On well-posedness of semilinear Rayleigh-Stokes problem with fractional derivative on ℝ^N

1 Introduction

Fractional calculus has proved a powerful tool to describe the viscoelasticity of fluids and anomalous diffusion phenomena, such as the constitutive relationship of the fluid models [21], basic random walk models [25], and so on. Besides in recent years, mainly due to the nonlocal characteristic of the fractional derivative, there are some excellent works on stochastic processes driven by fractional Brownian motion [13] and on physical phenomena like inverse problems for heat equation [19] and memory effect [2]. It is worth mentioning some solid works about time-fractional derivatives [1, 4, 7, 12, 14, 17, 37, 38, 39] and space-fractional derivatives [15, 16, 33] and the references therein, most of conclusions in these works commuted with fractional models are quite different from the situation of integer derivative, for instance, decay and asymptotical behaviors, blow-up analysis, well-posedness analysis, stability and Liouville property etc..

It is known that many significant complex media are non-Newtonian and exhibit time-dependent behavior of thixotropy and rheopecty [23, 28]. The behavior in non-Newtonian fluid often follows the power law [24]. Time-dependent non-Newtonian properties are more closely linked to fractional viscoelasticity than previously thought. Especially in a generalized second grade fluid, it is also more difficult to construct a simple mathematical model for describing many different behavior of non-Newtonian fluids. Form this physics point of view, in the second grade fluid, the employed constitutive relationship has the following form:

where σ is the Cauchy stress tensor, p is the hydrostatic pressure, I is the identity tensor. μ ≥ 0, ϱ₁ and ϱ₂ are normal stress moduli. ϵ₁ and ϵ₂ are the kinematical tensors defined by

where V is the velocity, ∇ is the gradient operator and the superscript T denotes a transpose operation. When we consider the time-dependent of time derivative in the kinematical tensor ϵ₁, generally the constitutive relationship of viscoelastic second grade fluids has the form as follows:

where ∂tα is the Riemann-Liouville fractional derivative of order α ∈ (0, 1) defined by

provided the right-hand side is pointwise defined, where Γ(⋅) stands for the Euler’s Gamma function. The form of the model was selected for its ability to portray accurately the temperature distribution in a generalized second grade fluid that subject to a linear flow on a heated flat plate and within a heated edge, see [32]. In this paper, we concern with the following semilinear time-fractional Rayleigh-Stokes problem

where Δ is the Laplacain operator, f is a semilinear function and φ is a given initial condition in L^p(ℝ^N).

Such type of problems play a central role in describing the viscoelasticity of non-Newtonian fluids behavior and characteristic. Many researchers showed a strong interest in this issue and they also obtained some satisfactory results. Here is a short description on the closely related works and comparision to our results. The Rayleigh-Stokes problem for a generalized second grade fluid subject to a flow on a heated flat plate and within a heated edge was introduced by Shen et al. [32] where their considered the exact solutions of the velocity and temperature fields. As for a viscous Newtonian fluid, their revealed that the solutions of the Stokes' first problem appear in the limiting for these exact solutions. The results in [32] were generalized later by Xue and Nie [34] in a porous half-space with a heated flat plate, they also obtained an exact solution of the velocity field and temperature fields, from which some classical results can be recovered. Both methods of two above papers are based on the Fourier sine transform and the fractional Laplace transform. For the smooth and nonsmooth initial data on a bounded domain Ω ⊂ ℝ^N (N = 1, 2, 3), Bazhlekova et al. [5] considered the solutions of the homogeneous problem on C([0, T]; L²(Ω)) ∩ C((0, T]; H01 (Ω) ∩ H²(Ω)) for the initial value φ ∈ L²(Ω), moreover, some operator theory and spectrum technique are used to also establish the related Sobolev regularity of the solutions. In the meantime, Bazhlekova [6] obtained a well-posedness result under the abstract analysis framework of a subordination identity relating to the solution operator associated with a bounded C₀-semigroup and a two parameters probability density function. Zhou and Wang [40] also established the existence results on C([0, T]; L²(Ω))of nonlinear problem in operator theory on a bounded domain Ω with smooth boundary. As for some numerical solution of Rayleigh-Stokes problem with fractional derivatives, several scholars have considered and developed it, for example Zaky [36], Chen et al. [10, 11], Yang and Jiang [35] etc.. Additionally, most of fluid flows and transport processes use more distribution parameters to establish equations, the inverse problem of parameter identification has been proposed to deal with this matter, see Nguyen et al. [26, 27].

In view of the results in above works, it is natural to consider whether the well-posedness result on ℝ^N can be extended to the mixed norm L^p(L^q) spaces and whether it is possible to obtain the integrability of solutions. Unfortunately, it turns out that these extensions cannot be established by applying the technique of subordination principle in [6]. Moreover, some useful L^p − L^q inequality estimates about solution operator generated by the Eq. (1.1) are not easy to obtain, in which it is not immediately suitable to build an integral equation when we consider that the time-fractional derivative depends on all the past states, and also we cannot apply the approach of classical solution operators to derive the relevant estimates, while it is readily to achieve at the classical solution operators of type heat operator, like fractional diffusion equations [22] and fractional Navier-Stokes equations [30]. To overcome the difficulty from the effect of time-fractional derivative, we propose a different technique to estimate the solution operator by means of the Gagliardo-Nirenberg inequality and generalized Gagliardo-Nirenberg inequality. For the proof, by using an admissible triplet concept that depends on the time-fractional derivative order α ∈ (0, 1) and the exponents p, q in L^p(L^q) spaces and their dimensions, we shall use the standard fixed point argument to establish main well-posedness results. We also consider a contain special space to make the local existence, that is due to the decay exponent just depends on the order of time-fractional derivative deriving from the solution operator. We find that the local solution will blow up in L^r(ℝ^N), and then the rate of the blow up solution may depend on the exponent of nonlinearity and r ≥ 2 in L^r(ℝ^N). Also, based on the standard harmonic analysis methods, such as Marcinkiewicz interpolation theorem and doubly weighted Hardy-Littlewood-Sobolev inequality, some new conclusions likely the integrability of global mild solutions in Lebesgue spaces L^μ(0, ∞; L^r(ℝ^N)) are investigated.

