ Ω u(0, x)
= = = =
−∇ p + f ,
0, 0, a,
t > 0, (1.1)
where ∂tα is the Caputo fractional derivative of order α ∈ (0, 1), u = (u1 (t , x), u2 (t , x), . . . , un (t , x)) represents the velocity field at a point x ∈ Ω and time t > 0, p = p(t , x) is the pressure, ν the viscosity, f = f (t , x) is the external force and a = a(x) is the initial velocity. From now on, we assume that Ω has a smooth boundary. Firstly, we get rid of the pressure term by applying Helmholtz projector P to Eq. (1.1), which converts Eq. (1.1) to
α ∂ u − ν P 1u + P (u · ∇)u t ∇ ·u u|∂ Ω u(0, x)
= = = =
Pf , 0, 0, a.
t > 0,
The operator −ν P ∆ with Dirichlet boundary conditions is, basically, the Stokes operator A in the divergence-free function space under consideration. Then we rewrite (1.1) as the following abstract form Dαt u = −Au + F (u, u) + Pf , u(0) = a,
C
t > 0,
(1.2)
where F (u, v) = −P (u ·∇)v . If one can give sense to the Helmholtz projection P and the Stokes operator A, then the solution of Eq. (1.2) is also the solution of Eq. (1.1). The objective of this paper is to establish the existence and uniqueness of global and local mild solutions of problem (1.2) in H β,q . Further, we prove the regularity results which state essentially that if Pf is Hölder continuous then there is a unique classical solution u(t ) such that Au and CDαt u(t ) are Hölder continuous in Jq . The paper is organized as follows. In Section 2 we recall some notations, definitions, and preliminary facts. Section 3 is devoted to the existence and uniqueness of global mild solution in H β,q of problem (1.2), then proceed to study the local mild solution in H β,q . In Section 4, we use the iteration method to obtain the existence and uniqueness of local mild solution in Jq of problem (1.2). Finally, Section 5 is concerned with the existence and regularity of classical solution in Jq of problem (1.2). 2. Preliminaries In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. Let Ω = {(x1 , . . . , xn ) : xn > 0} be open subset of Rn , where n ≥ 3. Let 1 < q < ∞. Then there is a bounded projection P called the Hodge projection on (Lq (Ω ))n , whose range is the closure of Cσ∞ (Ω ) := {u ∈ (C ∞ (Ω ))n : ∇ · u = 0, u has compact support in Ω }, and whose null space is the closure of
{u ∈ (C ∞ (Ω ))n : u = ∇φ, φ ∈ C ∞ (Ω )}. |·|q
For notational convenience, let Jq := Cσ∞ (Ω ) , which is a closed subspace of (Lq (Ω ))n . (W m,q (Ω ))n is a Sobolev space with the norm | · |m,q . A = −ν P ∆ denotes the Stokes operator in Jq whose domain is Dq (A) = Dq (∆) ∩ Jq ; here, Dq (∆) = {u ∈ (W 2,q (Ω ))n : u|∂ Ω = 0}. It is known that −A is a closed linear operator and generates the bounded analytic semigroup {e−tA } on Jq .
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So as to state our results, we need to introduce the definitions of the fractional power spaces associated with −A. For
β > 0 and u ∈ Jq , define ∞ 1 t β−1 e−tA udt . A−β u = Γ (β) 0
Then A−β is a bounded, one-to-one operator on Jq . Let Aβ be the inverse of A−β . For β > 0, we denote the space H β,q by the range of A−β with the norm
|u|H β,q = |Aβ u|q . It is easy to check that e−tA extends (or restricts) to a bounded analytic semigroup on H β,q . For more details, we refer to Von Wahl [4]. Let X be a Banach space and J be an interval of R. C (J , X ) denotes the set of all continuous X -valued functions. For 0 < ϑ < 1, C ϑ (J , X ) stands for the set of all functions which are Hölder continuous with the exponent ϑ . Let α ∈ (0, 1] and v : [0, ∞) → X . The fractional integral of order α with the lower limit zero for the function v is defined as t
Itα v(t ) =
gα (t − s)v(s)ds,
t > 0,
0
provided the right hand-side is point-wise defined on [0, ∞), where gα denotes the Riemann–Liouville kernel gα (t ) =
t α−1
,
Γ (α)
t > 0.
Further, CDαt stands the Caputo fractional derivative operator of order α ; it is defined by C α Dt
v(t ) =
d 1−α d It v(t ) − v(0) = dt dt
t
g1−α (t − s) v(t ) − v(0) ds ,
t > 0.
0
More generally, for u : [0, ∞) × Rn → Rn , Caputo fractional derivative with respect to time of the function u can be written as
∂tα u(t , x) = ∂t
t
g1−α (t − s) u(t , x) − u(0, x) ds ,
t > 0.
0
For more details, we refer the reader to Kilbas et al. [28]. Let us introduce the generalized Mittag-Leffler special functions: Eα (−t α A) =
∞
α
Mα (s)e−st A ds,
Eα,α (−t α A) =
∞
0
α
α sMα (s)e−st A ds, 0
where Mα (θ ) is Mainardi’s wright type function defined by
Mα (θ ) =
∞ k=0
θn . n!Γ (1 − α(1 + n))
Proposition 2.1. (i) Eα,α (−t α A) = 21π i Γ Eα,α (−µt α )(µI + A)−1 dµ; θ (ii) Aγ Eα,α (−t α A) = 21π i Γ µγ Eα,α (−µt α )(µI + A)−1 dµ.
θ
Proof. (i) In view of Eα,α (−t α A) =
∞ 0
α sMα (s)e−st ds = Eα,α (−t ) and Fubini theorem, we get
∞
α
α sMα (s)e−st A ds ∞ 1 α α sM α ( s) e−µst (µI + A)−1 dµds = 2π i 0 Γθ 1 α = Eα,α (−µt )(µI + A)−1 dµ, 2π i Γθ 0
where Γθ is a suitable integral path.
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(ii) A similar argument shows that
∞
α
α sMα (s)Aγ e−st A ds 0 ∞ 1 α = µγ e−µst (µI + A)−1 dµds α sM α ( s) 2π i 0 Γθ 1 = µγ Eα,α (−µt α )(µI + A)−1 dµ. 2π i Γθ
Aγ Eα,α (−t α A) =
Moreover, we have the following results. Lemma 2.1 ([29]). For t > 0, Eα (−t α A) and Eα,α (−t α A) are continuous in the uniform operator topology. Moreover, for every r > 0, the continuity is uniform on [r , ∞). Lemma 2.2 ([29]). Let 0 < α < 1. Then (i) (ii) (iii) (iv)
for all u ∈ X , limt →0+ Eα (−t α A)u = u; for all u ∈ D(A) and t > 0, CDαt Eα (−t α A)u = −AEα (−t α A)u; for all u ∈ X , Eα′ (−t α A)u = −t α−1 AEα,α (−t α A)u; for t > 0, Eα (−t α A)u = It1−α t α−1 Eα,α (−t α A)u .
Before presenting the definition of mild solution of problem (1.2), we give the following lemma for a given function h : [0, ∞) → X . For more details we refer to Zhou [30,31]. Lemma 2.3. If u(t ) = a +
t
1
Γ (α)
(t − s)α−1 Au(s) + h(s) ds,
for t ≥ 0
(2.1)
0
holds, then we have u(t ) = Eα (−t α A)a +
t
(t − s)α−1 Eα,α (−(t − s)α A)h(s)ds.
0
We rewrite (1.2) as u(t ) = a +
t
1
Γ (α)
(t − s)α−1 Au(s) + F (u(s), u(s)) + Pf (s) ds,
for t ≥ 0.
0
Inspired by above discussion, we adopt the following concepts of mild solution to problem (1.2). Definition 2.1. A function u : [0, ∞) → H β,q is called a global mild solution of problem (1.2) in H β,q , if u ∈ C ([0, ∞), H β,q ) and for t ∈ [0, ∞) u(t ) = Eα (−t α A)a +
t
(t − s)α−1 Eα,α (−(t − s)α A)F (u(s), u(s))ds +
0
t
(t − s)α−1 Eα,α (−(t − s)α A)Pf (s)ds.
(2.2)
0
Definition 2.2. Let 0 < T < ∞. A function u : [0, T ] → H β,q (or Jq ) is called a local mild solution of problem (1.2) in H β,q (or Jq ), if u ∈ C ([0, T ], H β,q ) (or C ([0, T ], Jq )) and u satisfies (2.2) for t ∈ [0, T ]. For convenience, we define two operators Φ and G as follows:
Φ (t ) =
t
(t − s)α−1 Eα,α (−(t − s)α A)f (s)ds,
0
G(u, v)(t ) =
t
(t − s)α−1 Eα,α (−(t − s)α A)F (u(s), v(s))ds.
0
In subsequent proof we use the following fixed point result. Lemma 2.4 ([1]). Let (X , ∥ · ∥X ) be a Banach space, G : X × X → X a bilinear operator and L a positive real number such that
∥G(u, v)∥X ≤ L∥u∥X ∥v∥X , Then for any u0 ∈ X with ∥u0 ∥X <
∀ u, v ∈ X . 1 , 4L
the equation u = u0 + G(u, u) has a unique solution u ∈ X .
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3. Global and local existence in H β,q Our main purpose in this section is to establish sufficient conditions for existence and uniqueness of mild solution to problem (1.2) in H β,q . To this end we assume that: (f) Pf is continuous for t > 0 and |Pf (t )|q = o(t −α(1−β) ) as t → 0 for 0 < β < 1. Lemma 3.1 ([32] (see also [10])). Let 1 < q < ∞ and β1 ≤ β2 . Then there is a constant C = C (β1 , β2 ) such that
|e−tA v|H β2 ,q ≤ Ct −(β2 −β1 ) |v|H β1 ,q ,
t >0
for v ∈ H β1 ,q . Furthermore, limt →0 t (β2 −β1 ) |e−tA v|H β2 ,q = 0. Now, we study an important technical lemma, that helps us to prove the final main theorems of this section. Lemma 3.2. Let 1 < q < ∞ and β1 ≤ β2 . Then for any T > 0, there exists a constant C1 = C1 (β1 , β2 ) > 0 such that
|Eα (−t α A)v|H β2 ,q ≤ C1 t −α(β2 −β1 ) |v|H β1 ,q and |Eα,α (−t α A)v|H β2 ,q ≤ C1 t −α(β2 −β1 ) |v|H β1 ,q for all v ∈ H β1 ,q and t ∈ (0, T ]. Furthermore, lim t α(β2 −β1 ) |Eα (−t α A)v|H β2 ,q = 0.
t →0
Proof. Let v ∈ H β1 ,q . By Lemma 3.1, we estimate
|Eα (−t α A)v|H β2 ,q ≤
∞
α
Mα (s)|e−st A v|H β2 ,q ds 0
∞
Mα (s)s
≤ C
−(β2 −β1 )
ds t −α(β2 −β1 ) |v|H β1 ,q
0
≤ C1 t −α(β2 −β1 ) |v|H β1 ,q . More precisely, Lebesgue’s dominated convergence theorem shows lim t α(β2 −β1 ) |Eα (−t α A)v|H β2 ,q ≤
t →0
∞
α
Mα (s) lim t α(β2 −β1 ) |e−st A v|H β2 ,q ds = 0. t →0
0
Similarly,
|Eα,α (−t α A)v|H β2 ,q ≤
∞
α
α sMα (s)|e−st A v|H β2 ,q ds 0 ∞ ≤ αC Mα (s)s1−(β2 −β1 ) ds t −α(β2 −β1 ) |v|H β1 ,q 0
≤ C1 t −α(β2 −β1 ) |v|H β1 ,q , where the constant C1 = C1 (α, β1 , β2 ) is such that
C1 ≥ C max
α Γ (2 − β2 + β1 ) Γ (1 − β2 + β1 ) , . Γ (1 + α(β1 − β2 )) Γ (1 + α(1 + β1 − β2 ))
3.1. Global existence in H β,q This subsection is concerned with the global mild solution of problem (1.2) in H β,q . For convenience, we denote M (t ) = sup {sα(1−β) |Pf (s)|q }, s∈(0,t ]
B1 = C1 max{B(α(1 − β), 1 − α(1 − β)), B(α(1 − γ ), 1 − α(1 − β))},
L ≥ MC1 max B(α(1 − β), 1 − 2α(γ − β)), B(α(1 − γ ), 1 − 2α(γ − β)) , where M is given later.
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Theorem 3.1. Let 1 < q < ∞, 0 < β < 1 and (f) hold. For every a ∈ H β,q , suppose that C1 |a|H β,q + B1 M∞ <
1 4L
,
(3.1)
where M∞ := sups∈(0,∞) {sα(1−β) Pf (s)}. If satisfying:
n 2q
− 21 < β , then there is a γ > max{β, 12 } and a unique function u : [0, ∞) → H β,q
(a) u : [0, ∞) → H β,q is continuous and u(0) = a; (b) u : (0, ∞) → H γ ,q is continuous and limt →0 t α(γ −β) |u(t )|H γ ,q = 0; (c) u satisfies (2.2) for t ∈ [0, ∞). Proof. Let γ =
(1+β) 2
. Define X∞ = X [∞] as the space of all curves u : (0, ∞) → H β,q such that:
(i) u : [0, ∞) → H β,q is bounded and continuous; (ii) u : (0, ∞) → H γ ,q is bounded and continuous, moreover, limt →0 t α(γ −β) |u(t )|H γ ,q = 0; with its natural norm
∥u∥X∞ = max sup |u(t )|H β,q , sup t α(γ −β) |u(t )|H γ ,q . t ≥0
t ≥0
It is obvious that X∞ is a non-empty complete metric space. From an argument of Weissler [10], we know that F : H γ ,q × H γ ,q → Jq is a bounded bilinear map, then there exists M such that for u, v ∈ H γ ,q
|F (u, v)|q ≤ M |u|H γ ,q |v|H γ ,q , |F (u, u) − F (v, v)|q ≤ M (|u|H γ ,q + |v|H γ ,q )|u − v|H γ ,q .
