1. Introduction
Abstract fractional differential equations have been applied to many fields in science and engineering, such as in viscoelastic mechanics, anomalous diffusion phenomena, materials science, electrochemistry etc.; for more details we refer to the books and the papers [
1,
2,
3,
4,
5,
6,
7]. It is known that the abstract fractional differential equations can be used to study some partial differential equations with fractional derivatives in an appropriate work space using an operator-theoretic approach. When considering a nonlinear constitutive relationship between shear stress and shear strain rate in fluids, non-Newtonian fluids appear in human blood, oil, and mud-rock flow etc. that cannot be described in a single model, contrasted to the Newtonian fluids. As mentioned in [
8], fractional calculus has proved an effective tool for describing viscoelastic fluids; a fractional Rayleigh–Stokes problem in non-Newtonian fluids is more suitable for describing its qualitative properties and behaviors. It is reasonable to analyze the properties and structures of solutions using the operator -theoretic approach.
The exact solutions of fractional Rayleigh–Stokes equations in second grade fluid [
4,
9,
10], Maxwell fluid [
3,
11], and Oldroyed fluid [
12] were obtained by virtue of the Fourier sine transform and fractional Laplace transform. Under the conditions of a non-local integral term, Luc et al. [
13] obtained the existence and uniqueness of solutions for nonlinear equations; by using the Fourier truncation method, they constructed a regularization solution to tackle the ill-posedness of solutions. Wang et al. [
14] obtained the well-posedness for nonlinear Rayleigh–Stokes equations in view of the fixed point arguments and they also showed the blow-up results. Nguyen et al. [
15] obtained some regularity properties of the solutions to the backward problem of determining initial conditions. Lan [
16] analyzed some sufficient conditions to ensure the global regularity of solutions and, if the nonlinearity is Lipschizian, then the mild solution of the given problem becomes a classical one. Wang et al. [
17] obtained the existence, uniqueness, and regularity of a weak solution in
by using the Galerkin method and they also proved an improved regularity result of a weak solution in the case of non-homogeneous term
and initial value
. Bao et al. [
18] studied an inverse problem with a nonlinear source and obtained some results on the existence and regularity of mild solutions. By using an operator-theoretic approach, Bazhlekova et al. [
19] obtained the well-posedness and Sobolev regularity of the homogenous Rayleigh–Stokes problem and Bazhlekova [
20] showed a well-posed result associated with the bounded
a-semigroup by means of the subordination principle. Pham et al. [
21] studied a final-value problem involving weak-valued nonlinearities and obtained the existence and Hölder regularity by using the regularity of the resolvent operators. Tran and Nguyen [
22] obtained the solvability and Hölder regularity on the embeddings of fractional Sobolev spaces.
In this paper, we considered the following abstract fractional differential equations:
where
is the Riemann–Liouville fractional derivative of order
,
is a positive parameter, and operator
A generates a bounded analytic semigroup on a Banach
X within some sectors
and
, in which
,
is an initial value and
f is a continuous function. A prototype example is given by the Rayleigh–Stokes problem on
for
, a Riemann–Liouville fractional partial derivative. Replacing
with a semilinear function
, the global well-posed result with a small initial value and a local well-posed result on
for some positive parameters
were considered by He et al. [
23].
We list several highlights in the following. Firstly, we note that there are few works concerned with the classical solutions of abstract evolution problem (
1), even with the fractional Rayleigh–Stokes problem on
or bounded domain
with smooth boundary
. The Hölder regularity of solutions is also still a considerable problem because the Hölder regularity of solutions plays an important role in the structure of solutions. Li [
24] studied the Hölder regularities of mild solutions for a class of fractional evolution equations with an order of
and the author showed that a mild solution is the classical one for
(
) especially. In [
25], Li and Li also considered the case of the order of
for the Hölder regularities of mild solutions. Alam et al. [
26] established the Hölder regularity of mild/strict solutions of fractional abstract differential equations of the order of
; the obtained results improved the existing results presented in [
25]. Allen et al. [
27] established a Hölder regularity theorem of De Giorgi–Nash–Moser type for a fractional diffusion equation, see e.g., [
28,
29,
30]. Secondly, when operator
A acts on an analytical semigroup
, it appears that
is bounded near
for
x in a Banach space
X (and it goes to 0 as
if
), and
is bounded in
for
. This means that, for studying the properties of classical solutions, the concept of an intermediate space is naturally introduced in order to reduce the requirement for
(
). For the consideration (
1), we showed that
is bounded in (0, 1) for
(
is defined in Equation (
2)); it is not suitable for discussing the requirements of
since
is no longer an analytic semigroup essentially, and should construct a new interpolation space to lower the spatial regularity on initial condition for the classical solutions. Thirdly, we also proved that a mild solution to problem (
1) is also a classical solution if
and
, even if it is a strict solution with zero initial value data. In particular, the results we obtained reflect the relevant properties and the structure of solutions of problem (
1).
