A study of the time fractional Navier-Stokes equations for vertical flow

1.   Introduction

To explain the flow of incompressible fluids in fluid mechanics, Navier-Stokes (NS) equations are applied as partial differential equations in order to discuss such fluids. Such equations constitute a modification of the equations composed by the Swiss mathematician Leonhard. In the 18th century, Euler illustrated the movement of frictionless and incompressible fluids. For the most realistic and challenging viscous fluid issue in 1821 Claude-Louis Navier, a French engineer, presented the element of viscosity (friction). Sir George Gabriel Stokes, a British physicist and mathematician, expanded this work in the mid-nineteenth century, however, comprehensive solutions have only been achieved for the case of 2-D flows. The complex vortexes and disturbance, or disruption, that arise in 3-D fluid flows (including gas) as velocities rise are demonstrated to be uncontrolled for every numerical analysis approach except approximation.

Because they characterize the physics of many scientific and engineering events, NS equations are useful. They might be models of climate, ocean waves, water flow in a pump, or air flow. The NS equations, in both their complete and simplified versions are applied for assistance with aviation and automobile design, blood flow studies, the design of power plants, pollution monitoring and a variety of other activities.

The NS equation, in modern notation for vertical flow:

where $\tilde{u}$ is defined as the fluid velocity vector, $P$ is the fluid pressure, $\rho$ is the fluid density, $\mu$ is the dynamic viscosity and $\nabla^2$ is the Laplacian operator.

The NS equations in mathematics demonstrate the conservation of speed and mass for Newtonian fluids. Sometimes the condition applied on equation associated with stress, temperature and body weight. They result from the application of Isaac Newton's second law to water movement, as well as the concept that the fluid stress is the sum of several viscous time (rate and velocity) and pressure respectively which describes the viscous flow. The difference between Euler equations and Navier equations is that the Euler equations model is just an inviscid flow whereas NS equations are viscosity sensitive. For that reason, NS equations are a parabolic with improved analytical properties at the cost of a constrained mathematical framework (for example, they are never fully integrated).

These equations are the central part of fluid flow modeling. Solving them for a set of specific boundary conditions (entrances, exits and walls) predicts fluid velocity and pressure for a given geometry. In view of the fact that complexity recognizes only a limited number of these equations analytical solutions ^[1], it is relatively convenient, for instance, to solve these equations for a flow between two parallel plates, or for the flow in a circular pipe. Abbas et al. ^[2] and Mehmood et al. ^[3] solved ordinary differential equations. Niazi et al.^[4], Iqbal et al. ^[5], Shafqat et al. ^[6], Alnahdi et al. ^[7], Khan et al. ^[8] and Abuasbeh et al. ^[9,10,11,12] investigated the existence and uniqueness of the fuzzy fractional evolution equations.

The Cauchy problem for NS equations in incompressible fluids is given by

If the fluid strikes orthogonally, i.e., if $\gamma = \dfrac{\pi}{2}$ then we have

Therefore we have,

while $\partial^\gamma_\upsilon$ denotes the fractional order Caputo derivative. At a point $x\in\omega$ and time $\upsilon > 0$ the velocity field vector is represented by $\tilde{u} = (\tilde{u}_1(\upsilon, x), \tilde{u}_2(\upsilon, x), ..., \tilde{u}_n(\upsilon, x))$, $\rho = \rho(\upsilon, x)$ denotes the pressure and $\upsilon$ is the viscosity. The gravitational force or body force is represented by $g = g(\upsilon, x)$ while the initial velocity is represented by $ax$. First of all we assume that $\Omega$ has a smooth boundary so in order to remove the pressure term we must apply, Helmholtz-Leray projector P in 1.3, which gives

In the divergence-free function space under discussion, the operator $-w{P}\triangle$ under Dirichlet boundary conditions is mainly the Stokes operator $A$. So, writing 1.3 in abstract form, we have

where $F(\tilde{u}, w)$ = $-P(\tilde{u}.\nabla)w$. If the Helmholtz-Leray projection $P$ and the Stokes operator $A$ makes, the solution of (1.5) is also the solution of (1.3). The fundamental goal is to prove the existence and uniqueness of global and local mild solutions of problem (1.5) in ${H^{\gamma, r}}$. Moreover, we also establish the regularity conclusions which claim that if $Pg$ is Hölder continuous then there is a unique classical solution $\tilde{u}(\upsilon)$ such that $A\tilde{u}$ and ${c}D_{\upsilon}^{\gamma}\tilde{u}(\upsilon)$ hold Hölder continuity in $J_r$.