(3.2)
Step 1. Let u, v ∈ X∞ . We show that the operator G(u(t ), v(t )) belongs to C ([0, ∞), H β,q ) as well as C ((0, ∞), H γ ,q ). For arbitrary t0 ≥ 0 fixed and ε > 0 enough small, consider t > t0 (the case t < t0 follows analogously), we have
|G(u(t ), v(t )) − G(u(t0 ), v(t0 ))|H β,q t ≤ (t − s)α−1 |Eα,α (−(t − s)α A)F (u(s), v(s))|H β,q ds t0 t0
+ 0
(t − s)α−1 − (t0 − s)α−1 Eα,α (−(t − s)α A)F (u(s), v(s)) β,q ds H
t0 −ε
+ 0
t0
+ t0 −ε
(t0 − s)α−1 Eα,α (−(t − s)α A) − Eα,α (−(t0 − s)α A) F (u(s), v(s))H β,q ds
(t0 − s)α−1 Eα,α (−(t − s)α A) − Eα,α (−(t0 − s)α A) F (u(s), v(s))H β,q ds
:= I11 (t ) + I12 (t ) + I13 (t ) + I14 (t ). We estimate each of the four terms separately. For I11 (t ), in view of Lemma 3.2, we obtain I11 (t ) ≤ C1
t
(t − s)α(1−β)−1 |F (u(s), v(s))|q ds
t0 t
≤ MC1
(t − s)α(1−β)−1 |u(s)|H γ ,q |v(s)|H γ ,q ds
t0 t
≤ MC1
(t − s)α(1−β)−1 s−2α(γ −β) ds sup {s2α(γ −β) |u(s)|H γ ,q |v(s)|H γ ,q } s∈[0,t ]
t0
= MC1
1 t0 /t
(1 − s)α(1−β)−1 s−2α(γ −β) ds sup {s2α(γ −β) |u(s)|H γ ,q |v(s)|H γ ,q }. s∈[0,t ]
By the properties of the Beta function, there exists δ > 0 small enough such that for 0 < t − t0 < δ ,
1 t0 /t
(1 − s)α(1−β)−1 s−2α(γ −β) ds → 0,
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which follows that I11 (t ) tends to 0 as t − t0 → 0. For I12 (t ), since I12 (t ) ≤ C1
t0
(t0 − s)α−1 − (t − s)α−1 (t − s)−αβ |F (u(s), v(s))|q ds
0 t0
≤ MC1
(t0 − s)α−1 − (t − s)α−1 (t − s)−αβ s−2α(γ −β) ds sup {s2α(γ −β) |u(s)|H γ ,q |v(s)|H γ ,q }, s∈[0,t0 ]
0
noting that
t0
|(t0 − s)α−1 − (t − s)α−1 |(t − s)−αβ s−2α(γ −β) ds t0 t0 (t0 − s)α−1 (t − s)−αβ s−2α(γ −β) ds (t − s)α−1 (t − s)−αβ s−2α(γ −β) ds + ≤ 0 0 t0 (t0 − s)α(1−β)−1 s−2α(γ −β) ds ≤2
0
0
= 2B(α(1 − β), 1 − 2α(γ − β)), then by Lebesgue’s dominated convergence theorem, we have t0
(t0 − s)α−1 − (t − s)α−1 (t − s)−αβ s−2α(γ −β) ds → 0,
as t → t0 ,
0
one deduces that limt →t0 I12 (t ) = 0. For I13 (t ), since I13 (t ) ≤
t0 −ε
0
(t0 − s)α−1 Eα,α (−(t − s)α A) + Eα,α (−(t0 − s)α A) F (u(s), v(s))H β,q ds
t0 −ε
(t0 − s)α−1 (t − s)−αβ + (t0 − s)−αβ |F (u(s), v(s))|q ds 0 t0 −ε ≤ 2MC1 (t0 − s)α(1−β)−1 s−2α(γ −β) ds sup {s2α(γ −β) |u(s)|H γ ,q |v(s)|H γ ,q },
≤
s∈[0,t0 ]
0
using Lebesgue’s dominated convergence theorem again, the fact from the uniform continuity of the operator Eα,α (−t α A) due to Lemma 2.1 shows lim I13 (t ) =
t0 −ε
t → t0
(t0 − s)α−1 lim Eα,α (−(t − s)α A) − Eα,α (−(t0 − s)α A) F (u(s), v(s))H β,q ds t →t0
0
= 0. For I14 (t ), by immediate calculation, we estimate I14 (t ) ≤
t0
(t0 − s)α−1 (t − s)−αβ + (t0 − s)−αβ |F (u(s), v(s))|q ds
t0 −ε
t0
≤ 2MC1
(t0 − s)α(1−β)−1 s−2α(γ −β) ds
t0 −ε
→ 0,
sup
{s2α(γ −β) |u(s)|H γ ,q |v(s)|H γ ,q }
s∈[t0 −ε,t0 ]
as ε → 0
according to the properties of the Beta function. Thenceforth, it follows
|G(u(t ), v(t )) − G(u(t0 ), v(t0 ))|H β,q → 0,
as t → t0 .
The continuity of the operator G(u, v) evaluated in C ((0, ∞), H γ ,q ) follows by the similar discussion as above. So, we omit the details.
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Step 2. We show that the operator G : X∞ × X∞ → X∞ is a continuous bilinear operator. By Lemma 3.2, we have
t |G(u(t ), v(t ))|H β,q ≤ (t − s)α−1 Eα,α (−(t − s)α A)F (u(s), v(s))ds 0 H β,q t (t − s)α(1−β)−1 |F (u(s), v(s))|q ds ≤ C1 0 t (t − s)α(1−β)−1 s−2α(γ −β) ds sup {s2α(γ −β) |u(s)|H γ ,q |v(s)|H γ ,q } ≤ MC1 s∈[0,t ]
0
= MC1 B(α(1 − β), 1 − 2α(γ − β))∥u∥X∞ ∥v∥X∞ and
t α−1 α ≤ (t − s) Eα,α (−(t − s) A)F (u(s), v(s))ds 0 H γ ,q t ≤ C1 (t − s)α(1−γ )−1 |F (u(s), v(s))|q ds 0 t ≤ MC1 (t − s)α(1−γ )−1 s−2α(γ −β) ds sup {s2α(γ −β) |u(s)|H γ ,q |v(s)|H γ ,q }
|G(u(t ), v(t ))|H γ ,q
s∈[0,t ]
0
= MC1 t −α(γ −β) B(α(1 − γ ), 1 − 2α(γ − β))∥u∥X∞ ∥v∥X∞ , it follows that sup t α(γ −β) |G(u(t ), v(t ))|H γ ,q ≤ MC1 B(α(1 − γ ), 1 − 2α(γ − β))∥u∥X∞ ∥v∥X∞ .
t ∈[0,∞)
More precisely, lim t α(γ −β) |G(u(t ), v(t ))|H γ ,q = 0.
t →0
Hence, G(u, v) ∈ X∞ and ∥G(u(t ), v(t ))∥X∞ ≤ L∥u∥X∞ ∥v∥X∞ . Step 3. We verify that (c) holds. Let 0 < t0 < t. Since
|Φ (t ) − Φ (t0 )|H β,q t ≤ (t − s)α−1 |Eα,α (−(t − s)α A)Pf (s)|H β,q ds + t0
t0
(t0 − s)α−1 − (t − s)α−1 |Eα,α (−(t − s)α A)Pf (s)|H β,q ds
0 t0 −ε
+ 0
t0
+
t0 −ε t
≤ C1
(t0 − s)α−1 Eα,α (−(t − s)α A) − Eα,α (−(t0 − s)α A) Pf (s)H β,q ds
(t0 − s)α−1 Eα,α (−(t − s)α A) − Eα,α (−(t0 − s)α A) Pf (s)H β,q ds
(t − s)α(1−β)−1 |Pf (s)|q ds + C1
t0
t0
(t0 − s)α−1 − (t − s)α−1 (t − s)−αβ |Pf (s)|q ds
0 t0 −ε
(t0 − s)α−1 Eα,α (−(t − s)α A) − Eα,α (−(t0 − s)α A) Pf (s)H β,q ds + 2C1
+ C1 0
≤ C1 M (t )
+ C1 M (t )
t
(t − s)α(1−β)−1 s−α(1−β) ds + C1 M (t )
t0
t0
t0
(t0 − s)α(1−β)−1 |Pf (s)|q ds
t0 −ε
(t − s)α−1 − (t0 − s)α−1 s−α(1−β) ds
0
t0 −ε
(t0 − s)α−1 Eα,α (−(t − s)α A) − Eα,α (−(t0 − s)α A) Pf (s)
0
+ 2C1 M (t )
t0
H β,q
ds
(t0 − s)α(1−β)−1 s−α(1−β) ds.
t0 −ε
By the properties of the Beta function, the first two integrals and the last integral tend to 0 as t → t0 as well as ε → 0. In view of Lemma 2.1, the third integral also goes to 0 as t → t0 , which implies
|Φ (t ) − Φ (t0 )|H β,q → 0 as t → t0 . The continuity of Φ (t ) evaluated in H γ ,q follows by the similar argument as above.
Y. Zhou, L. Peng / Computers and Mathematics with Applications (
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On the other hand, we have
|Φ (t )|H β,q
t α−1 α ≤ (t − s) Eα,α (−(t − s) A)Pf (s)ds 0 H β,q t (t − s)α(1−β)−1 |Pf (s)|q ds ≤ C1 0 t ≤ C1 M (t ) (t − s)α(1−β)−1 s−α(1−β) ds 0
= C1 M (t )B(α(1 − β), 1 − α(1 − β)),
(3.3)
and
t |Φ (t )|H γ ,q ≤ (t − s)α−1 Eα,α (−(t − s)α A)Pf (s)ds 0 H γ ,q t ≤ C1 (t − s)α(1−γ )−1 |Pf (s)|q ds 0 t ≤ C1 M (t ) (t − s)α(1−γ )−1 s−α(1−β) ds 0
= t −α(γ −β) C1 M (t )B(α(1 − γ ), 1 − α(1 − β)). More precisely, t α(γ −β) |Φ (t )|H γ ,q ≤ C1 M (t )B(α(1 − γ ), 1 − α(1 − β)) → 0,
as t → 0,
since M (t ) → 0 as t → 0 due to assumption (f). This ensures that Φ (t ) ∈ X∞ and ∥Φ (t )∥∞ ≤ B1 M∞ . For a ∈ H β,q . By Lemma 2.1, it is easy to see that Eα (−t α A)a ∈ C ([0, ∞), H β,q )
and Eα (−t α A)a ∈ C ((0, ∞), H γ ,q ).
This, together with Lemma 3.2, implies that for all t ∈ (0, T ], Eα (−t α A)a ∈ X∞ , t
α(γ −β)
Eα (−t α A)a ∈ C ([0, ∞), H γ ,q ),
∥Eα (−t α A)a∥X∞ ≤ C1 |a|H β,q . According to (3.1), the inequality
∥Eα (−t α A)a + Φ (t )∥X∞ ≤ ∥Eα (−t α A)a∥X∞ + ∥Φ (t )∥X∞ ≤
1 4L
holds, which yields that F has a unique fixed point. Step 4. To show that u(t ) → a in H β,q as t → 0. We need to verify t
lim
t →0
t
lim
t →0
(t − s)α−1 Eα,α (−(t − s)α A)Pf (s)ds = 0,
0
(t − s)α−1 Eα,α (−(t − s)α A)F (u(s), u(s))ds = 0
0
in H β,q . In fact, it is obvious that limt →0 Φ (t ) = 0 (limt →0 M (t ) = 0) owing to (3.3). In addition,
t (t − s)α−1 Eα,α (−(t − s)α A)F (u(s), u(s))ds β,q 0 H t α(1−β)−1 ≤ C1 ( t − s) |F (u(s), u(s))|q ds 0 t ≤ MC1 (t − s)α(1−β)−1 |u(s)|2H γ ,q ds 0 t ≤ MC1 (t − s)α(1−β)−1 s−2α(γ −β) ds sup {s2α(γ −β) |u(s)|2H γ ,q } 0
s∈[0,t ]
= MC1 B(α(1 − β), 1 − 2α(γ − β)) sup {s2α(γ −β) |u(s)|2H γ ,q } → 0 as t → 0. s∈[0,t ]
10
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3.2. Local existence in H β,q In this subsection, we study the local mild solution of problem (1.2) in H β,q . Theorem 3.2. Let 1 < q < ∞, 0 < β < 1 and (f) hold. Suppose n 2q
−
1
< β.
2
(3.4)
Then there is a γ > max{β, 12 } such that for every a ∈ H β,q there exist T∗ > 0 and a unique function u : [0, T∗ ] → H β,q satisfying: (a) u : [0, T∗ ] → H β,q is continuous and u(0) = a; (b) u : (0, T∗ ] → H γ ,q is continuous and limt →0 t α(γ −β) |u(t )|H γ ,q = 0; (c) u satisfies (2.2) for t ∈ [0, T∗ ]. Proof. Let γ =
(1+β) 2
. Fix a ∈ H β,q . Let X = X [T ] be the space of all curves u : (0, T ] → H β,q such that:
(i) u : [0, T ] → H β,q is continuous; (ii) u : (0, T ] → H γ ,q is continuous and limt →0 t α(γ −β) |u(t )|H γ ,q = 0; with its natural norm
∥u∥X = sup {t α(γ −β) |u(t )|H γ ,q }. t ∈[0,T ]
Similar to the proof of Theorem 3.1, it is easy to claim that G : X × X → X is continuous linear map and Φ (t ) ∈ X . By Lemma 2.1, it is easy to see that for all t ∈ (0, T ], Eα (−t α A)a ∈ C ([0, T ], H β,q ),
Eα (−t α A)a ∈ C ((0, T ], H γ ,q ). From Lemma 3.2, it follows that Eα (−t α A)a ∈ X , t α(γ −β) Eα (−t α A)a ∈ C ([0, T ], H γ ,q ). Hence, let T∗ > 0 be sufficiently small such that
∥Eα (−t α A)a + Φ (t )∥X [T∗ ] ≤ ∥Eα (−t α A)a∥X [T∗ ] + ∥Φ (t )∥X [T∗ ] < which implies that F has a unique fixed point due to Lemma 2.4.
1 4L
,
4. Local existence in Jq This section is devoted to consideration of local mild solution to problem (1.2) in Jq by means of the iteration method. Let
γ =
(1+β) 2
.
Theorem 4.1. Let 1 < q < ∞, 0 < β < 1 and (f) hold. Suppose that a ∈ H β,q
with
n 2q
−
1 2
< β.
Then problem (1.2) has a unique mild solution u in Jq for a ∈ H β,q . Moreover, u is continuous on [0, T ], Aγ u is continuous in (0, T ] and t α(γ −β) Aγ u(t ) is bounded as t → 0. Proof. Step 1. Set K (t ) := sup sα(γ −β) |Aγ u(s)|q s∈(0,t ]
and
Ψ (t ) := G(u, u)(t ) =
t
(t − s)α−1 Eα,α (−(t − s)α A)F (u(s), u(s))ds.
0
As an immediate consequence of Step 2 in Theorem 3.1, Ψ (t ) is continuous in [0, T ], Aγ Ψ (t ) exists and is continuous in (0, T ] with
|Aγ Ψ (t )|q ≤ MC1 B(α(1 − γ ), 1 − 2α(γ − β))K 2 (t )t −α(γ −β) .
(4.1)
Y. Zhou, L. Peng / Computers and Mathematics with Applications (
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We also consider the integral Φ (t ). Since (f) holds, the inequality
|Pf (s)|q ≤ M (t )sα(1−β) is satisfied with a continuous function M (t ). From Step 3 in Theorem 3.1, we derive that Aγ Φ (t ) is continuous in (0, T ] with
|Aγ Φ (t )|q ≤ C1 M (t )B(α(1 − γ ), 1 − α(1 − β))t −α(γ −β) .