For these targets, we established the existence and Hölder regularity of solutions to problem (
1) under an analytic resolvent
determined by
A as follows:
where
and for
and
, the contour
is defined by
the circular arc is oriented counterclockwise. We showed that the mild solution is Hölder-continuous for
,
. Additionally, the solution shall be singular at
for considering the Hölder continuous, in order to lower the regularity of the initial value data. By using the
K-method, we introduced a new interpolation space
compared to the classical one
driven by the analytic semigroup, and we proved that these two spaces are isometric isomorphic. In particular, if
and
, the mild solution is indeed a classical solution. If
, the solution will still be a strict solution. Especially, it possesses a Hölder regularity with an exponent of
for
and
. Our proofs of the main results are based on the analytic properties of
and the operator approach.
The present paper is constructed as follows. In
Section 2, in view of the Hardy type inequality, we showed several main properties of the analytic resolvent
. In
Section 3, we constructed a new interpolation space in terms of the analytic resolvent and we analyzed its properties. In
Section 4, we proved the existence and uniqueness of the solutions of the problem (
1), and we established the Hölder regularity of solutions. Finally, some examples are presented to check the main results.
2. Preliminaries
Let X and be two Banach spaces; the notation denotes the space of all bounded linear operators mapping from X into with the norm —for short, by . We denote by the space of continuous functions that from an interval to X. Let A be a linear closed operator; we set and by the resolvent set and spectral set of A, respectively, and the resolvent operator of A is given by . I is an identity operator. The notation ∧ denotes for any constant . For convenience, the notation C will denote a positive constant.
For
, the Hölder continuous function space
is defined by
equipped with the norm
.
For
, denote a space by
, equipped with norm
and
.
It is known that the
K-method is a classical method for producing real interpolation spaces; for every
and
, let
For any
,
, denote the following space
with its norm
. Then, the real interpolation
is a Banach space.
Note that
as
; in order to check
, it is sufficient to show that
for some fixed
. In terms of the analytic semigroup
generated by
A,
also has the following expression:
with norm
, see e.g., [
31].
Consider the weak singular kernel in Riemann–Liouville fractional integral
,
,
where
is the Gamma function. And let ∗ denote the convolution of functions
by
Definition 1. Let The Riemann–Liouville fractional integral of order is defined by Definition 2. Let The Riemann–Liouville fractional derivative of order is defined by Recall that the following definition of analytic resolvent
is introduced by Prüss [
32].
Definition 3. A resolvent is called analytic if the function admits an analytic extension to a sector for some . An analytic resolvent is said to be of analyticity type if, for each and , there is such that for .
Suppose that (
1) admits a solution, then the problem can be rewritten as the following integral equation:
By the variation of the parameters formula—for example, see [
20,
23]—the solution is given by
where
is defined in (
2). Note that, since
admits meromorphic extension to
,
, and
for
from [
23], it yields that the analytic resolvent
is an analyticity type
by Prüss ([
32], Theorem 2.1).
We observe that
A is the infinitesimal generator of an analytic semigroup
,
, we know that
, where
is the Laplace transform of
. From the identity of the Laplace transform,
it follows that
where the probability density function
satisfies the inverse Laplace integral,
Therefore—for example, see [
20]—by the uniqueness of the Laplace transform, it also yields
Lemma 1. Let be defined in Equation (2). Then, - (i)
For every , and ;
- (ii)
For each , ;
- (iii)
For any , and ;
- (iv)
For every , .
Proof. The proof is similar to ([
23], Lemma 2.2), so we omit it. □
Let be an arbitrary piecewise smooth simple curve in running from to and ; we have the following Lemma.
Lemma 2 ([
33], Lemma 4.1.1)
. Assume that the map , satisfying:- (i)
For , is holomorphic;
- (ii)
For , ;
- (iii)
For , there exists constant such that
Lemma 3. Let be defined in Equation (2). Then, the following results hold: - (i)
For each , for ;
- (ii)
The mapping and for , there holds - (iii)
uniformly in a compact interval.
Proof. Since
is an analytic semigroup, and for
,
, it can readily be seen that
is true by (
4).