2.   Preliminaries

In this section, we define the Gamma function, fractional order integral, Riemann-Liouville (RL) fractional derivative, Caputo fractional derivative and some more definitions, lemmas and theorems. For a brief review of fractional calculus definitions and properties, see ^[13].

Let the half space in ${\bf{R^n}}$, i.e., $\Omega = \mathcal{H} = {(x_1, ..., x_n) : x_n > 0}$ be the open subset of ${\bf{R^n}}$, where $n\geq3$. Let $1 < r < \infty$. Then we have the Hödge projection which is a bounded projection $P$ on $(L^r(\Omega)^n)$, the range for which is as follows:

and the null space of which is given as

For a suitable approach, let $J_r$ = $\overline{{C}^\infty_\sigma(\mathcal{H})}^{|.|_r}$, which is closed subspace of $(L^r(\mathcal{H}))^n$, $A = -\upsilon{P}\Delta$ is the Stokes operator in $J_r$ containing the domain $D_r(A) = D_r(\Delta)\cap J_r$. The Stokes operator, named George Gabriel Stokes is an unbounded linear operator which is used in the theory of partial differential equations and specifically in the fields of fluid dynamics and electro magnetics;

We have to introduce definitions of fractional power spaces that are related to $-A$. For $\gamma > 0$ and $v \in {J_r}$, define the following:

Therefore, $A^{-\gamma}$ is bounded ^[14], just like the injective operator on $J_r$. Suppose that $A^{-\gamma}$ is the inverse of $A^{-\gamma}$. For $\gamma > 0$ we symbolize the space ${H^{\gamma, r}}$ by the extent of $A^{-\gamma}$ with the norm

Definition 2.1. The fractional integration of order $\gamma > 0$ for a function $f$ is defined as

The RL fractional derivative for a function $v:[0, \infty) \rightarrow {\bf{R^n}}$ of order $\gamma\in{\bf{R^n}}$ is defined by

The RL fractional order integral is defined as

Thus by using the definitions of the RL fractional-order integral, we construct the Caputo fractional order differential operator.

Definition 2.2. ^[15] The Caputo fractional order derivative is defined as follows

Definition 2.3. ^[16] The Mittag-Leffler function was introduced by Magnus Gustaf (Gösta) Mittag-Leffler (Swedish mathematician) in 1902. It is a simple conclusion of the exponential function. Recently, some researchers have been focusing on the Mittag-Leffler function because of its application in the analysis of fractional differential equations (FDEs). It occurs often in the solutions of FDEs and fractional integral equations. The Mittag-Leffler function with the one-parameter $E_{\gamma}(t)$ is defined as

Now, let us consider the generalized Mittag-Leffler functions:

and

where

The function $M_\gamma$ is also known as the Mainardi function. In order to distinguish between the fundamental solutions for some standard boundary value problems, Mainardi introduced such a type of functions that is a special type of Wright function. It is impressive that the Mainardi function plays a role as a bridge between the classical abstract theories and fractional theories.

Theorem 2.1. Suppose that ${f_n}:{\bf{R^n}}\rightarrow [-\infty, \infty]$ denotes (Lebesgue) measurable functions such that the pointwise limit $f(x) = \lim_{n\rightarrow \infty}f_n(x)$ exists. Assume that there is an integrable $g:{\bf{R^n}}\rightarrow [0, \infty]$ with $|f_n(x)| \leq g(x)$ for each $x\in {\bf{R^n}}$. Then, $f$ is integrable as is $f_n$ for each n, and

(ⅰ) $E_{\gamma, \gamma}(-\upsilon^\gamma A) = \dfrac{1}{2\pi\iota}\int_{\Gamma_\theta}^ {E_{\gamma, \gamma}}{(-\mu{\upsilon}^\gamma)} {{(\mu{I} + A)}^{-1}} d\mu$;

(ⅱ) ${A^\gamma}{E_{\gamma, \gamma}}(-\upsilon^\gamma A) = \dfrac{1}{2\pi\iota}\int_{\Gamma_\theta}^{\mu^\gamma}{E_{\gamma, \gamma}}{(-\mu{\upsilon}^\gamma)} {{(\mu{I} + A)}^{-1}} d\mu$.

Proof. See the results in ^[17].