(4.2) γ
For |Pf (t )|q = o(t ) as t → 0, we have M (t ) = 0. Here (4.2) means |A Φ (t )|q = o(t We prove that Φ is continuous in Jq . In fact, take 0 ≤ t0 < t < T , we have −α(1−β)
|Φ (t ) − Φ (t0 )|q t (t − s)α−1 |Pf (s)|q ds + C3 ≤ C3
t0
−α(γ −β)
) as t → 0.
(t0 − s)α−1 − (t − s)α−1 |Pf (s)|q ds
0
t0 t0 −ε
+ C3
α−1
(t0 − s)
∥Eα,α (−(t − s)α A) − Eα,α (−(t0 − s)α A)∥ |Pf (s)|q ds
0 t0
(t0 − s)α−1 |Pf (s)|q ds
+ 2C3 t0 −ε
≤ C3 M (t )
t
(t − s)α−1 s−α(1−β) ds + C3 M (t )
t0
+ C3 M (t )
t0
(t − s)α−1 − (t0 − s)α−1 s−α(1−β) ds
0 t0 −ε
(t0 − s)α−1 s−α(1−β) ds sup ∥Eα,α (−(t − s)α A) − Eα,α (−(t0 − s)α A)∥ s∈[0,t −ε]
0
+ 2C3 M (t )
t0
(t0 − s)α−1 s−α(1−β) ds → 0,
as t → t0
t0 −ε
by previous discussion. Further, we consider the function Eα (−t α A)a. It is obvious by Lemma 3.2 that
|Aγ Eα (−t α A)a|q ≤ C1 t −α(γ −β) |Aβ a|q = C1 t −α(γ −β) |a|H β,q , lim t α(γ −β) |Aγ Eα (−t α A)a|q = lim t α(γ −β) |Eα (−t α A)a|H γ ,q = 0.
t →0
t →0
Step 2. Now we construct the solution by the successive approximation: u0 (t ) = Eα (−t α A)a + Φ (t ), un+1 (t ) = u0 (t ) + G(un , un )(t ),
n = 0, 1, 2 . . . .
(4.3)
Making use of above results, we know that Kn (t ) := sup sα(γ −β) |Aγ un (s)|q s∈(0,t ]
are continuous and increasing functions on [0, T ] with Kn (0) = 0. Furthermore, in virtue of (4.2) and (4.2), Kn (t ) fulfils the following inequality Kn+1 (t ) ≤ K0 (t ) + MC1 B(α(1 − γ ), 1 − 2α(γ − β))Kn2 (t ).
(4.4)
For K0 (0) = 0, we choose a T > 0 such that 4MC1 B(α(1 − γ ), 1 − 2α(γ − β))K0 (T ) < 1.
(4.5)
Then a fundamental consideration of (4.4) ensures that the sequence {Kn (T )} is bounded, i.e., Kn (T ) ≤ ρ(T ),
n = 0, 1, 2, . . . ,
where
ρ(t ) =
1−
√
1 − 4MC1 B(α(1 − γ ), 1 − 2α(γ − β))K0 (t ) 2MC1 B(α(1 − γ ), 1 − 2α(γ − β))
.
Analogously, for any t ∈ (0, T ], Kn (t ) ≤ ρ(t ) holds. In the same way we note that ρ(t ) ≤ 2K0 (t ). Let us consider the equality
wn+1 (t ) =
t
0
(t − s)α−1 Eα,α (−(t − s)α A)[F (un+1 (s), un+1 (s)) − F (un (s), un (s))]ds,
12
Y. Zhou, L. Peng / Computers and Mathematics with Applications (
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where wn = un+1 − un , n = 0, 1, . . . , and t ∈ (0, T ]. Writing Wn (t ) := sup sα(γ −β) |Aγ wn (s)|q . s∈(0,t ]
On account of (3.1), we have
|F (un+1 (s), un+1 (s)) − F (un (s), un (s))|q ≤ M (Kn+1 (s) + Kn (t ))Wn (s)s−2α(γ −β) , which follows from Step 2 in Theorem 3.1 that t α(γ −β) |Aγ wn+1 (t )|q ≤ 2MC1 B(α(1 − γ ), 1 − α(1 − β))ρ(t )Wn (t ). This inequality gives Wn+1 (T ) ≤ 2MC1 B(α(1 − γ ), 1 − 2α(γ − β))ρ(T )Wn (T )
≤ 4MC1 B(α(1 − γ ), 1 − 2α(γ − β))K0 (T )Wn (T ).
(4.6)
According to (4.5) and (4.6), it is easy to see that lim
Wn+1 (T )
n→0
Wn (T )
< 4MC1 B(α(1 − γ ), 1 − 2α(γ − β)) < 1,
α(γ −β) γ thus the series A wn (t ) converges uniformly for t ∈ (0, T ], n=0 Wn (T ) converges. It shows that the series n=0 t therefore, the sequence {t α(γ −β) Aγ un (t )} converges uniformly in (0, T ]. This implies that
∞
∞
lim un (t ) = u(t ) ∈ D(Aγ )
n→∞
and lim t α(γ −β) Aγ un (t ) = t α(γ −β) Aγ u(t ) uniformly,
n→∞
since A−γ is bounded and Aγ is closed. Accordingly, the function K (t ) = sups∈(0,t ] sα(γ −β) |Aγ u(s)|q also satisfies K (t ) ≤ ρ(t ) ≤ 2K0 (t ),
t ∈ (0, t ].
(4.7)
and
ςn := sup s2α(γ −β) |F (un (s), un (s)) − F (u(s), u(s))|q s∈(0,T ]
≤ M (Kn (T ) + K (T )) sup sα(γ −β) |Aγ (un (s) − u(s))|q → 0, s∈(0,T ]
as n → ∞.
Finally, it remains to verify that u is a mild solution of problem (1.2) in [0, T ]. Since
|G(un , un )(t ) − G(u, u)(t )|q ≤
t
(t − s)α−1 ςn s−2α(γ −β) ds = t αβ ςn → 0,
(n → ∞),
0
we have G(un , un )(t ) → G(u, u)(t ). Taking the limits on both sides of (4.2), we derive u(t ) = u0 (t ) + G(u, u)(t ).
(4.8)
Let u(0) = a, we find that (4.8) holds for t ∈ [0, T ] and u ∈ C ([0, T ], Jq ). What is more, the uniform convergence of t α(γ −β) Aγ un (t ) to t α(γ −β) Aγ u(t ) derives the continuity of Aγ u(t ) on (0, T ]. From (4.7) and K0 (0) = 0, we get that |Aγ u(t )|q = o(t −α(γ −β) ) is obvious. Step 3. We prove that the mild solution is unique. Suppose that u and v are mild solutions of problem (1.2). Let w = u −v , we consider the equality
w(t ) =
t
(t − s)α−1 Eα,α (−(t − s)α A)[F (u(s), u(s)) − F (v(s), v(s))]ds.
0
Introducing the functions
K (t ) := max{ sup sα(γ −β) |Aγ u(s)|q , sup sα(γ −β) |Aγ v(s)|q }. s∈(0,t ]
s∈(0,t ]
By (3.1) and Lemma 3.2, we get
|Aγ w(t )|q ≤ MC1 K (t )
t
(t − s)α(1−γ )−1 s−α(γ −β) |Aγ w(s)|q ds.
0
Gronwall inequality shows that Aγ w(t ) = 0 for t ∈ (0, T ]. This implies that w(t ) = u(t ) − v(t ) ≡ 0 for t ∈ [0, T ]. Therefore the mild solution is unique.
Y. Zhou, L. Peng / Computers and Mathematics with Applications (
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5. Regularity In this section, we consider the regularity of a solution u which satisfies problem (1.2). Throughout this part we assume that:
(f1 ) Pf (t ) is Hölder continuous with an exponent ϑ ∈ (0, α(1 − γ )), that is, |Pf (t ) − Pf (s)|q ≤ L|t − s|ϑ ,
for all 0 < t , s ≤ T .
Definition 5.1. A function u : [0, T ] → Jq is called a classical solution of problem (1.2), if u ∈ C ([0, T ], Jq ) with (t ) ∈ C ((0, T ], Jq ), which takes values in D(A) and satisfies (1.2) for all t ∈ (0, T ].
C t Dt u
Lemma 5.1. Let (f1 ) be satisfied. If
Φ1 (t ) :=
t
(t − s)α−1 Eα,α (−(t − s)α A) Pf (s) − Pf (t ) ds,
for t ∈ (0, T ],
0
then Φ1 (t ) ∈ D(A) and AΦ1 (t ) ∈ C ϑ ([0, T ], Jq ). Proof. For fixed t ∈ (0, T ], from Lemma 3.2 and (f1 ), we have
(t − s)α−1 |AEα,α (−(t − s)α A) Pf (s) − Pf (t ) |q ≤ (t − s)−1 |Pf (s) − Pf (t )|q ≤ C1 L(t − s)ϑ−1 ∈ L1 ([0, T ], Jq ),
(5.1)
then
t
(t − s)α−1 AEα,α (−(t − s)α A) Pf (s) − Pf (t ) q ds 0 t C1 K ϑ ≤ C1 L (t − s)ϑ−1 ds ≤ t < ∞. ϑ 0
|AΦ1 (t )|q ≤
By the closeness of A, we obtain Φ1 (t ) ∈ D(A). We need to show that AΦ1 (t ) is Hölder continuous. Since
d α−1 t Eα,α (−µt α ) = t α−2 Eα,α−1 (−µt α ), dt then
d α−1 1 t AEα,α (−t α A) = dt 2π i = =
1
Γθ
2π i 1 2π i
Γθ
t α−2 Eα,α−1 (−µt α )A(µI + A)−1 dµ t α−2 Eα,α−1 (−µt α )dµ −
−t
α−2
Γθ′
1
1
2π i 1
Eα,α−1 (ξ ) α dξ − t 2π i
Γθ
t α−2 µEα,α−1 (−µt α )(µI + A)−1 dµ
t Γθ′
α−2
ξ
Eα,α−1 (ξ ) α t
−1 ξ 1 − αI +A dξ . α t
t
C In view of ∥(µI + A)−1 ∥ ≤ |µ| , we derive that
d α−1 α ≤ C α t −2 , t AE (− t A ) α,α dt
0 < t ≤ T.
By the mean value theorem, for every 0 < s < t ≤ T , we have
t d α−1 α ∥t α−1 AEα,α (−t α A) − sα−1 AEα,α (−sα A)∥ = τ AE (−τ A ) d τ α,α s dτ t d α−1 α ≤ dτ τ AEα,α (−τ A) dτ s t ≤ Cα τ −2 d τ = C α s −1 − t −1 . s
(5.2)
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Y. Zhou, L. Peng / Computers and Mathematics with Applications (
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Let h > 0 be such that 0 < t < t + h ≤ T , then AΦ1 (t + h) − AΦ1 (t ) =
t
(t + h − s)α−1 AEα,α (−(t + h − s)α A) 0 − (t − s)α−1 AEα,α (−(t − s)α A) Pf (s) − Pf (t ) ds t (t + h − s)α−1 AEα,α (−(t + h − s)α A) Pf (t ) − Pf (t + h) ds + 0 t +h
+
(t + h − s)α−1 AEα,α (−(t + h − s)α A) Pf (s) − Pf (t + h) ds
t
:= h1 (t ) + h2 (t ) + h3 (t ).
(5.3)
We estimate each of the three terms separately. For h1 (t ), from (5.2) and (f1 ), we have t
∥(t + h − s)α−1 AEα,α (−(t + h − s)α A) − (t − s)α−1 AEα,α (−(t − s)α A)∥ |Pf (s) − Pf (t )|q ds t ≤ Cα Lh (t + h − s)−1 (t − s)ϑ−1 ds 0 t ≤ Cα Lh (s + h)−1 (t − s)ϑ−1 ds
|h1 (t )|q ≤
0
0 h
h
≤ Cα L
s+h
0
sϑ−1 ds + Cα Lh
∞
h
s s+h
sϑ−1 ds
≤ Cα Lhϑ .
(5.4)
For h2 (t ), we use Lemma 3.2 and (f1 ), t
(t + h − s)α−1 |AEα,α (−(t + h − s)α A) Pf (t ) − Pf (t + h) |q ds 0 t ≤ C1 (t + h − s)−1 |Pf (t ) − Pf (t + h)|q ds 0 t ≤ C1 Lhϑ (t + h − s)−1 ds
|h2 (t )|q ≤
0
= C1 L[ln h − ln(t + h)]hϑ .
(5.5)
Furthermore, for h3 (t ), by Lemma 3.2 and (f1 ), we have
|h3 (t )|q ≤
t +h
(t + h − s)α−1 |AEα,α (−(t + h − s)α A) Pf (s) − Pf (t + h) |q ds
t t +h
(t + h − s)−1 |Pf (s) − Pf (t + h)|q ds
≤ C1 t
t +h
≤ C1 L
(t + h − s)ϑ−1 ds = C1 L
t
hϑ
ϑ
.
Combining (5.4), (5.5) with (5.6), we deduce that AΦ1 (t ) is Hölder continuous.
(5.6)
Theorem 5.1. Let the assumptions of Theorem 4.1 be satisfied. If (f1 ) holds, then for every a ∈ D(A), the mild solution of (1.2) is a classical one. Proof. For a ∈ D(A). Then Lemma 2.2(ii) ensures that u(t ) = Eα (−t α A)a (t > 0) is a classical solution to the following problem Dαt u = −Au, u(0) = a.
C
t > 0,
Step 1. We verify that
Φ (t ) =
t
0
(t − s)α−1 Eα,α (−(t − s)α A)Pf (s)ds
Y. Zhou, L. Peng / Computers and Mathematics with Applications (
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15
is a classical solution to the problem Dαt u = −Au + Pf (t ), u(0) = 0.
C
t > 0,
It follows from Theorem 4.1 that Φ ∈ C ([0, T ], Jq ). We rewrite Φ (t ) = Φ1 (t ) + Φ2 (t ), where
Φ1 (t ) =
Φ2 (t ) =
t
(t − s)α−1 Eα,α (−(t − s)α A) Pf (s) − Pf (t ) ds,
0 t
(t − s)α−1 Eα,α (−(t − s)α A)Pf (t )ds.
0
According to Lemma 5.1, we know that Φ1 (t ) ∈ D(A). To prove the same conclusion for Φ2 (t ). By Lemma 2.2(iii), we notice that AΦ2 (t ) = Pf (t ) − Eα (−t α A)Pf (t ). Since (f1 ) holds, it follows that
|AΦ2 (t )|q ≤ (1 + C1 )|Pf (t )|q , thus
Φ2 (t ) ∈ D(A) for t ∈ (0, T ] and AΦ2 (t ) ∈ C ν ((0, T ], Jq ).