Let us check
. From ([
23], Lemma 2.2), we know that
and
for
, also
. Thus, we get
. Function
is continuous for
,
, it yields from Lemma 2 that
belongs to
. Hence, it follows that
. Moreover, by the analyticity of
in
, we get that
Therefore, for
, from the identity of the inequality
, we have
For
, by Lemma 1, we note that
Therefore, we also obtain uniformly in a compact interval.
By the Laplace transform and its uniqueness, obverse that the integrals
and
are uniformly bounded on a compact interval. For
, by
, we have
From the identity
for
, it follows that:
which shows Equation (
5). The proof is completed. □
Corollary 1. Let be defined in Equation (2). Then,and . In particular, it follows that: Proof. Obverse that the integral,
is uniform in a compact interval, and then
By using Lemma 2 and applying the similar proof in Lemma 3 and the definition of operator , we get for any . Another conclusion follows the same approach. We thus obtain this corollary. □
Remark 1. By the similar proof in Corollary 1, from the analytic resolvent of , then for , , it also yields that .
Remark 2. Since A generates a bounded analytic semigroup on a Banach X within some sector and , the equation is valid on X for every , which derives that analytic resolvent is a solution operator of the homogeneous equation to problem (1) and also for , . Next, we introduce the concepts of solutions as follows:
Definition 4. A function is a mild solution of problem (1) on if the function u defined in Equation (3) belongs to . Definition 5. A function is called a classical solution of problem (1) on if u is continuous on , continuously differentiable on , and for , and (1) is satisfied on . Definition 6. A function is called a strict solution of problem (1) on if u is continuous on and continuously differentiable on , for and (1) is satisfied on . Clearly, the relations satisfy: the strict solution ⇒, the classical solution ⇒, and the mild solution. Next, we will use the Hardy-type inequalities involving the Riemann–Liouville fractional integral.
Lemma 4 ([
34])
. Let and . Then, for non-negative weight functions u and v, it yieldsiff for , Lemma 5 ([
35])
. Let u, v be non-negative weight functions. If there is constant , such that Let us recall the Hardy–Young inequality.
Lemma 6 ([
36])
. Let and . Then, for any and measurable function , it yields 4. The Existence and Hölder Regularity
Theorem 3. Let for and . Then, problem (1) admits a unique mild solution. Proof. It is clear from Lemma 1 and Hölder’s inequality that
From Lemma 1, we know that
. For
, we have
Moreover, Lemma 3 implies that
By Lemma 3, for
with small
, we have
We also show that
for
with small
. Similarly, we can obtain
for
with small
. Therefore, we obtain the continuity of
u and the uniqueness follows (
16). This means that
u is a unique mild solution to problem (
1). □
A basic computation shows that the Riemann–Liouville fractional integral has the following property:
Lemma 7. For , , let with , then .
Proof. We show the Hölder continuity of
for
. In fact, for any
, it yields
Since
with
, then
By the inequality
for
and
, we obtain
integrating the above inequality a.e.
, and for
, we get
Therefore, we derive the conclusion as follows:
The proof is completed. □
Remark 4. Note that in Lemma 4.1 in [24], for , , let with , then . In the sequel, we show the Hölder regularity of the mild solution.
Theorem 4. Let , for . Then, for every , for . If, moreover, for , , then , and if for , , , then . Especially, if , then for , . If , , then for any , .
Proof. The existence of the mild solution
u to problem (
1) follows Theorem 3 and it satisfies
. By Lemma 3, we know that
for all
. Hence, for every
,
, by the mean value theorem,
is Lipschitz-continuous on
. For
, from Lemma 1, we obtain
this implies that
. Consequently,
.
For
, by Remark 2, we know that
Obviously, is Hölder-continuous with exponent on . Let . Clearly, for . Lemma 7 shows that . Consequently, . Based on , the second result is shown.
Due to
, it yields that
for
. In fact, by Hölder inequality, let
; we derive that
which means that
. By Remark 4, one can check that
, thus
.
In particular, for
, it suffices to check that
for
. In fact, for
, we have
Since
for
. By Lemma 1 and Hölder inequality, we have
As for
, by Hölder inequality, we get
Combined with the above arguments, the fourth conclusion follows.
For , , from the proof of the first result, we know that is Lipschitz-continuous on . Similar to the proof of Theorem 3, we obtain . The proof is completed. □
The following means that the mild solution is a classical solution.
Theorem 5. Let , . Then, the following descriptions hold:
- (i)
, , and
- (ii)
u is a classical solution of (1).