Let $\gamma\in(0, 1)$ and $-1 < r < \infty$, $\lambda > 0$. Then Mainardi function possess the following properties:

(ⅰ) $M_\gamma(\upsilon) \geq 0$ for all $\upsilon \geq 0$;

(ⅱ) $\int_{0}^{\infty} {\upsilon^r M_\gamma(\upsilon)}d\upsilon = \dfrac{\Gamma(r+1)}{\Gamma(\gamma r+1)}$;

(ⅲ) $\mathcal{L}\{\gamma \upsilon M_\gamma(\upsilon)\}(z) = E_{\gamma, \gamma}(-z)$;

(ⅳ) $\mathcal{L}\{ M_\gamma(\upsilon)\}(z) = E_\gamma(-z)$;

(ⅴ) $\mathcal{L}\{\gamma \upsilon^{-(1+\gamma)}M_\gamma(\upsilon^{-\gamma})\}(\lambda) = e^{-\lambda^\gamma}$.

Proof. The proof of this proposition can be found in ^[18].

Lemma 2.1. For $\upsilon > 0$, the operators ${E}_\gamma{(-{\upsilon}^\gamma{A})}$ and ${E}_{\gamma, \gamma}{(-{\upsilon}^\gamma{A})}$ in the uniform operator topology are continuous and well defined from X to X. Then continuity is uniform on $[r, \infty)$ for every $r > 0$.

Lemma 2.2. ^[19] Let $0 < \gamma < 1$. Then

(ⅰ) $\forall \; v \in X, lim_{t\rightarrow {0}^+} \; {E}_\gamma{(-{\upsilon}^\gamma{A})v} = v;$

(ⅱ) $\forall \; v\in {D(A)}$ and $\upsilon > 0, \; ^C{D}^\gamma_{\upsilon}{ E_\gamma}{(-{\upsilon}^\gamma{A})v} = -{A} \; {E}_\gamma{(-{\upsilon}^\gamma{A})v};$

(ⅲ) $\forall \; v \in X$ and ${E}^\prime_\gamma{(-{\upsilon}^\gamma{A})v} = - {\upsilon}^{\gamma-1} A {E_{\gamma, \gamma}}(-\upsilon^\gamma A)v;$

(ⅳ) $for \; \upsilon > 0\; , {E}_\gamma{(-{\upsilon}^\gamma{A})v} = I^{1-\gamma}_{\upsilon} \big\{({\upsilon}^{\gamma-1} {E_{\gamma, \gamma}}(-\upsilon^\gamma A)u)\big\}.$

Lemma 2.3. Suppose that $1 < r < \infty$ and $\gamma_1 \leq \gamma_2$. Then, there exist a constant $C = C(\gamma_1, \gamma_2)$ such that

Moreover, $\lim\limits_{\upsilon \rightarrow 0} \upsilon^{(\gamma_2 - \gamma_1)}|e^{-\upsilon A} v|_H^{\gamma_2, r} = 0$.

Lemma 2.4. Suppose that $1 < r < \infty$ and $\gamma_1 \leq \gamma_2$. For any $\Im > 0$, there exists a constant $C_1 = C_1(\gamma_1, \gamma_2)$ such that

and

for all $v\in {H^{\gamma_1, r}}$ and $\upsilon \in(0, T]$. Therefore, $\lim_{\upsilon \rightarrow 0} \upsilon^{\alpha(\gamma_2 - \gamma_1)}|E_\gamma(-\upsilon^\gamma A) v|_{H^{\gamma_2, r}} = 0$.

Proof. The proof of this lemma can be studied in ^[17].

Theorem 2.2. If $f(\upsilon)$ is defined on the interval $[c, d]$ is Riemann-integrable, then $|f(\upsilon)|$ is also Riemann integrable defined on the interval $[c, d]$ and

Theorem 2.3. Suppose that $f:[a, b]\rightarrow {\bf{R^n}}$ is continuous and $g:I\rightarrow {\bf{R^n}}$ is continuously differentiable with image $g(I)\subset [a, b]$, where $I\subset {\bf{R^n}}$ is some open interval showing that the function

is continuously differentiable on $I$.

Theorem 2.4. Let $\Im(\upsilon) : \upsilon\geq 0 \subset \mathcal{X}$ be a $C_0$ semi group on X. Then, the following holds

${\rm(i)}$ If $C:D(G)\subset X\rightarrow X$, then $G$ is said to be dense and close defined by linear operators. Therefore, $\upsilon\in[0, \infty)\rightarrow \Im(\upsilon)x \in X$ is continuously differentiable for any $x\in D(G)$:

${\rm(ii)}$ Then there exists $\sigma > 0$ such that $Re(\lambda) > 0$; given $\lambda\in \rho(C)$, we have

Theorem 2.5. Let $\gamma\in(0, 1]$ and suppose that the positive sectorial operator is $A:D(A)\subset X\rightarrow X$. Thus, the operators $\{E_\gamma (-\upsilon^\gamma A): \upsilon\geq 0\}$ and $\{E_{\gamma, \gamma} (-\upsilon^\gamma A): \upsilon \geq 0\}$ as follows:

and

where $\Im(\upsilon) : \upsilon\geq 0$} defines the $C_0$ semigroup which is generated by $-A$.