(5.7)
Next, we prove CDαt Φ ∈ C ((0, T ], Jq ). In view of Lemma 2.2(iv) and Φ (0) = 0, we have C α Dt Φ
(t ) =
d d 1−α It Φ ( t ) = (Eα (−t α A) ∗ Pf ). dt dt
It remains to prove that Eα (t α A) ∗ Pf is continuously differentiable in Jq . Let 0 < h ≤ T − t, one derives the following: 1 h
α
α
Eα (−(t + h) A) ∗ Pf − Eα (−t A) ∗ Pf =
t
0
1
+
1 Eα (−(t + h − s)α A)Pf (s) − Eα (−(t − s)α A)Pf (s) ds h t +h
h
Eα (−(t + h − s)α A)Pf (s)ds.
t
Notice that t
1
|Eα (−(t + h − s)α A)Pf (s) − Eα (−(t − s)α A)Pf (s)|q ds 1 t 1 t α |Eα (−(t + h − s) A)Pf (s)|q ds + C1 |Eα (−(t − s)α A)Pf (s)|q ds ≤ C1 h 0 h 0 1 t 1 t ≤ C1 M (t ) (t + h − s)−α s−α(1−β) ds + C1 M (t ) (t − s)−α s−α(1−β) ds h
0
≤ C1 M (t )
h 1 h
h
0
(t + h)1−α + t
1−α
0
B(1 − α, 1 − α(1 − β)),
then using the dominated convergence theorem, we find t
1
lim
h→0
h
0
Eα (−(t + h − s)α A)Pf (s) − Eα (−(t − s)α A)Pf (s) ds =
t
(t − s)α−1 AEα,α (−(t − s)α A)Pf (s)ds
0
= AΦ (t ). On the other hand, 1 h
t +h
Eα (−(t + h − s)α A)Pf (s)ds
t
= =
1 h 1 h
h
Eα (−sα A)Pf (t + h − s)ds
0 h
Eα (−sα A) Pf (t + h − s) − Pf (t − s) ds +
0
1 h
h
Eα (−sα A) Pf (t − s) − Pf (t ) ds +
0
1 h
h
0
Eα (−sα A)Pf (t )ds.
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Y. Zhou, L. Peng / Computers and Mathematics with Applications (
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From Lemmas 2.1, 3.2 and (f1 ), we have
h 1 α Eα (−s A) Pf (t + h − s) − Pf (t − s) ds ≤ C1 Lhϑ , h 0 q h 1 hϑ Eα (−sα A) Pf (t − s) − Pf (t ) ds ≤ C1 L . h ϑ +1 0
q
Also Lemma 2.2(i) gives that limh→0 lim
h→0
1 h
t +h
h 1 h
0
Eα (sα A)Pf (t )ds = Pf (t ). Hence
Eα ((t + h − s)α A)Pf (s)ds = Pf (t ).
t
We deduce that Eα (t α A) ∗ Pf is differentiable at t+ and dtd Eα (t α A) ∗ Pf + = AΦ (t ) + Pf (t ). Similarly, Eα (t α A) ∗ Pf is differentiable at t− and dtd Eα (t α A) ∗ Pf − = AΦ (t ) + Pf (t ).
We show that AΦ = AΦ1 + AΦ2 ∈ C ((0, T ], Jq ). In fact, it is clear that Φ2 (t ) = Pf (t )− Eα (t α A)Pf (t ) due to Lemma 2.2(iii), which is continuous in view of Lemma 2.1. Furthermore, according to Lemma 5.1, we know that AΦ1 (t ) is also continuous. Consequently, CDαt Φ ∈ C ((0, T ], Jq ). Step 2. Let u be the mild solution of (1.2). To prove that F (u, u) ∈ C ϑ ((0, T ], Jq ), in view of (3.1), we have to verify that Aγ u is Hölder continuous in Jq . Take h > 0 such that 0 < t < t + h. Denote ϕ(t ) := Eα (−t α A)a, by Lemmas 2.2(iv) and 3.2, then
t +h −sα−1 Aγ Eα,α (−sα A)ads |Aγ ϕ(t + h) − Aγ ϕ(t )|q = t q t +h ≤ sα−1 |Aγ −β Eα,α (−sα A)Aβ a|q ds t t +h
≤ C1
sα(1+β−γ )−1 ds|Aβ a|q
t
=
C1 |a|H β,q
(t + h)α(1+β−γ ) − t α(1+β−γ )
α(1 + β − γ ) C1 |a|H β,q hα(1+β−γ ) . ≤ α(1 + β − γ )
Thus, Aγ ϕ ∈ C ϑ ((0, T ], Jq ). For every small ε > 0, take h such that ε ≤ t < t + h ≤ T , since
|Aγ Φ (t + h) − Aγ Φ (t )|q t +h α−1 γ α ≤ (t + h − s) A Eα,α (−(t + h − s) A)Pf (s)ds t q t + Aγ (t + h − s)α−1 Eα,α (−(t + h − s)α A) − (t − s)α−1 Eα,α (−(t − s)α A) Pf (s)ds 0
q
= φ1 (t ) + φ2 (t ). Applying Lemma 3.2 and (f), we get
φ1 (t ) ≤ C1
t +h
(t + h − s)α(1−γ )−1 |Pf (s)|q ds
t
≤ C1 M ( t )
t +h
(t + h − s)α(1−γ )−1 s−α(1−β) ds
t
≤ M (t ) ≤ M (t )
C1
α(1 − γ ) C1
α(1 − γ )
hα(1−γ ) t −α(1−β) hα(1−γ ) ε −α(1−β) .
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To estimate φ2 , we give the inequality
d α−1 γ 1 t A Eα,α (−t α A) = dt 2π i
µγ t α−2 Eα,α−1 (−µt α )(µI + A)−1 dµ
Γ
1
=
2π i
Γ′
−1 1 ξ ξ γ α−2 t Eα,α−1 (ξ ) − α I + A dξ , − − α α t
t
t
this yields that ∥ dt t α−1 Aγ Eα,α (−t α A) ∥ ≤ Cα t α(1−γ )−2 . The mean value theorem shows t d α−1 γ α−1 γ α α−1 γ α α ∥t A Eα,α (−t A) − s A Eα,α (−s A)∥ ≤ dτ τ A Eα,α (−τ A) dτ s t τ α(1−γ )−2 dτ = Cα sα(1−γ )−1 − t α(1−γ )−1 , ≤ Cα d
s
thus
φ2 (t ) ≤
t
0
γ A (t + h − s)α−1 Eα,α (−(t + h − s)α A) − (t − s)α−1 Eα,α (−(t − s)α A) Pf (s) ds q
t
(t − s)α(1−γ )−1 − (t + h − s)α(1−γ )−1 |Pf (s)|q ds 0 t t +h ≤ Cα M (t ) (t − s)α(1−γ )−1 s−α(1−β) ds − (t − s + h)α(1−γ )−1 s−α(1−β) ds
≤
0
0 t +h
α(1−γ )−1 −α(1−β)
+ Cα M ( t ) (t − s + h) s ds t ≤ Cα M (t ) t α(β−γ ) − (t + h)α(β−γ ) B(α(1 − γ ), 1 − α(1 − β)) + Cα M (t )hα(1−γ ) t −α(1−β) ≤ Cα M (t )hα(γ −β) [ε(ε + h)]α(β−γ ) + Cα M (t )hα(1−γ ) ε−α(1−β) , which ensures that Aγ Φ ∈ C ϑ ([ε, T ], Jq ). Therefore Aγ Φ ∈ C ϑ ((0, T ], Jq ) due to arbitrary ε . Recall
Ψ (t ) =
t
(t − s)α−1 Eα,α (−(t − s)α A)F (u(s), u(s))ds.
0
Since |F (u(s), u(s))|q ≤ MK 2 (t )s−2α(γ −β) , where K (t ) := sups∈[0,t ] sα(γ −β) |u(s)|H γ ,q is continuous and bounded in (0, T ]. A similar argument enable us to give the Hölder continuity of Aγ Ψ in C ϑ ((0, T ], Jq ). Therefore, we have Aγ u(t ) = Aγ ϕ(t ) + Aγ Φ (t ) + Aγ Ψ (t ) ∈ C ϑ ((0, T ], Jq ). Since F (u, u) ∈ C ϑ ((0, T ], Jq ) is proved, according to Step 2, this yields that CDαt Ψ ∈ C ((0, T ], Jq ), AΨ ∈ C ((0, T ], Jq ) and C α Dt Ψ = −AΨ + F (u, u). In this way we obtain that CDαt u ∈ C ((0, T ], Jq ), Au ∈ C ((0, T ], Jq ) and CDαt u = −Au + F (u, u) + Pf , we conclude that u is a classical solution. Theorem 5.2. Assume that (f1 ) holds. If u is a classical solution of (1.2), then Au ∈ C ν ((0, T ], Jq ) and CDαt u ∈ C ν ((0, T ], Jq ). Proof. If u is a classical solution of (1.2), then u(t ) = ϕ(t ) + Φ (t ) + Ψ (t ). It remains to show that Aϕ ∈ C α(1−β) ((0, T ], Jq ), it suffices to prove that Aϕ ∈ C α(1−β) ([ε, T ], Jq ) for every ε > 0. In fact, take h such that ε ≤ t < t + h ≤ T , by Lemma 2.2(iii), t +h
|Aϕ(t + h) − Aϕ(t )|q =
−s
A Eα,α (−s A)ads
α−1 2
t
α
q
t +h
s−α(1−β)−1 ds|a|H β,q
≤ C1 t
=
C1 |a|H β,q −α(1−β) t − (t + h)−α(1−β)
≤
C1 |a|H β,q
hα(1−β)
α
[ε(ε + h)]α(1−β)
α
.
Similar to Lemma 5.1, we write Φ (t ) as
Φ (t ) = Φ1 (t ) + Φ2 (t ) =
t
(t − s) 0
α−1
α
Eα,α (−(t − s) A) Pf (s) − Pf (t ) ds +
t
(t − s)α−1 Eα,α (−(t − s)α A)Pf (t )ds,
0
for t ∈ (0, T ]. It follows from Lemma 5.1 and (5.7) that AΦ1 (t ) ∈ C ν ([0, T ], Jq ) and AΦ2 (t ) ∈ C ϑ ((0, T ], Jq ), respectively.
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Since F (u, u) ∈ C ϑ ((0, T ], Jq ), the result related to the function Ψ (t ) is proved by similar argument, which means that AΨ ∈ C ν ((0, T ], Jq ). Therefore Au ∈ C ν ((0, T ], Jq ) and CDαt u = Au + F (u, u)+ Pf ∈ C ν ((0, T ], Jq ). The proof is completed. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
M. Cannone, Ondelettes, Paraproduits et Navier–Stokes, Diderot Editeur, 1995. W. Varnhorn, The Stokes Equations, Akademie Verlag, Berlin, 1994. P.G. Lemarié-Rieusset, Recent Developments in the Navier–Stokes Problem, Chapman CRC Press, 2002. W. Von Wahl, The Equations of Navier–Stokes and Abstract Parabolic Equations, Springer-Verlag, 2013. M.F. de Almeida, L.C.F. Ferreira, On the Navier–Stokes equations in the half-space with initial and boundary rough data in Morrey spaces, J. Differential Equations 254 (2013) 1548–1570. H. Heck, H. Kim, H. Kozono, Weak solutions of the stationary Navier–Stokes equations for a viscous incompressible fluid past an obstacle, Math. Ann. 356 (2013) 653–681. T. Iwabuchi, R. Takada, Global solutions for the Navier–Stokes equations in the rotational framework, Math. Ann. 357 (2013) 727–741. G. Koch, N. Nadirashvili, G.A. Seregin, V. Šverák, Liouville theorems for the Navier–Stokes equations and applications, Acta Math. 203 (2009) 83–105. N. Masmoudi, T.K. Wong, Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods, Comm. Pure Appl. Math. 68 (2015) 1683–1741. F.B. Weissler, The Navier–Stokes initial value problem in Lp , Arch. Ration. Mech. Anal. 74 (1980) 219–230. C. Amrouche, A. Rejaiba, Lp -theory for Stokes and Navier–Stokes equations with Navier boundary condition, J. Differential Equations 256 (2014) 1515–1547. J.Y. Chemin, I. Gallagher, Large, global solutions to the Navier–Stokes equations, slowly varying in one direction, Trans. Amer. Math. Soc. 362 (2010) 2859–2873. J.Y. Chemin, I. Gallagher, M. Paicu, Global regularity for some classes of large solutions to the Navier–Stokes equations, Ann. of Math. 173 (2011) 983–1012. H.J. Choe, Boundary regularity of suitable weak solution for the Navier–Stokes equations, J. Funct. Anal. 268 (2015) 2171–2187. R. Danchin, Global existence in critical spaces for compressible Navier–Stokes equations, Invent. Math. 141 (2000) 579–614. Y. Giga, S. Yoshikazu, Abstract Lp estimates for the Cauchy problem with applications to the Navier–Stokes equations in exterior domains, J. Funct. Anal. 102 (1991) 72–94. H. Kozono, L1 -solutions of the Navier–Stokes equations in exterior domains, Math. Ann. 312 (1998) 319–340. G. Raugel, G.R. Sell, Navier–Stokes equations on thin 3D domains, I, global attractors and global regularity of solutions, J. Amer. Math. Soc. 6 (1993) 503–568. R. Herrmann, Fractional Calculus: An Introduction for Physicists, World Scientific, Singapore, 2011. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. Y. Zhou, F. Jiao, J. Pecaric, Abstract Cauchy problem for fractional functional differential equations in Banach spaces, Topol. Methods Nonlinear Anal. 42 (2013) 119–136. Y. Zhou, V. Vijayakumar, R. Murugesu, Controllability for fractional evolution inclusions without compactness, Evol. Equ. Control Theory 4 (2015) 507–524. Y. Zhou, L. Zhang, X.H. Shen, Existence of mild solutions for fractional evolution equations, J. Integral Equations Appl. 25 (2013) 557–586. Z.Z. Ganji, D.D. Ganji, A. Ganji, M. Rostamian, Analytical solution of time-fractional Navier–Stokes equation in polar coordinate by homotopy perturbation method, Numer. Methods Partial Differential Equations 26 (2010) 117–124. M. El-Shahed, A. Salem, On the generalized Navier–Stokes equations, Appl. Math. Comput. 156 (1) (2004) 287–293. S. Momani, O. Zaid, Analytical solution of a time-fractional Navier–Stokes equation by Adomian decomposition method, Appl. Math. Comput. 177 (2006) 488–494. P.M. De Carvalho-Neto, P. Gabriela, Mild solutions to the time fractional Navier–Stokes equations in RN , J. Differential Equations 259 (2015) 2948–2980. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, in: North-Holland Math. Stud., vol. 204, Elsevier Science B.V., Amsterdam, 2006. R.N. Wang, D.H. Chen, T.J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, J. Differential Equations 252 (2012) 202–235. Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014. Y. Zhou, Fractional Evolution Equations and Inclusions: Analysis and Control, Academic Press, 2016. G.P. Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations: Nonlinear Steady Problems, Springer Tracts in Natural Philosophy, New York, 1998.