Proof. By Lemma 1, it follows that
for all
. In fact, one can derive that
Since the integral
holds for all
, we see that
which shows that
From the assumptions and Theorem 3, we know that
u is the mild solution of problem (
1) and obviously
due to
for
,
, and
from Lemma 3, and
. Corollary 1 and Remark 1 show that
and
for
,
. This means that
is the classical solution of the homogeneous equation by Remark 2. Consequently, it suffices to check that the remaining is a classical solution of the nonhomogeneous equation.
From
, we have
. In fact, it yields
where we have used
and the inequality
Hence,
. Since
, by Remark 2, we also have
and
The definition of the fractional derivative shows that
combined with
. Thus,
u is a classical solution of (
1). □
The following corollary is immediate.
Corollary 2. Let , . Then, u is a strict solution of (1). Lemma 8. Let . Then .
Proof. By Lemma 1, we have
for
. For
, from Lemma 3, we see that
Thus, . □
In the sequel, we obtain the Hölder regularity of the classical solution.
Theorem 6. Let , for . Then, there exists a classical solution u of (1) satisfying for . Moreover, there holds Proof. Obviously, Theorem 3 implies that there is a unique mild solution
u. Lemma 1 and Corollary 1 show that
and
for any
. Hence, it suffices to check that
. Note that
for any
,
, it follows from Remark 2 that
From
, we have
Hence,
for any
. Therefore, by
, we obtain
for any
. Thus,
u is the classical solution of (
1).
Additionally, by Lemma 8, we get that
. From Lemma 3 (
iii), we next check
. In fact, for any
, by Remark 2 and Corollary 1, we see that
Thus, for Then, The inequality easily follows the above arguments. The proof is completed. □
Theorem 7. Let and let , , . If , for , then the classical solution . If, moreover, , then . In particular, if , , then the strict solution and .
Proof. From Theorem 5, we know that the mild solution
u is a classical one for
. Using the proof of Theorem 4, we get that
. Moreover, by the proof of Theorem 3, we see from Hölder inequality that
for
. Consequently,
.
By using the proof of Lemma 8, it follows that
. By
and Lemma 3, we have
From the proof of the third conclusion of Theorem 4, we know that for . Thus, by Remark 4, it follows that . Furthermore, is Hölder-continuous with exponent . Hence, . Therefore, .
Let us check the case . In fact, Theorem 6 and the mentioned arguments in the current proof imply that . Since , from the proof of Theorem 5, we obtain . For , by Remark 2 and the proof of Theorem 4, we get . Hence, . The proof is completed. □
5. Applications
Let
. We consider the following fractional partial differential equation:
where
is the Caputo fractional partial derivative of order
,
is the Laplace operator, and
f takes the
data.
Note from ([
37], Theorem 2.3.2) that the Laplace operator
with maximal domain
generates a bounded analytic semigroup of the spectral angle that is less than or equal to
on
with
. We set
. By He et al. [
23], the problem (
18) can be reformulated as problem (
1). It follows that the analytic resolvent
generated by
A is defined in (
2). For the
data of
, we know that there exists a unique mild solution of (
18) from Theorem 3. Due to the interpolation
and if further
for
,
,
,
then
from Theorem 4. In addition, by Theorem 7, if
, then the mild solution will be a classical solution which possesses the Hölder continuity with the same exponent
and the estimate
For another consideration of the following initial-boundary value problem,
where
is a bounded domain and boundary
is
,
. It is clear that
is a realization of the Laplace operator
on
under Neumann type boundary condition
. It is known that
generates an analytic semigroup on
of the spectral angle that is less than
; by (
4) and the definition of
, the problem (
19) can be reformulated as problem (
1). Then, by a direct application of Theorem 6, we know that for any
,
for
, problem (
19) possesses a unique classical solution in the function space
with the estimate
In particular, we consider an initial-boundary value problem with a Dirichlet boundary value condition as follows:
where
and
,
.
Let
and let us consider the spectral problem on
, i.e.,
,
. It is known that
is the eigenvalue of
corresponding to the eigenfunction
. According to the discussion in [
14,
19], the problem (
20) can be rewritten as the sum of the Fourier series for problem
It follows that the analytic resolvent
is given by
where
Since
, the Corollary 2 shows that problem (
20) has a strict solution. In fact, one can check that
meets all equations in (
20) and
u is indeed the strict solution. Also, in view of
, it is clear that the strict solution
and
for a suitable number
by Theorem 7.