This paper is organized as follows. In Section 2, we give some concepts about fractional calculus in Banach space and we introduce several useful analytic properties of Laplacian operator. By a rigorous analysis of solution operator S(t), we establish two crucial estimates that will be used throughout this paper. After introducing a definition of mild solution in Section 3, the first subsection shows the global and local well-posedness of the semilinear problem (1.1). Further, we obtain continuation and blow-up alternative of local mild solution of problem. In the last subsection, we show several integrability results of the global mild solution in Lebesgue space.

2 Preliminaries

Let (X, ∥⋅∥) be a Banach space and let 𝓛(X, Y) stand for the space of all linear bounded operators maps Banach space X into Banach space Y, we remark that C_b(ℝ₊, X) stands for the space of bounded continuous functions which is defined on ℝ₊ and takes values in X, equipped with the norm sup_t∈ℝ+∥⋅∥_X and C(J, X) stands for the space of continuous functions which is defined on an interval J ⊆ ℝ₊ and takes values in X. If A is a linear closed operator, the symbols ρ(A) and σ(A) are called the resolvent set and the spectral set of A, respectively, identity R(λ; A) = (λ I − A)⁻¹ is the resolvent operator of A. We will denote by D(A^α), α ∈ (0, 1), the fractional power spaces associated with the linear closed operator A.

An operator A is called the sectorial operator, if it follows the next concept.

Additionally, from [8, Theorem 2.3.2], it is not difficult to check that the Laplacian operator Δ with maximal domain D(Δ) = {u ∈ X : Δ u ∈ X} generates a bounded analytic semigroup of the spectral angle less than or equal to π/2 on X := L^p(ℝ^N) with 1 ≤ p < + ∞. Moreover, the spectrum is given by σ(− Δ) = [0, +∞) for 1 < p < +∞.

For δ > 0 and θ ∈ (0, π/2) we introduce the contour Γ_δ,θ defined by

where the circular arc is oriented counterclockwise, and the two rays are oriented with an increasing imaginary part. In the sequel, let A = − Δ, then A is a densely defined linear closed operator on Banach space L^p(ℝ^N) with 1 < p < +∞, we define a linear operator S(t) by means of Dunford integral as follows

where

Recall that for any θ > 0

It should be noticed that estimates of the operator S(t) are standard in the theory of analytic semigroups as follows.

The inequality in (2.2) enables us to get another estimate about S(t)x in fractional power spaces. To do this, we need the following inequality.

Now, we introduce the concept of admissible triplet.

Thenceforth, we give some useful L^p − L^q estimates about the linear operator S(t).

In the sequel, we set a function W_f with respect to f ∈ L¹(0, T; X) with any T > 0 (or T = +∞), given by

in which we will prove some properties of this function.

3 Well-posedness

Let u be a solution of problem (1.1), taking the Laplace transform into (1.1) yields

by means of the inverse Laplace transform, we thus derive an integral representation of problem (1.1) by

following this, we regard S(t) defined in (2.1) as the solution operator of problem (1.1). Next, we introduce the definitions of global/local mild solutions to the problem (1.1).

In the sequel, the well-posedness of problem (1.1) will be considered, in order to achieve this goal, the following general hypothesis of the semilinear function introduced by [9] will be also considered. Let r′, r be the conjugate indices.

1. We suppose that f(0) = 0 and f : L^r(ℝ^N) → L^r′(ℝ^N), for some

   r∈2,2NN−2,ifN≥2,(r∈[2,∞],ifN=1).

   Additionally, we suppose that there exist constants σ ≥ 0 and K > 0 such that

   ∥f(u)−f(v)∥Lr′(RN)≤K(∥u∥Lr(RN)σ+∥v∥Lr(RN)σ)∥u−v∥Lr(RN),

   for all u, v ∈ L^r(ℝ^N).

We first consider the case T = +∞, i.e., the global well-posedness of the problem for mild solutions. For any α ∈ (0, 1), let (p, r, μ) be an admissible triplet such that 1 < r′ ≤ p < r ≤ +∞, consider the Banach space X_pr of continuous functions v : [0, ∞) → L^p(ℝ^N) equipped with its natural norm

In particular, from the restriction on μ in Theorem 3.1 and the admissible triplet (2, N + 2, μ), the dimension N in Corollary 3.1 is N=41−α for some suitable α ∈ (0, 1).

Now, let us turn to the case T > 0 and discuss the local well-posedness of the problem. Let α ∈ (0, 1) and (p, r, μ) be an admissible triplet such that 1 < p ≤ r < +∞ and p = r′ < N, and

Consider the Banach space Y_pr[T] of continuous functions v : [0, T] → L^p(ℝ^N) under this admissible triplet by satisfying

equipped with its natural norm