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Contents lists available at ScienceDirect
Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa
On the time-fractional Navier–Stokes equations✩ Yong Zhou a,b,∗ , Li Peng a a
Faculty of Mathematics and Computational Science, Xiangtan University, Hunan 411105, PR China
b
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
article
info
Article history: Available online xxxx Keywords: Navier–Stokes equations Caputo fractional derivative Mittag-Leffler functions Mild solutions Regularity
abstract This paper is concerned with the Navier–Stokes equations with time-fractional derivative of order α ∈ (0, 1). This type of equations can be used to simulate anomalous diffusion in fractal media. We establish the existence and uniqueness of local and global mild solutions in H β,q . Meanwhile, we also give local mild solutions in Jq . Moreover, we prove the existence and regularity of classical solutions for such equations in Jq . © 2016 Elsevier Ltd. All rights reserved.
1. Introduction The Navier–Stokes equations describe the motion of the incompressible Newtonian fluid flows ranging from large scale atmospheric motions to the lubrication of ball bearings, and express the conservation of mass and momentum. For more details we refer to the monographs of Cannone [1] and Varnhorn [2]. We find this system which is so rich in phenomena that the whole power of mathematical theory is needed to discuss the existence, regularity and boundary conditions; see, e.g., Lemarié-Rieusset [3] and Von Wahl [4]. It is worth mentioning that Leray carried out an initial study that a boundary-value problem for the time-dependent Navier–Stokes equations possesses a unique smooth solution on some intervals of time provided the data are sufficiently smooth. Since then many results on the existence for weak, mild and strong solutions for the Navier–Stokes equations have been investigated intensively by many authors; see, e.g., Almeida and Ferreira [5], Heck et al. [6], Iwabuchi and Takada [7], Koch et al. [8], Masmoudi and Wong [9], and Weissler [10]. Moreover, one can find results on regularity of weak and strong solution from Amrouche and Rejaiba [11], Chemin and Gallagher [12], Chemin et al. [13], Choe [14], Danchin [15], Giga [16], Kozono [17], Raugel and Sell [18] and the references therein. On the other hand, fractional calculus has gained considerable popularity during the past decades due mainly to its demonstrated applications in numerous seemingly diverse and wide-spread fields of science and engineering, including fluid flow, rheology, dynamical processes and porous structures, diffusive transport akin to diffusion, control theory of dynamical systems, viscoelasticity and so on; see, e.g., Herrmann [19], Hilfer [20] and Zhou et al. [21–23]. The most important among such models are those described by partial differential equations with fractional derivatives. Such models are interesting for not only physicists but also pure mathematicians. Recent theoretical analysis and experimental data have shown that classical diffusion equation fails to describe diffusion phenomenon in heterogeneous porous media that exhibits fractal characteristics. How is the classical diffusion equation
✩ Project supported by National Natural Science Foundation of China (11271309).
∗
Corresponding author at: Faculty of Mathematics and Computational Science, Xiangtan University, Hunan 411105, PR China. E-mail address: [email protected] (Y. Zhou).
http://dx.doi.org/10.1016/j.camwa.2016.03.026 0898-1221/© 2016 Elsevier Ltd. All rights reserved.
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Y. Zhou, L. Peng / Computers and Mathematics with Applications (
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modified to make it appropriate to depict anomalous diffusion phenomena? This problem is interesting for researchers. Fractional calculus have been found effective in modelling anomalous diffusion processes since it has been recognized as one of the best tools to characterize the long memory processes. Consequently, it is reasonable and significative to propose the generalized Navier–Stokes equations with Caputo fractional derivative operator, which can be used to simulate anomalous diffusion in fractal media. Its evolutions behave in a much more complex way than in classical inter-order case and the corresponding investigation becomes more challenging. The main effort on time-fractional Navier–Stokes equations has been put into attempts to derive numerical solutions and analytical solutions; see Ganji et al. [24], El-Shahed et al. [25], and Momani and Zaid [26]. However, to the best of our knowledge, there are very few results on the existence and regularity of mild solutions for time-fractional Navier–Stokes equations. Recently, Carvalho-Neto [27] dealt with the existence and uniqueness of global and local mild solutions for the time-fractional Navier–Stokes equations. Motivated by above discussion, in this paper we study the following time-fractional Navier–Stokes equations in an open set Ω ⊂ Rn (n ≥ 3):
α ∂ u − ν 1u + (u · ∇)u t ∇ ·u u|∂ Ω u(0, x)
= = = =
−∇ p + f ,
0, 0, a,
t > 0, (1.1)
where ∂tα is the Caputo fractional derivative of order α ∈ (0, 1), u = (u1 (t , x), u2 (t , x), . . . , un (t , x)) represents the velocity field at a point x ∈ Ω and time t > 0, p = p(t , x) is the pressure, ν the viscosity, f = f (t , x) is the external force and a = a(x) is the initial velocity. From now on, we assume that Ω has a smooth boundary. Firstly, we get rid of the pressure term by applying Helmholtz projector P to Eq. (1.1), which converts Eq. (1.1) to
α ∂ u − ν P 1u + P (u · ∇)u t ∇ ·u u|∂ Ω u(0, x)
= = = =
Pf , 0, 0, a.
t > 0,
The operator −ν P ∆ with Dirichlet boundary conditions is, basically, the Stokes operator A in the divergence-free function space under consideration. Then we rewrite (1.1) as the following abstract form Dαt u = −Au + F (u, u) + Pf , u(0) = a,
C
t > 0,
(1.2)
where F (u, v) = −P (u ·∇)v . If one can give sense to the Helmholtz projection P and the Stokes operator A, then the solution of Eq. (1.2) is also the solution of Eq. (1.1). The objective of this paper is to establish the existence and uniqueness of global and local mild solutions of problem (1.2) in H β,q . Further, we prove the regularity results which state essentially that if Pf is Hölder continuous then there is a unique classical solution u(t ) such that Au and CDαt u(t ) are Hölder continuous in Jq . The paper is organized as follows. In Section 2 we recall some notations, definitions, and preliminary facts. Section 3 is devoted to the existence and uniqueness of global mild solution in H β,q of problem (1.2), then proceed to study the local mild solution in H β,q . In Section 4, we use the iteration method to obtain the existence and uniqueness of local mild solution in Jq of problem (1.2). Finally, Section 5 is concerned with the existence and regularity of classical solution in Jq of problem (1.2). 2. Preliminaries In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. Let Ω = {(x1 , . . . , xn ) : xn > 0} be open subset of Rn , where n ≥ 3. Let 1 < q < ∞. Then there is a bounded projection P called the Hodge projection on (Lq (Ω ))n , whose range is the closure of Cσ∞ (Ω ) := {u ∈ (C ∞ (Ω ))n : ∇ · u = 0, u has compact support in Ω }, and whose null space is the closure of
{u ∈ (C ∞ (Ω ))n : u = ∇φ, φ ∈ C ∞ (Ω )}. |·|q
For notational convenience, let Jq := Cσ∞ (Ω ) , which is a closed subspace of (Lq (Ω ))n . (W m,q (Ω ))n is a Sobolev space with the norm | · |m,q . A = −ν P ∆ denotes the Stokes operator in Jq whose domain is Dq (A) = Dq (∆) ∩ Jq ; here, Dq (∆) = {u ∈ (W 2,q (Ω ))n : u|∂ Ω = 0}. It is known that −A is a closed linear operator and generates the bounded analytic semigroup {e−tA } on Jq .
Y. Zhou, L. Peng / Computers and Mathematics with Applications (
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So as to state our results, we need to introduce the definitions of the fractional power spaces associated with −A. For
β > 0 and u ∈ Jq , define ∞ 1 t β−1 e−tA udt . A−β u = Γ (β) 0
Then A−β is a bounded, one-to-one operator on Jq . Let Aβ be the inverse of A−β . For β > 0, we denote the space H β,q by the range of A−β with the norm
|u|H β,q = |Aβ u|q . It is easy to check that e−tA extends (or restricts) to a bounded analytic semigroup on H β,q . For more details, we refer to Von Wahl [4]. Let X be a Banach space and J be an interval of R. C (J , X ) denotes the set of all continuous X -valued functions. For 0 < ϑ < 1, C ϑ (J , X ) stands for the set of all functions which are Hölder continuous with the exponent ϑ . Let α ∈ (0, 1] and v : [0, ∞) → X . The fractional integral of order α with the lower limit zero for the function v is defined as t
Itα v(t ) =
gα (t − s)v(s)ds,
t > 0,
0
provided the right hand-side is point-wise defined on [0, ∞), where gα denotes the Riemann–Liouville kernel gα (t ) =
t α−1
,
Γ (α)
t > 0.
Further, CDαt stands the Caputo fractional derivative operator of order α ; it is defined by C α Dt
v(t ) =
d 1−α d It v(t ) − v(0) = dt dt
t
g1−α (t − s) v(t ) − v(0) ds ,
t > 0.
0
More generally, for u : [0, ∞) × Rn → Rn , Caputo fractional derivative with respect to time of the function u can be written as
∂tα u(t , x) = ∂t
t
g1−α (t − s) u(t , x) − u(0, x) ds ,
t > 0.
0
For more details, we refer the reader to Kilbas et al. [28]. Let us introduce the generalized Mittag-Leffler special functions: Eα (−t α A) =
∞
α
Mα (s)e−st A ds,
Eα,α (−t α A) =
∞
0
α
α sMα (s)e−st A ds, 0
where Mα (θ ) is Mainardi’s wright type function defined by
Mα (θ ) =
∞ k=0
θn . n!Γ (1 − α(1 + n))
Proposition 2.1. (i) Eα,α (−t α A) = 21π i Γ Eα,α (−µt α )(µI + A)−1 dµ; θ (ii) Aγ Eα,α (−t α A) = 21π i Γ µγ Eα,α (−µt α )(µI + A)−1 dµ.
θ
Proof. (i) In view of Eα,α (−t α A) =
∞ 0
α sMα (s)e−st ds = Eα,α (−t ) and Fubini theorem, we get
∞
α
α sMα (s)e−st A ds ∞ 1 α α sM α ( s) e−µst (µI + A)−1 dµds = 2π i 0 Γθ 1 α = Eα,α (−µt )(µI + A)−1 dµ, 2π i Γθ 0
where Γθ is a suitable integral path.
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(ii) A similar argument shows that
∞
α
α sMα (s)Aγ e−st A ds 0 ∞ 1 α = µγ e−µst (µI + A)−1 dµds α sM α ( s) 2π i 0 Γθ 1 = µγ Eα,α (−µt α )(µI + A)−1 dµ. 2π i Γθ
Aγ Eα,α (−t α A) =
Moreover, we have the following results. Lemma 2.1 ([29]). For t > 0, Eα (−t α A) and Eα,α (−t α A) are continuous in the uniform operator topology. Moreover, for every r > 0, the continuity is uniform on [r , ∞). Lemma 2.2 ([29]). Let 0 < α < 1. Then (i) (ii) (iii) (iv)
for all u ∈ X , limt →0+ Eα (−t α A)u = u; for all u ∈ D(A) and t > 0, CDαt Eα (−t α A)u = −AEα (−t α A)u; for all u ∈ X , Eα′ (−t α A)u = −t α−1 AEα,α (−t α A)u; for t > 0, Eα (−t α A)u = It1−α t α−1 Eα,α (−t α A)u .
Before presenting the definition of mild solution of problem (1.2), we give the following lemma for a given function h : [0, ∞) → X . For more details we refer to Zhou [30,31]. Lemma 2.3. If u(t ) = a +
t
1
Γ (α)
(t − s)α−1 Au(s) + h(s) ds,
for t ≥ 0
(2.1)
0
holds, then we have u(t ) = Eα (−t α A)a +
t
(t − s)α−1 Eα,α (−(t − s)α A)h(s)ds.
0
We rewrite (1.2) as u(t ) = a +
t
1
Γ (α)
(t − s)α−1 Au(s) + F (u(s), u(s)) + Pf (s) ds,
for t ≥ 0.
0
Inspired by above discussion, we adopt the following concepts of mild solution to problem (1.2). Definition 2.1. A function u : [0, ∞) → H β,q is called a global mild solution of problem (1.2) in H β,q , if u ∈ C ([0, ∞), H β,q ) and for t ∈ [0, ∞) u(t ) = Eα (−t α A)a +
t
(t − s)α−1 Eα,α (−(t − s)α A)F (u(s), u(s))ds +
0
t
(t − s)α−1 Eα,α (−(t − s)α A)Pf (s)ds.
(2.2)
0
Definition 2.2. Let 0 < T < ∞. A function u : [0, T ] → H β,q (or Jq ) is called a local mild solution of problem (1.2) in H β,q (or Jq ), if u ∈ C ([0, T ], H β,q ) (or C ([0, T ], Jq )) and u satisfies (2.2) for t ∈ [0, T ]. For convenience, we define two operators Φ and G as follows:
Φ (t ) =
t
(t − s)α−1 Eα,α (−(t − s)α A)f (s)ds,
0
G(u, v)(t ) =
t
(t − s)α−1 Eα,α (−(t − s)α A)F (u(s), v(s))ds.
0
In subsequent proof we use the following fixed point result. Lemma 2.4 ([1]). Let (X , ∥ · ∥X ) be a Banach space, G : X × X → X a bilinear operator and L a positive real number such that
∥G(u, v)∥X ≤ L∥u∥X ∥v∥X , Then for any u0 ∈ X with ∥u0 ∥X <
∀ u, v ∈ X . 1 , 4L
the equation u = u0 + G(u, u) has a unique solution u ∈ X .