Let $\gamma\in(0, 1)$ and consider the $A:D(A)\subset X \rightarrow X$ {is a positive} sectorial operator. Then for any $x\in X$, it holds that

Proof. The first equality can be proved {analogously the second equality is for any} $x\in X$; observe that by Theorem 2.4,

Now by using $s = \omega \upsilon^{-\gamma}$, we conclude the following

Choose

By taking $\upsilon = \tau\omega^{\frac{1}{\gamma}}$ from Lemma 2.4, we have that

Therefore, by Theorem 2.5, we have

Lemma 2.5. If $\tilde{u}(\upsilon)$ is the solution of (1.5) for $\tilde{u}(0) = ax$, then $\tilde{u}(\upsilon)$ is given as

therefore we get

where

Proof. ^[20] By using the above lemma rewriting (1.5) and applying the RL derivative, we get

By applying the Laplace transform, we have the following

Simplification yields

By taking the inverse Laplace transform and applying the convolution theorem, we get

Definition 2.4 A Problem (1.5) has global mild solution of a function $\tilde{u}: [0, \infty)\to {H^{\gamma, r}}$ in ${H^{\gamma, r}}$, if $\tilde{u}\in \mathcal{C}{([0, \infty)}, {H^{\gamma, r}})$ and, for $\upsilon\in {[0, \infty)}$,

Definition 2.5 Let $0 < T < \infty$. A Problem (1.5) has local mild solution of a function $\tilde{u}:[0, T]\to {H^{\gamma, r}}$ in ${H^{\gamma, r}}$ if $\tilde{u}\in{([0, T], {H^{\gamma, r}})}$ and for $\upsilon\in[0, T]$, $\tilde{u}$ satisfies the Definition [4]. Conveniently, we respectively define three operators $\tilde{\psi}{(\upsilon)}$, $\tilde{\varphi}{(\upsilon)}$ and $\tilde{\varpi}{(\tilde{u}, w)}{(\upsilon)}:$

3.   Global and local existence of mild solutions in ${H^{\gamma, r}}$

To provide suitable circumstances for the existence and uniqueness of mild solutions ^[21] to the Problem (1.5) in ${H^{\gamma, r}}$, we consider this section. Suppose that we have (K). For $\upsilon > 0$, $Pg$ is continuous and

3.1.   Global existence in ${H^{\gamma, r}}$

To deal with the mild solution of Problem (1.5) with global existence in ${H^{\gamma, r}}$, let

and

Theorem 3.1. Let $1 < r < \infty$, $0 < \gamma < 1$ and (3.1) hold; then, for all $at \in {H^{\gamma, r}}$. Suppose that

If a unique function $\tilde{u}:[0, \infty)\to {H^{\gamma, r}}$ is satisfied and we choose $\dfrac{n}{2r}-\dfrac{1}{2} < \beta$, then there is $\alpha > max \big(\beta, \dfrac{1}{2}\big)$ where $\mathcal{M}_\infty = \sup\limits_{s\in[0, \infty)}\big(s^{\gamma(1-\beta)}Pg(s)\big)$:

(ⅰ) $\tilde{u}(0) = ax$ and $\tilde{u}:[0, \infty)\to {H^{\gamma, r}}$ shows continuity;

(ⅱ) $\tilde{u}:[0, \infty)\to {H^{\alpha, r}}$ is continuous and $\lim_{\upsilon \rightarrow 0}\upsilon^{\gamma(\alpha-\beta)}|\tilde{u}(\upsilon)|_{H^{\alpha, r}} = 0$;

(ⅲ) For $\upsilon \in[0, \infty)$, $\tilde{u}$ satisfies Definition 2.4.

Proof. Here we explain $X_\infty$ which is subspace of all of the curves and $X_\infty = X[\infty]$, $\tilde{u}:(0, \infty)\to {H^{\gamma, r}}$. Now suppose that $\alpha = \dfrac{1+\beta}{2}$ such that the following is true:

(ⅰ) $\tilde{u}:[0, \infty)\to {H^{\gamma, r}}$ is continuous and bounded.

