Remarks on the Systems of Semilinear Fractional Rayleigh- Stokes Equation

In this paper, we study the Cauchy problem for a system of Rayleigh-Stokes equations. In this system of equations, we use derivatives in the classical Riemann-Liouville sense. This system has many applications in some non-Newtonian fluids. We obtained results for the existence, uniqueness, and frequency of the solution. We discuss the stability of the solutions and find the solution spaces. Our main technique is to use the Banach mapping theorem combined with some techniques in Fourier analysis.

In fluid dynamics, the Rayleigh problem is the first Stokes problem, which determines the flow generated by the sudden motion of an infinitely long plate from the resting state, named after Lord Rayleigh and Sir George Stokes. This problem is considered to be one of the simplest problems with the correct solution for the Navier-Stokes equation. In recent times, with the development of fractions, a number of authors such as Shen et al. [21] investigated Rayleigh-Stokes, which is a more general form than the classical model. The fractional Rayleigh-Stokes equation (1) has applications in non-Newtonian behavior of fluids [21], and other applications of this equation can be given in [21,22]. We list some papers on fractional Rayleigh stokes in the following.
(i) The initial and boundary values for the Rayleigh-Stokes problem in the case of homogeneity have been explored in a number of interesting papers; see for example [23][24][25][26][27] and its references (ii) The authors in [21,28,29] used the Fourier transform and the fractional Laplace transform to obtain the exact solution (iii) Numerical solutions for Problem (1) has been studied by many authors in [4,23,24,30,31] (iv) In [32], the authors concerned with the following problem for a following stochastic Rayleigh-Stokes equation The existence and uniqueness of mild solution in each case are established separately by applying a standard method that is Banach fixed point theorem. In [22], Caraballo et al. investigated the following time-fractional Rayleigh-Stokes stochastic equation where fW ðt,:Þg t∈ J represents a standard Wiener process.
To the best of the author's knowledge, the problem of the system of equations for a fractional Rayleigh-Stokes with a nonlinear source, i.e., Problem (1), has yet to be studied. The goal of this paper is to develop a theory of the existence and regularity estimate for the mild solution to the Problem (1). Our main technique is to use the Banach mapping theorem combined with some techniques in Fourier analysis.

The Existence and Regularity of the Solution
In this section, we consider the existence and mild solution of Problem (1). Before going into the main theorem of this section, we briefly discuss spectral, eigenvalues, and related functional spaces on the Laplacian operator.
The definition of the negative fractional power A −s can be found in [33]. Its domain DðA −s Þ is a Hilbert space endowed with the dual inner product h:,:i −s,s taken between DðA −s Þ and DðA s Þ. This generates the norm A couple ðu, vÞ of functions uðx, tÞ, vðx, tÞ: , TÞ are called a function of two variables x, t u, v ð Þ: Here, the norm of ðu, vÞ ∈ X × X (for any space X) is defined for any v 1 , v 1 , v 2 , v 2 ∈ L 2 ðΩÞ. Then, problem (1) has a unique solution ðu, vÞ belongs to L ∞ ð0, T ; L 2 ðΩÞÞ × L ∞ ð0, T ; L 2 ðΩÞÞ and regularity estimates hold Proof. Assume that the mild solution u is described by a Fourier series Thanks to the results of [24], we deduce that the solution of Problem (1) with the initial condition uðx, 0Þ = φðxÞ is given by where K is represented as follows 2

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Hence, the mild solution of Problem (1) is given by We get the following estimate Now, we continue to show that (1) has a unique mild solution.
For any a ≥ 0, denote by ðL ∞ a ð0, T ; H m ðΩÞÞÞ 2 the function space ðL ∞ ð0, T ; H m ðΩÞÞÞ 2 associated with the norm for any Let us give the following operator where Q 1 and Q 2 are defined by the following From two equality as above, if ðχ 1 , χ 2 Þ = ð0, 0Þ, we find that two following equality Hence, we get that for any a > 0 Since the fact that we know that Combining (22), (24), and (25), we find that which allows us to deduce that 3 Advances in Mathematical Physics Let two functions ðχ 1 , χ 2 Þ and ð χ 1 , χ 2 Þ in the space ðL ∞ a ð0, T ; L 2 ðΩÞÞÞ 2 . Then, using Parseval's equality, we get Noting that Hence, and noting that λ Let us emphasize that for 0 ≤ s ≤ T then Combining (30) and (31), we find that where we denote Next, we need to deal with the integral quantity Int1 = Ð t 0 ðt − sÞ α−1 e −aðt−sÞ ds. By change variable s = tγ, we find that Thank the inequality y ≤ e y for y ≥ 0, we know that Combining with the fact that Ð 1 0 ð1 − γÞ α/2−1 dγ = 2/α, we deduce that the following inequality This inequality together with (32) leads to By a similar way, we also get that Therefore, we can deduce that

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Since the limitation we know that there exists a positive a * such that 4C 1 ðα, β, d, mÞ/αðT/a * Þ α/2 < 1: Thus, we can deduce that Q is contractive on ðL ∞ a * ð0, T ; H m ðΩÞÞÞ 2 . Applying Banach fixed point theorem, we get that Q has a fixed point ðu, vÞ, so, the function ðu, vÞ is also the unique solution of (1). Since (14), we find that By looking at (24) and (29), we follow from (41) that where we have used that Gð0, 0Þ = 0 and Hence, we derive that By a similar way, we also obtain that the following estimate Combining (44) and (45), we get that By applying Gronwall's inequality as in [34], we obtain that Hence, we can deduce that ðu, vÞ belongs to L ∞ ð0, T ; L 2 ðΩÞÞ × L ∞ ð0, T ; L 2 ðΩÞÞ, and furthermore, we also derive that Proof. For k > 0, we set the following space It is easy to see that First, we look at the second term J 2 ðx, t′ − tÞ. Using the inequality, we find that Using (16), we obtain that Combining (52) and (53) and using the inequality ðc + dÞ α ≤ c α + d α for any 0 < α < 1, we get that Next, we consider the second term J 2 ðx, t ′ − tÞ. It is easy to observe that ð55Þ