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3. Global and local existence in H β,q Our main purpose in this section is to establish sufficient conditions for existence and uniqueness of mild solution to problem (1.2) in H β,q . To this end we assume that: (f) Pf is continuous for t > 0 and |Pf (t )|q = o(t −α(1−β) ) as t → 0 for 0 < β < 1. Lemma 3.1 ([32] (see also [10])). Let 1 < q < ∞ and β1 ≤ β2 . Then there is a constant C = C (β1 , β2 ) such that
|e−tA v|H β2 ,q ≤ Ct −(β2 −β1 ) |v|H β1 ,q ,
t >0
for v ∈ H β1 ,q . Furthermore, limt →0 t (β2 −β1 ) |e−tA v|H β2 ,q = 0. Now, we study an important technical lemma, that helps us to prove the final main theorems of this section. Lemma 3.2. Let 1 < q < ∞ and β1 ≤ β2 . Then for any T > 0, there exists a constant C1 = C1 (β1 , β2 ) > 0 such that
|Eα (−t α A)v|H β2 ,q ≤ C1 t −α(β2 −β1 ) |v|H β1 ,q and |Eα,α (−t α A)v|H β2 ,q ≤ C1 t −α(β2 −β1 ) |v|H β1 ,q for all v ∈ H β1 ,q and t ∈ (0, T ]. Furthermore, lim t α(β2 −β1 ) |Eα (−t α A)v|H β2 ,q = 0.
t →0
Proof. Let v ∈ H β1 ,q . By Lemma 3.1, we estimate
|Eα (−t α A)v|H β2 ,q ≤
∞
α
Mα (s)|e−st A v|H β2 ,q ds 0
∞
Mα (s)s
≤ C
−(β2 −β1 )
ds t −α(β2 −β1 ) |v|H β1 ,q
0
≤ C1 t −α(β2 −β1 ) |v|H β1 ,q . More precisely, Lebesgue’s dominated convergence theorem shows lim t α(β2 −β1 ) |Eα (−t α A)v|H β2 ,q ≤
t →0
∞
α
Mα (s) lim t α(β2 −β1 ) |e−st A v|H β2 ,q ds = 0. t →0
0
Similarly,
|Eα,α (−t α A)v|H β2 ,q ≤
∞
α
α sMα (s)|e−st A v|H β2 ,q ds 0 ∞ ≤ αC Mα (s)s1−(β2 −β1 ) ds t −α(β2 −β1 ) |v|H β1 ,q 0
≤ C1 t −α(β2 −β1 ) |v|H β1 ,q , where the constant C1 = C1 (α, β1 , β2 ) is such that
C1 ≥ C max
α Γ (2 − β2 + β1 ) Γ (1 − β2 + β1 ) , . Γ (1 + α(β1 − β2 )) Γ (1 + α(1 + β1 − β2 ))
3.1. Global existence in H β,q This subsection is concerned with the global mild solution of problem (1.2) in H β,q . For convenience, we denote M (t ) = sup {sα(1−β) |Pf (s)|q }, s∈(0,t ]
B1 = C1 max{B(α(1 − β), 1 − α(1 − β)), B(α(1 − γ ), 1 − α(1 − β))},
L ≥ MC1 max B(α(1 − β), 1 − 2α(γ − β)), B(α(1 − γ ), 1 − 2α(γ − β)) , where M is given later.
6
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Theorem 3.1. Let 1 < q < ∞, 0 < β < 1 and (f) hold. For every a ∈ H β,q , suppose that C1 |a|H β,q + B1 M∞ <
1 4L
,
(3.1)
where M∞ := sups∈(0,∞) {sα(1−β) Pf (s)}. If satisfying:
n 2q
− 21 < β , then there is a γ > max{β, 12 } and a unique function u : [0, ∞) → H β,q
(a) u : [0, ∞) → H β,q is continuous and u(0) = a; (b) u : (0, ∞) → H γ ,q is continuous and limt →0 t α(γ −β) |u(t )|H γ ,q = 0; (c) u satisfies (2.2) for t ∈ [0, ∞). Proof. Let γ =
(1+β) 2
. Define X∞ = X [∞] as the space of all curves u : (0, ∞) → H β,q such that:
(i) u : [0, ∞) → H β,q is bounded and continuous; (ii) u : (0, ∞) → H γ ,q is bounded and continuous, moreover, limt →0 t α(γ −β) |u(t )|H γ ,q = 0; with its natural norm
∥u∥X∞ = max sup |u(t )|H β,q , sup t α(γ −β) |u(t )|H γ ,q . t ≥0
t ≥0
It is obvious that X∞ is a non-empty complete metric space. From an argument of Weissler [10], we know that F : H γ ,q × H γ ,q → Jq is a bounded bilinear map, then there exists M such that for u, v ∈ H γ ,q
|F (u, v)|q ≤ M |u|H γ ,q |v|H γ ,q , |F (u, u) − F (v, v)|q ≤ M (|u|H γ ,q + |v|H γ ,q )|u − v|H γ ,q .
(3.2)
Step 1. Let u, v ∈ X∞ . We show that the operator G(u(t ), v(t )) belongs to C ([0, ∞), H β,q ) as well as C ((0, ∞), H γ ,q ). For arbitrary t0 ≥ 0 fixed and ε > 0 enough small, consider t > t0 (the case t < t0 follows analogously), we have
|G(u(t ), v(t )) − G(u(t0 ), v(t0 ))|H β,q t ≤ (t − s)α−1 |Eα,α (−(t − s)α A)F (u(s), v(s))|H β,q ds t0 t0
+ 0
(t − s)α−1 − (t0 − s)α−1 Eα,α (−(t − s)α A)F (u(s), v(s)) β,q ds H
t0 −ε
+ 0
t0
+ t0 −ε
(t0 − s)α−1 Eα,α (−(t − s)α A) − Eα,α (−(t0 − s)α A) F (u(s), v(s))H β,q ds
(t0 − s)α−1 Eα,α (−(t − s)α A) − Eα,α (−(t0 − s)α A) F (u(s), v(s))H β,q ds
:= I11 (t ) + I12 (t ) + I13 (t ) + I14 (t ). We estimate each of the four terms separately. For I11 (t ), in view of Lemma 3.2, we obtain I11 (t ) ≤ C1
t
(t − s)α(1−β)−1 |F (u(s), v(s))|q ds
t0 t
≤ MC1
(t − s)α(1−β)−1 |u(s)|H γ ,q |v(s)|H γ ,q ds
t0 t
≤ MC1
(t − s)α(1−β)−1 s−2α(γ −β) ds sup {s2α(γ −β) |u(s)|H γ ,q |v(s)|H γ ,q } s∈[0,t ]
t0
= MC1
1 t0 /t
(1 − s)α(1−β)−1 s−2α(γ −β) ds sup {s2α(γ −β) |u(s)|H γ ,q |v(s)|H γ ,q }. s∈[0,t ]
By the properties of the Beta function, there exists δ > 0 small enough such that for 0 < t − t0 < δ ,
1 t0 /t
(1 − s)α(1−β)−1 s−2α(γ −β) ds → 0,
Y. Zhou, L. Peng / Computers and Mathematics with Applications (
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which follows that I11 (t ) tends to 0 as t − t0 → 0. For I12 (t ), since I12 (t ) ≤ C1
t0
(t0 − s)α−1 − (t − s)α−1 (t − s)−αβ |F (u(s), v(s))|q ds
0 t0
≤ MC1
(t0 − s)α−1 − (t − s)α−1 (t − s)−αβ s−2α(γ −β) ds sup {s2α(γ −β) |u(s)|H γ ,q |v(s)|H γ ,q }, s∈[0,t0 ]
0
noting that
t0
|(t0 − s)α−1 − (t − s)α−1 |(t − s)−αβ s−2α(γ −β) ds t0 t0 (t0 − s)α−1 (t − s)−αβ s−2α(γ −β) ds (t − s)α−1 (t − s)−αβ s−2α(γ −β) ds + ≤ 0 0 t0 (t0 − s)α(1−β)−1 s−2α(γ −β) ds ≤2
0
0
= 2B(α(1 − β), 1 − 2α(γ − β)), then by Lebesgue’s dominated convergence theorem, we have t0
(t0 − s)α−1 − (t − s)α−1 (t − s)−αβ s−2α(γ −β) ds → 0,
as t → t0 ,
0
one deduces that limt →t0 I12 (t ) = 0. For I13 (t ), since I13 (t ) ≤
t0 −ε
0
(t0 − s)α−1 Eα,α (−(t − s)α A) + Eα,α (−(t0 − s)α A) F (u(s), v(s))H β,q ds
t0 −ε
(t0 − s)α−1 (t − s)−αβ + (t0 − s)−αβ |F (u(s), v(s))|q ds 0 t0 −ε ≤ 2MC1 (t0 − s)α(1−β)−1 s−2α(γ −β) ds sup {s2α(γ −β) |u(s)|H γ ,q |v(s)|H γ ,q },
≤
s∈[0,t0 ]
0
using Lebesgue’s dominated convergence theorem again, the fact from the uniform continuity of the operator Eα,α (−t α A) due to Lemma 2.1 shows lim I13 (t ) =
t0 −ε
t → t0
(t0 − s)α−1 lim Eα,α (−(t − s)α A) − Eα,α (−(t0 − s)α A) F (u(s), v(s))H β,q ds t →t0
0
= 0. For I14 (t ), by immediate calculation, we estimate I14 (t ) ≤
t0
(t0 − s)α−1 (t − s)−αβ + (t0 − s)−αβ |F (u(s), v(s))|q ds
t0 −ε
t0
≤ 2MC1
(t0 − s)α(1−β)−1 s−2α(γ −β) ds
t0 −ε
→ 0,
sup
{s2α(γ −β) |u(s)|H γ ,q |v(s)|H γ ,q }
s∈[t0 −ε,t0 ]
as ε → 0
according to the properties of the Beta function. Thenceforth, it follows
|G(u(t ), v(t )) − G(u(t0 ), v(t0 ))|H β,q → 0,
as t → t0 .
The continuity of the operator G(u, v) evaluated in C ((0, ∞), H γ ,q ) follows by the similar discussion as above. So, we omit the details.
8
Y. Zhou, L. Peng / Computers and Mathematics with Applications (
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Step 2. We show that the operator G : X∞ × X∞ → X∞ is a continuous bilinear operator. By Lemma 3.2, we have
t |G(u(t ), v(t ))|H β,q ≤ (t − s)α−1 Eα,α (−(t − s)α A)F (u(s), v(s))ds 0 H β,q t (t − s)α(1−β)−1 |F (u(s), v(s))|q ds ≤ C1 0 t (t − s)α(1−β)−1 s−2α(γ −β) ds sup {s2α(γ −β) |u(s)|H γ ,q |v(s)|H γ ,q } ≤ MC1 s∈[0,t ]
0
= MC1 B(α(1 − β), 1 − 2α(γ − β))∥u∥X∞ ∥v∥X∞ and
t α−1 α ≤ (t − s) Eα,α (−(t − s) A)F (u(s), v(s))ds 0 H γ ,q t ≤ C1 (t − s)α(1−γ )−1 |F (u(s), v(s))|q ds 0 t ≤ MC1 (t − s)α(1−γ )−1 s−2α(γ −β) ds sup {s2α(γ −β) |u(s)|H γ ,q |v(s)|H γ ,q }
|G(u(t ), v(t ))|H γ ,q
s∈[0,t ]
0
= MC1 t −α(γ −β) B(α(1 − γ ), 1 − 2α(γ − β))∥u∥X∞ ∥v∥X∞ , it follows that sup t α(γ −β) |G(u(t ), v(t ))|H γ ,q ≤ MC1 B(α(1 − γ ), 1 − 2α(γ − β))∥u∥X∞ ∥v∥X∞ .
t ∈[0,∞)
More precisely, lim t α(γ −β) |G(u(t ), v(t ))|H γ ,q = 0.
t →0
Hence, G(u, v) ∈ X∞ and ∥G(u(t ), v(t ))∥X∞ ≤ L∥u∥X∞ ∥v∥X∞ . Step 3. We verify that (c) holds. Let 0 < t0 < t. Since
|Φ (t ) − Φ (t0 )|H β,q t ≤ (t − s)α−1 |Eα,α (−(t − s)α A)Pf (s)|H β,q ds + t0
t0
(t0 − s)α−1 − (t − s)α−1 |Eα,α (−(t − s)α A)Pf (s)|H β,q ds
0 t0 −ε
+ 0
t0
+
t0 −ε t
≤ C1
(t0 − s)α−1 Eα,α (−(t − s)α A) − Eα,α (−(t0 − s)α A) Pf (s)H β,q ds
(t0 − s)α−1 Eα,α (−(t − s)α A) − Eα,α (−(t0 − s)α A) Pf (s)H β,q ds
(t − s)α(1−β)−1 |Pf (s)|q ds + C1
t0
t0
(t0 − s)α−1 − (t − s)α−1 (t − s)−αβ |Pf (s)|q ds
0 t0 −ε
(t0 − s)α−1 Eα,α (−(t − s)α A) − Eα,α (−(t0 − s)α A) Pf (s)H β,q ds + 2C1
+ C1 0
≤ C1 M (t )
+ C1 M (t )
t
(t − s)α(1−β)−1 s−α(1−β) ds + C1 M (t )
t0
t0
t0
(t0 − s)α(1−β)−1 |Pf (s)|q ds
t0 −ε
(t − s)α−1 − (t0 − s)α−1 s−α(1−β) ds
0
t0 −ε
(t0 − s)α−1 Eα,α (−(t − s)α A) − Eα,α (−(t0 − s)α A) Pf (s)
0
+ 2C1 M (t )
t0
H β,q
ds
(t0 − s)α(1−β)−1 s−α(1−β) ds.
t0 −ε
By the properties of the Beta function, the first two integrals and the last integral tend to 0 as t → t0 as well as ε → 0. In view of Lemma 2.1, the third integral also goes to 0 as t → t0 , which implies
|Φ (t ) − Φ (t0 )|H β,q → 0 as t → t0 . The continuity of Φ (t ) evaluated in H γ ,q follows by the similar argument as above.
Y. Zhou, L. Peng / Computers and Mathematics with Applications (
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On the other hand, we have
|Φ (t )|H β,q
t α−1 α ≤ (t − s) Eα,α (−(t − s) A)Pf (s)ds 0 H β,q t (t − s)α(1−β)−1 |Pf (s)|q ds ≤ C1 0 t ≤ C1 M (t ) (t − s)α(1−β)−1 s−α(1−β) ds 0
= C1 M (t )B(α(1 − β), 1 − α(1 − β)),
(3.3)
and
t |Φ (t )|H γ ,q ≤ (t − s)α−1 Eα,α (−(t − s)α A)Pf (s)ds 0 H γ ,q t ≤ C1 (t − s)α(1−γ )−1 |Pf (s)|q ds 0 t ≤ C1 M (t ) (t − s)α(1−γ )−1 s−α(1−β) ds 0
= t −α(γ −β) C1 M (t )B(α(1 − γ ), 1 − α(1 − β)). More precisely, t α(γ −β) |Φ (t )|H γ ,q ≤ C1 M (t )B(α(1 − γ ), 1 − α(1 − β)) → 0,
as t → 0,
since M (t ) → 0 as t → 0 due to assumption (f). This ensures that Φ (t ) ∈ X∞ and ∥Φ (t )∥∞ ≤ B1 M∞ . For a ∈ H β,q . By Lemma 2.1, it is easy to see that Eα (−t α A)a ∈ C ([0, ∞), H β,q )
and Eα (−t α A)a ∈ C ((0, ∞), H γ ,q ).
This, together with Lemma 3.2, implies that for all t ∈ (0, T ], Eα (−t α A)a ∈ X∞ , t
α(γ −β)
Eα (−t α A)a ∈ C ([0, ∞), H γ ,q ),
∥Eα (−t α A)a∥X∞ ≤ C1 |a|H β,q . According to (3.1), the inequality
∥Eα (−t α A)a + Φ (t )∥X∞ ≤ ∥Eα (−t α A)a∥X∞ + ∥Φ (t )∥X∞ ≤
1 4L
holds, which yields that F has a unique fixed point. Step 4. To show that u(t ) → a in H β,q as t → 0. We need to verify t
lim
t →0
t
lim
t →0
(t − s)α−1 Eα,α (−(t − s)α A)Pf (s)ds = 0,
0
(t − s)α−1 Eα,α (−(t − s)α A)F (u(s), u(s))ds = 0
0
in H β,q . In fact, it is obvious that limt →0 Φ (t ) = 0 (limt →0 M (t ) = 0) owing to (3.3). In addition,
t (t − s)α−1 Eα,α (−(t − s)α A)F (u(s), u(s))ds β,q 0 H t α(1−β)−1 ≤ C1 ( t − s) |F (u(s), u(s))|q ds 0 t ≤ MC1 (t − s)α(1−β)−1 |u(s)|2H γ ,q ds 0 t ≤ MC1 (t − s)α(1−β)−1 s−2α(γ −β) ds sup {s2α(γ −β) |u(s)|2H γ ,q } 0
s∈[0,t ]
= MC1 B(α(1 − β), 1 − 2α(γ − β)) sup {s2α(γ −β) |u(s)|2H γ ,q } → 0 as t → 0. s∈[0,t ]
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3.2. Local existence in H β,q In this subsection, we study the local mild solution of problem (1.2) in H β,q . Theorem 3.2. Let 1 < q < ∞, 0 < β < 1 and (f) hold. Suppose n 2q
−
1
< β.
2
(3.4)
Then there is a γ > max{β, 12 } such that for every a ∈ H β,q there exist T∗ > 0 and a unique function u : [0, T∗ ] → H β,q satisfying: (a) u : [0, T∗ ] → H β,q is continuous and u(0) = a; (b) u : (0, T∗ ] → H γ ,q is continuous and limt →0 t α(γ −β) |u(t )|H γ ,q = 0; (c) u satisfies (2.2) for t ∈ [0, T∗ ]. Proof. Let γ =
(1+β) 2
. Fix a ∈ H β,q . Let X = X [T ] be the space of all curves u : (0, T ] → H β,q such that:
(i) u : [0, T ] → H β,q is continuous; (ii) u : (0, T ] → H γ ,q is continuous and limt →0 t α(γ −β) |u(t )|H γ ,q = 0; with its natural norm
∥u∥X = sup {t α(γ −β) |u(t )|H γ ,q }. t ∈[0,T ]
Similar to the proof of Theorem 3.1, it is easy to claim that G : X × X → X is continuous linear map and Φ (t ) ∈ X . By Lemma 2.1, it is easy to see that for all t ∈ (0, T ], Eα (−t α A)a ∈ C ([0, T ], H β,q ),
Eα (−t α A)a ∈ C ((0, T ], H γ ,q ). From Lemma 3.2, it follows that Eα (−t α A)a ∈ X , t α(γ −β) Eα (−t α A)a ∈ C ([0, T ], H γ ,q ). Hence, let T∗ > 0 be sufficiently small such that
∥Eα (−t α A)a + Φ (t )∥X [T∗ ] ≤ ∥Eα (−t α A)a∥X [T∗ ] + ∥Φ (t )∥X [T∗ ] < which implies that F has a unique fixed point due to Lemma 2.4.
1 4L
,
4. Local existence in Jq This section is devoted to consideration of local mild solution to problem (1.2) in Jq by means of the iteration method. Let
γ =
(1+β) 2
.
Theorem 4.1. Let 1 < q < ∞, 0 < β < 1 and (f) hold. Suppose that a ∈ H β,q
with
n 2q
−
1 2
< β.
Then problem (1.2) has a unique mild solution u in Jq for a ∈ H β,q . Moreover, u is continuous on [0, T ], Aγ u is continuous in (0, T ] and t α(γ −β) Aγ u(t ) is bounded as t → 0. Proof. Step 1. Set K (t ) := sup sα(γ −β) |Aγ u(s)|q s∈(0,t ]
and
Ψ (t ) := G(u, u)(t ) =
t
(t − s)α−1 Eα,α (−(t − s)α A)F (u(s), u(s))ds.
0
As an immediate consequence of Step 2 in Theorem 3.1, Ψ (t ) is continuous in [0, T ], Aγ Ψ (t ) exists and is continuous in (0, T ] with
|Aγ Ψ (t )|q ≤ MC1 B(α(1 − γ ), 1 − 2α(γ − β))K 2 (t )t −α(γ −β) .
(4.1)
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We also consider the integral Φ (t ). Since (f) holds, the inequality
|Pf (s)|q ≤ M (t )sα(1−β) is satisfied with a continuous function M (t ). From Step 3 in Theorem 3.1, we derive that Aγ Φ (t ) is continuous in (0, T ] with
|Aγ Φ (t )|q ≤ C1 M (t )B(α(1 − γ ), 1 − α(1 − β))t −α(γ −β) .
(4.2) γ
For |Pf (t )|q = o(t ) as t → 0, we have M (t ) = 0. Here (4.2) means |A Φ (t )|q = o(t We prove that Φ is continuous in Jq . In fact, take 0 ≤ t0 < t < T , we have −α(1−β)
|Φ (t ) − Φ (t0 )|q t (t − s)α−1 |Pf (s)|q ds + C3 ≤ C3
t0
−α(γ −β)
) as t → 0.
(t0 − s)α−1 − (t − s)α−1 |Pf (s)|q ds
0
t0 t0 −ε
+ C3
α−1
(t0 − s)
∥Eα,α (−(t − s)α A) − Eα,α (−(t0 − s)α A)∥ |Pf (s)|q ds
0 t0
(t0 − s)α−1 |Pf (s)|q ds
+ 2C3 t0 −ε
≤ C3 M (t )
t
(t − s)α−1 s−α(1−β) ds + C3 M (t )
t0
+ C3 M (t )
t0
(t − s)α−1 − (t0 − s)α−1 s−α(1−β) ds
0 t0 −ε
(t0 − s)α−1 s−α(1−β) ds sup ∥Eα,α (−(t − s)α A) − Eα,α (−(t0 − s)α A)∥ s∈[0,t −ε]
0
+ 2C3 M (t )
t0
(t0 − s)α−1 s−α(1−β) ds → 0,
as t → t0
t0 −ε
by previous discussion. Further, we consider the function Eα (−t α A)a. It is obvious by Lemma 3.2 that
|Aγ Eα (−t α A)a|q ≤ C1 t −α(γ −β) |Aβ a|q = C1 t −α(γ −β) |a|H β,q , lim t α(γ −β) |Aγ Eα (−t α A)a|q = lim t α(γ −β) |Eα (−t α A)a|H γ ,q = 0.
t →0
t →0
Step 2. Now we construct the solution by the successive approximation: u0 (t ) = Eα (−t α A)a + Φ (t ), un+1 (t ) = u0 (t ) + G(un , un )(t ),
n = 0, 1, 2 . . . .
(4.3)
Making use of above results, we know that Kn (t ) := sup sα(γ −β) |Aγ un (s)|q s∈(0,t ]
are continuous and increasing functions on [0, T ] with Kn (0) = 0. Furthermore, in virtue of (4.2) and (4.2), Kn (t ) fulfils the following inequality Kn+1 (t ) ≤ K0 (t ) + MC1 B(α(1 − γ ), 1 − 2α(γ − β))Kn2 (t ).
(4.4)
For K0 (0) = 0, we choose a T > 0 such that 4MC1 B(α(1 − γ ), 1 − 2α(γ − β))K0 (T ) < 1.
(4.5)
Then a fundamental consideration of (4.4) ensures that the sequence {Kn (T )} is bounded, i.e., Kn (T ) ≤ ρ(T ),
n = 0, 1, 2, . . . ,
where
ρ(t ) =
1−
√
1 − 4MC1 B(α(1 − γ ), 1 − 2α(γ − β))K0 (t ) 2MC1 B(α(1 − γ ), 1 − 2α(γ − β))
.
Analogously, for any t ∈ (0, T ], Kn (t ) ≤ ρ(t ) holds. In the same way we note that ρ(t ) ≤ 2K0 (t ). Let us consider the equality
wn+1 (t ) =
t
0
(t − s)α−1 Eα,α (−(t − s)α A)[F (un+1 (s), un+1 (s)) − F (un (s), un (s))]ds,
12
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where wn = un+1 − un , n = 0, 1, . . . , and t ∈ (0, T ]. Writing Wn (t ) := sup sα(γ −β) |Aγ wn (s)|q . s∈(0,t ]
On account of (3.1), we have
|F (un+1 (s), un+1 (s)) − F (un (s), un (s))|q ≤ M (Kn+1 (s) + Kn (t ))Wn (s)s−2α(γ −β) , which follows from Step 2 in Theorem 3.1 that t α(γ −β) |Aγ wn+1 (t )|q ≤ 2MC1 B(α(1 − γ ), 1 − α(1 − β))ρ(t )Wn (t ). This inequality gives Wn+1 (T ) ≤ 2MC1 B(α(1 − γ ), 1 − 2α(γ − β))ρ(T )Wn (T )
≤ 4MC1 B(α(1 − γ ), 1 − 2α(γ − β))K0 (T )Wn (T ).
(4.6)
According to (4.5) and (4.6), it is easy to see that lim
Wn+1 (T )
n→0
Wn (T )
< 4MC1 B(α(1 − γ ), 1 − 2α(γ − β)) < 1,
α(γ −β) γ thus the series A wn (t ) converges uniformly for t ∈ (0, T ], n=0 Wn (T ) converges. It shows that the series n=0 t therefore, the sequence {t α(γ −β) Aγ un (t )} converges uniformly in (0, T ]. This implies that
∞
∞
lim un (t ) = u(t ) ∈ D(Aγ )
n→∞
and lim t α(γ −β) Aγ un (t ) = t α(γ −β) Aγ u(t ) uniformly,
n→∞
since A−γ is bounded and Aγ is closed. Accordingly, the function K (t ) = sups∈(0,t ] sα(γ −β) |Aγ u(s)|q also satisfies K (t ) ≤ ρ(t ) ≤ 2K0 (t ),
t ∈ (0, t ].
(4.7)
and
ςn := sup s2α(γ −β) |F (un (s), un (s)) − F (u(s), u(s))|q s∈(0,T ]
≤ M (Kn (T ) + K (T )) sup sα(γ −β) |Aγ (un (s) − u(s))|q → 0, s∈(0,T ]
as n → ∞.
Finally, it remains to verify that u is a mild solution of problem (1.2) in [0, T ]. Since
|G(un , un )(t ) − G(u, u)(t )|q ≤
t
(t − s)α−1 ςn s−2α(γ −β) ds = t αβ ςn → 0,
(n → ∞),
0
we have G(un , un )(t ) → G(u, u)(t ). Taking the limits on both sides of (4.2), we derive u(t ) = u0 (t ) + G(u, u)(t ).
(4.8)
Let u(0) = a, we find that (4.8) holds for t ∈ [0, T ] and u ∈ C ([0, T ], Jq ). What is more, the uniform convergence of t α(γ −β) Aγ un (t ) to t α(γ −β) Aγ u(t ) derives the continuity of Aγ u(t ) on (0, T ]. From (4.7) and K0 (0) = 0, we get that |Aγ u(t )|q = o(t −α(γ −β) ) is obvious. Step 3. We prove that the mild solution is unique. Suppose that u and v are mild solutions of problem (1.2). Let w = u −v , we consider the equality
w(t ) =
t
(t − s)α−1 Eα,α (−(t − s)α A)[F (u(s), u(s)) − F (v(s), v(s))]ds.
0
Introducing the functions
K (t ) := max{ sup sα(γ −β) |Aγ u(s)|q , sup sα(γ −β) |Aγ v(s)|q }. s∈(0,t ]
s∈(0,t ]
By (3.1) and Lemma 3.2, we get
|Aγ w(t )|q ≤ MC1 K (t )
t
(t − s)α(1−γ )−1 s−α(γ −β) |Aγ w(s)|q ds.
0
Gronwall inequality shows that Aγ w(t ) = 0 for t ∈ (0, T ]. This implies that w(t ) = u(t ) − v(t ) ≡ 0 for t ∈ [0, T ]. Therefore the mild solution is unique.
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5. Regularity In this section, we consider the regularity of a solution u which satisfies problem (1.2). Throughout this part we assume that:
(f1 ) Pf (t ) is Hölder continuous with an exponent ϑ ∈ (0, α(1 − γ )), that is, |Pf (t ) − Pf (s)|q ≤ L|t − s|ϑ ,
for all 0 < t , s ≤ T .
Definition 5.1. A function u : [0, T ] → Jq is called a classical solution of problem (1.2), if u ∈ C ([0, T ], Jq ) with (t ) ∈ C ((0, T ], Jq ), which takes values in D(A) and satisfies (1.2) for all t ∈ (0, T ].
C t Dt u
Lemma 5.1. Let (f1 ) be satisfied. If
Φ1 (t ) :=
t
(t − s)α−1 Eα,α (−(t − s)α A) Pf (s) − Pf (t ) ds,
for t ∈ (0, T ],
0
then Φ1 (t ) ∈ D(A) and AΦ1 (t ) ∈ C ϑ ([0, T ], Jq ). Proof. For fixed t ∈ (0, T ], from Lemma 3.2 and (f1 ), we have
(t − s)α−1 |AEα,α (−(t − s)α A) Pf (s) − Pf (t ) |q ≤ (t − s)−1 |Pf (s) − Pf (t )|q ≤ C1 L(t − s)ϑ−1 ∈ L1 ([0, T ], Jq ),
(5.1)
then
t
(t − s)α−1 AEα,α (−(t − s)α A) Pf (s) − Pf (t ) q ds 0 t C1 K ϑ ≤ C1 L (t − s)ϑ−1 ds ≤ t < ∞. ϑ 0
|AΦ1 (t )|q ≤
By the closeness of A, we obtain Φ1 (t ) ∈ D(A). We need to show that AΦ1 (t ) is Hölder continuous. Since
d α−1 t Eα,α (−µt α ) = t α−2 Eα,α−1 (−µt α ), dt then
d α−1 1 t AEα,α (−t α A) = dt 2π i = =
1
Γθ
2π i 1 2π i
Γθ
t α−2 Eα,α−1 (−µt α )A(µI + A)−1 dµ t α−2 Eα,α−1 (−µt α )dµ −
−t
α−2
Γθ′
1
1
2π i 1
Eα,α−1 (ξ ) α dξ − t 2π i
Γθ
t α−2 µEα,α−1 (−µt α )(µI + A)−1 dµ
t Γθ′
α−2
ξ
Eα,α−1 (ξ ) α t
−1 ξ 1 − αI +A dξ . α t
t
C In view of ∥(µI + A)−1 ∥ ≤ |µ| , we derive that
d α−1 α ≤ C α t −2 , t AE (− t A ) α,α dt
0 < t ≤ T.
By the mean value theorem, for every 0 < s < t ≤ T , we have
t d α−1 α ∥t α−1 AEα,α (−t α A) − sα−1 AEα,α (−sα A)∥ = τ AE (−τ A ) d τ α,α s dτ t d α−1 α ≤ dτ τ AEα,α (−τ A) dτ s t ≤ Cα τ −2 d τ = C α s −1 − t −1 . s
(5.2)
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Let h > 0 be such that 0 < t < t + h ≤ T , then AΦ1 (t + h) − AΦ1 (t ) =
t
(t + h − s)α−1 AEα,α (−(t + h − s)α A) 0 − (t − s)α−1 AEα,α (−(t − s)α A) Pf (s) − Pf (t ) ds t (t + h − s)α−1 AEα,α (−(t + h − s)α A) Pf (t ) − Pf (t + h) ds + 0 t +h
+
(t + h − s)α−1 AEα,α (−(t + h − s)α A) Pf (s) − Pf (t + h) ds
t
:= h1 (t ) + h2 (t ) + h3 (t ).
(5.3)
We estimate each of the three terms separately. For h1 (t ), from (5.2) and (f1 ), we have t
∥(t + h − s)α−1 AEα,α (−(t + h − s)α A) − (t − s)α−1 AEα,α (−(t − s)α A)∥ |Pf (s) − Pf (t )|q ds t ≤ Cα Lh (t + h − s)−1 (t − s)ϑ−1 ds 0 t ≤ Cα Lh (s + h)−1 (t − s)ϑ−1 ds
|h1 (t )|q ≤
0
0 h
h
≤ Cα L
s+h
0
sϑ−1 ds + Cα Lh
∞
h
s s+h
sϑ−1 ds
≤ Cα Lhϑ .
(5.4)
For h2 (t ), we use Lemma 3.2 and (f1 ), t
(t + h − s)α−1 |AEα,α (−(t + h − s)α A) Pf (t ) − Pf (t + h) |q ds 0 t ≤ C1 (t + h − s)−1 |Pf (t ) − Pf (t + h)|q ds 0 t ≤ C1 Lhϑ (t + h − s)−1 ds
|h2 (t )|q ≤
0
= C1 L[ln h − ln(t + h)]hϑ .
(5.5)
Furthermore, for h3 (t ), by Lemma 3.2 and (f1 ), we have
|h3 (t )|q ≤
t +h
(t + h − s)α−1 |AEα,α (−(t + h − s)α A) Pf (s) − Pf (t + h) |q ds
t t +h
(t + h − s)−1 |Pf (s) − Pf (t + h)|q ds
≤ C1 t
t +h
≤ C1 L
(t + h − s)ϑ−1 ds = C1 L
t
hϑ
ϑ
.
Combining (5.4), (5.5) with (5.6), we deduce that AΦ1 (t ) is Hölder continuous.
(5.6)
Theorem 5.1. Let the assumptions of Theorem 4.1 be satisfied. If (f1 ) holds, then for every a ∈ D(A), the mild solution of (1.2) is a classical one. Proof. For a ∈ D(A). Then Lemma 2.2(ii) ensures that u(t ) = Eα (−t α A)a (t > 0) is a classical solution to the following problem Dαt u = −Au, u(0) = a.
C
t > 0,
Step 1. We verify that
Φ (t ) =
t
0
(t − s)α−1 Eα,α (−(t − s)α A)Pf (s)ds
Y. Zhou, L. Peng / Computers and Mathematics with Applications (
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is a classical solution to the problem Dαt u = −Au + Pf (t ), u(0) = 0.
C
t > 0,
It follows from Theorem 4.1 that Φ ∈ C ([0, T ], Jq ). We rewrite Φ (t ) = Φ1 (t ) + Φ2 (t ), where
Φ1 (t ) =
Φ2 (t ) =
t
(t − s)α−1 Eα,α (−(t − s)α A) Pf (s) − Pf (t ) ds,
0 t
(t − s)α−1 Eα,α (−(t − s)α A)Pf (t )ds.
0
According to Lemma 5.1, we know that Φ1 (t ) ∈ D(A). To prove the same conclusion for Φ2 (t ). By Lemma 2.2(iii), we notice that AΦ2 (t ) = Pf (t ) − Eα (−t α A)Pf (t ). Since (f1 ) holds, it follows that
|AΦ2 (t )|q ≤ (1 + C1 )|Pf (t )|q , thus
Φ2 (t ) ∈ D(A) for t ∈ (0, T ] and AΦ2 (t ) ∈ C ν ((0, T ], Jq ).
(5.7)
Next, we prove CDαt Φ ∈ C ((0, T ], Jq ). In view of Lemma 2.2(iv) and Φ (0) = 0, we have C α Dt Φ
(t ) =
d d 1−α It Φ ( t ) = (Eα (−t α A) ∗ Pf ). dt dt
It remains to prove that Eα (t α A) ∗ Pf is continuously differentiable in Jq . Let 0 < h ≤ T − t, one derives the following: 1 h
α
α
Eα (−(t + h) A) ∗ Pf − Eα (−t A) ∗ Pf =
t
0
1
+
1 Eα (−(t + h − s)α A)Pf (s) − Eα (−(t − s)α A)Pf (s) ds h t +h
h
Eα (−(t + h − s)α A)Pf (s)ds.
t
Notice that t
1
|Eα (−(t + h − s)α A)Pf (s) − Eα (−(t − s)α A)Pf (s)|q ds 1 t 1 t α |Eα (−(t + h − s) A)Pf (s)|q ds + C1 |Eα (−(t − s)α A)Pf (s)|q ds ≤ C1 h 0 h 0 1 t 1 t ≤ C1 M (t ) (t + h − s)−α s−α(1−β) ds + C1 M (t ) (t − s)−α s−α(1−β) ds h
0
≤ C1 M (t )
h 1 h
h
0
(t + h)1−α + t
1−α
0
B(1 − α, 1 − α(1 − β)),
then using the dominated convergence theorem, we find t
1
lim
h→0
h
0
Eα (−(t + h − s)α A)Pf (s) − Eα (−(t − s)α A)Pf (s) ds =
t
(t − s)α−1 AEα,α (−(t − s)α A)Pf (s)ds
0
= AΦ (t ). On the other hand, 1 h
t +h
Eα (−(t + h − s)α A)Pf (s)ds
t
= =
1 h 1 h
h
Eα (−sα A)Pf (t + h − s)ds
0 h
Eα (−sα A) Pf (t + h − s) − Pf (t − s) ds +
0
1 h
h
Eα (−sα A) Pf (t − s) − Pf (t ) ds +
0
1 h
h
0
Eα (−sα A)Pf (t )ds.
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From Lemmas 2.1, 3.2 and (f1 ), we have
h 1 α Eα (−s A) Pf (t + h − s) − Pf (t − s) ds ≤ C1 Lhϑ , h 0 q h 1 hϑ Eα (−sα A) Pf (t − s) − Pf (t ) ds ≤ C1 L . h ϑ +1 0
q
Also Lemma 2.2(i) gives that limh→0 lim
h→0
1 h
t +h
h 1 h
0
Eα (sα A)Pf (t )ds = Pf (t ). Hence
Eα ((t + h − s)α A)Pf (s)ds = Pf (t ).
t
We deduce that Eα (t α A) ∗ Pf is differentiable at t+ and dtd Eα (t α A) ∗ Pf + = AΦ (t ) + Pf (t ). Similarly, Eα (t α A) ∗ Pf is differentiable at t− and dtd Eα (t α A) ∗ Pf − = AΦ (t ) + Pf (t ).
We show that AΦ = AΦ1 + AΦ2 ∈ C ((0, T ], Jq ). In fact, it is clear that Φ2 (t ) = Pf (t )− Eα (t α A)Pf (t ) due to Lemma 2.2(iii), which is continuous in view of Lemma 2.1. Furthermore, according to Lemma 5.1, we know that AΦ1 (t ) is also continuous. Consequently, CDαt Φ ∈ C ((0, T ], Jq ). Step 2. Let u be the mild solution of (1.2). To prove that F (u, u) ∈ C ϑ ((0, T ], Jq ), in view of (3.1), we have to verify that Aγ u is Hölder continuous in Jq . Take h > 0 such that 0 < t < t + h. Denote ϕ(t ) := Eα (−t α A)a, by Lemmas 2.2(iv) and 3.2, then
t +h −sα−1 Aγ Eα,α (−sα A)ads |Aγ ϕ(t + h) − Aγ ϕ(t )|q = t q t +h ≤ sα−1 |Aγ −β Eα,α (−sα A)Aβ a|q ds t t +h
≤ C1
sα(1+β−γ )−1 ds|Aβ a|q
t
=
C1 |a|H β,q
(t + h)α(1+β−γ ) − t α(1+β−γ )
α(1 + β − γ ) C1 |a|H β,q hα(1+β−γ ) . ≤ α(1 + β − γ )
Thus, Aγ ϕ ∈ C ϑ ((0, T ], Jq ). For every small ε > 0, take h such that ε ≤ t < t + h ≤ T , since
|Aγ Φ (t + h) − Aγ Φ (t )|q t +h α−1 γ α ≤ (t + h − s) A Eα,α (−(t + h − s) A)Pf (s)ds t q t + Aγ (t + h − s)α−1 Eα,α (−(t + h − s)α A) − (t − s)α−1 Eα,α (−(t − s)α A) Pf (s)ds 0
q
= φ1 (t ) + φ2 (t ). Applying Lemma 3.2 and (f), we get
φ1 (t ) ≤ C1
t +h
(t + h − s)α(1−γ )−1 |Pf (s)|q ds
t
≤ C1 M ( t )
t +h
(t + h − s)α(1−γ )−1 s−α(1−β) ds
t
≤ M (t ) ≤ M (t )
C1
α(1 − γ ) C1
α(1 − γ )
hα(1−γ ) t −α(1−β) hα(1−γ ) ε −α(1−β) .
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To estimate φ2 , we give the inequality
d α−1 γ 1 t A Eα,α (−t α A) = dt 2π i
µγ t α−2 Eα,α−1 (−µt α )(µI + A)−1 dµ
Γ
1
=
2π i
Γ′
−1 1 ξ ξ γ α−2 t Eα,α−1 (ξ ) − α I + A dξ , − − α α t
t
t
this yields that ∥ dt t α−1 Aγ Eα,α (−t α A) ∥ ≤ Cα t α(1−γ )−2 . The mean value theorem shows t d α−1 γ α−1 γ α α−1 γ α α ∥t A Eα,α (−t A) − s A Eα,α (−s A)∥ ≤ dτ τ A Eα,α (−τ A) dτ s t τ α(1−γ )−2 dτ = Cα sα(1−γ )−1 − t α(1−γ )−1 , ≤ Cα d
s
thus
φ2 (t ) ≤
t
0
γ A (t + h − s)α−1 Eα,α (−(t + h − s)α A) − (t − s)α−1 Eα,α (−(t − s)α A) Pf (s) ds q
t
(t − s)α(1−γ )−1 − (t + h − s)α(1−γ )−1 |Pf (s)|q ds 0 t t +h ≤ Cα M (t ) (t − s)α(1−γ )−1 s−α(1−β) ds − (t − s + h)α(1−γ )−1 s−α(1−β) ds
≤
0
0 t +h
α(1−γ )−1 −α(1−β)
+ Cα M ( t ) (t − s + h) s ds t ≤ Cα M (t ) t α(β−γ ) − (t + h)α(β−γ ) B(α(1 − γ ), 1 − α(1 − β)) + Cα M (t )hα(1−γ ) t −α(1−β) ≤ Cα M (t )hα(γ −β) [ε(ε + h)]α(β−γ ) + Cα M (t )hα(1−γ ) ε−α(1−β) , which ensures that Aγ Φ ∈ C ϑ ([ε, T ], Jq ). Therefore Aγ Φ ∈ C ϑ ((0, T ], Jq ) due to arbitrary ε . Recall
Ψ (t ) =
t
(t − s)α−1 Eα,α (−(t − s)α A)F (u(s), u(s))ds.
0
Since |F (u(s), u(s))|q ≤ MK 2 (t )s−2α(γ −β) , where K (t ) := sups∈[0,t ] sα(γ −β) |u(s)|H γ ,q is continuous and bounded in (0, T ]. A similar argument enable us to give the Hölder continuity of Aγ Ψ in C ϑ ((0, T ], Jq ). Therefore, we have Aγ u(t ) = Aγ ϕ(t ) + Aγ Φ (t ) + Aγ Ψ (t ) ∈ C ϑ ((0, T ], Jq ). Since F (u, u) ∈ C ϑ ((0, T ], Jq ) is proved, according to Step 2, this yields that CDαt Ψ ∈ C ((0, T ], Jq ), AΨ ∈ C ((0, T ], Jq ) and C α Dt Ψ = −AΨ + F (u, u). In this way we obtain that CDαt u ∈ C ((0, T ], Jq ), Au ∈ C ((0, T ], Jq ) and CDαt u = −Au + F (u, u) + Pf , we conclude that u is a classical solution. Theorem 5.2. Assume that (f1 ) holds. If u is a classical solution of (1.2), then Au ∈ C ν ((0, T ], Jq ) and CDαt u ∈ C ν ((0, T ], Jq ). Proof. If u is a classical solution of (1.2), then u(t ) = ϕ(t ) + Φ (t ) + Ψ (t ). It remains to show that Aϕ ∈ C α(1−β) ((0, T ], Jq ), it suffices to prove that Aϕ ∈ C α(1−β) ([ε, T ], Jq ) for every ε > 0. In fact, take h such that ε ≤ t < t + h ≤ T , by Lemma 2.2(iii), t +h
|Aϕ(t + h) − Aϕ(t )|q =
−s
A Eα,α (−s A)ads
α−1 2
t
α
q
t +h
s−α(1−β)−1 ds|a|H β,q
≤ C1 t
=
C1 |a|H β,q −α(1−β) t − (t + h)−α(1−β)
≤
C1 |a|H β,q
hα(1−β)
α
[ε(ε + h)]α(1−β)
α
.
Similar to Lemma 5.1, we write Φ (t ) as
Φ (t ) = Φ1 (t ) + Φ2 (t ) =
t
(t − s) 0
α−1
α
Eα,α (−(t − s) A) Pf (s) − Pf (t ) ds +
t
(t − s)α−1 Eα,α (−(t − s)α A)Pf (t )ds,
0
for t ∈ (0, T ]. It follows from Lemma 5.1 and (5.7) that AΦ1 (t ) ∈ C ν ([0, T ], Jq ) and AΦ2 (t ) ∈ C ϑ ((0, T ], Jq ), respectively.
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–
Since F (u, u) ∈ C ϑ ((0, T ], Jq ), the result related to the function Ψ (t ) is proved by similar argument, which means that AΨ ∈ C ν ((0, T ], Jq ). Therefore Au ∈ C ν ((0, T ], Jq ) and CDαt u = Au + F (u, u)+ Pf ∈ C ν ((0, T ], Jq ). The proof is completed. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
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