(ⅱ) $\tilde{u}:(0, \infty)\to {H^{\alpha, r}}$ is continuous and bounded; furthermore, $\lim_{\upsilon \rightarrow 0}\upsilon^{\gamma(\alpha-\beta)}|\tilde{u}(\upsilon)|_{H^{\alpha, r}} = 0$ and its genuine form is given by

Now, we know that there exists $\mathcal{M}$ such that $\tilde{u}, w \in {H^{\alpha, r}}$ such that $F:{H^{\alpha, r}} * {H^{\alpha, r}} \rightarrow J_r$ is bounded and bilinear mapping:

Step 1:

Consider that $\tilde{u}, w\in {X_\infty}$. The term $\tilde{\varpi}(\tilde{u}(\upsilon), w(\tau))$ is a part of $\mathcal{C}(0, \infty), {H^{\gamma, r}}$ and $\mathcal{C}\; {([0, \Im], {H^{\gamma, r}})}$. By considering $\varepsilon > 0$ that is very small and randomly fixing $\upsilon_0 \geq 0$, now again suppose that $\upsilon > \upsilon_0$ ${(\upsilon < \upsilon_0\; analogously)}$; we have

In the view of Lemma 2.4, we consider every term separately for $\mathcal{D}_{11}{(\upsilon)}$ and get the following

By applying the $\beta$ function properties, $\exists\; \delta > 0$ much smaller as $0 < \; \upsilon-\upsilon_{0} < \delta$; we have

which follows that as $\upsilon-\upsilon_{0}\rightarrow 0$, $\mathcal{D}_{11}{(\upsilon)}$ tends to $0$.

Now for $\mathcal{D}_{12}{(\upsilon)}$,

It is interesting to note that

We can show this by applying Lebesgue's dominated convergence Theorem 2.1, we get

We can say that $\lim\limits_{\upsilon \rightarrow \upsilon_0}\mathcal{D}_{12}{(\upsilon)} = 0$. For $\mathcal{D}_{13}{(\upsilon)}$, we have

Considering the uniform continuity, and by using Lebesgue's dominated convergence theorem, we get $\mathcal{D}_{13}{(\upsilon)}$ as follows

For $\mathcal{D}_{14}{(\upsilon)}$, using calculations, we estimate that, according to the ${\beta}$ function properties

It follows that

The operator's continuity $\tilde{\varpi}(\tilde{u}, w)$ can be demonstrated in $\mathcal{C}\big((0, \infty), {H^{\alpha, r}}\big)$ following the preceding debate.

Step 2:

To show that $\tilde{\varpi} :X_\infty \ast X_\infty\to X_\infty$ is a continuous bilinear operator, we consider that, according to Lemma 2.4, we have

and

As a result of this

To be more specific,

Therefore, ${\tilde{\varpi}(\tilde{u}, w)}\in{X_\infty}$ and $||{\tilde{\varpi}(\tilde{u}(\upsilon), w(\upsilon))}||_{X_\infty} \leq L||\tilde{u}||_{X_\infty} ||w||_{X_\infty}$.

Step 3:

Let $0 < \upsilon_0 < \upsilon$. We have that

The first two integrals and last integral tend to $0$ as $\upsilon\rightarrow {\upsilon}_0$; also, $\varepsilon\rightarrow 0$ by using the characteristics of the $\beta$ function. Now the third integral also tends to $0$ as $\upsilon\rightarrow {\upsilon}_0$ by using Lemma 2.1. This suggests that

In order to evaluate the continuity of ${\tilde{\varphi(\upsilon)}}$ in ${H^{\alpha, r}}$, we must go along with the same pattern as in ${H^{\gamma, r}}$. On the contrary,

To be more precise,

Because ${\tilde{M}(t)}\rightarrow 0$ is the same as $\upsilon\rightarrow 0$ owing to assumption, this result confirms that $||\tilde{\varphi}(\upsilon)||_\infty \leq \tilde{B}_{1}\tilde{M}_\infty$ as $\tilde{\varphi}(t)\in X_\infty$. Now for $a\upsilon\in {H^{\gamma, r}}$, according to Lemma 2.1 obvious that

and

Therefore by Theorem 2.3, we can say that

This implies that for all $\upsilon\in(0, \Im]$, and together with Theorem 2.2,

By using the above condition, we get

According to Theorem 3.1, the inequality

continues to hold, resulting in F having a unique fixed point.

Step 4:

In order to verify that $\tilde{u}(\upsilon)\rightarrow a\upsilon$ in ${H^{\gamma, r}}$ as $\upsilon\rightarrow 0$, we must show that

in ${H^{\gamma, r}}$. In fact $\lim_{\upsilon \rightarrow 0}{\tilde{\psi}(\upsilon)} = 0$ and $\lim_{\upsilon \rightarrow 0}\; {\tilde{\varphi}(\upsilon)} = 0{(\lim_{\upsilon \rightarrow 0}{\tilde{M}(\upsilon)} = 0)}$ due to (3.2). So,

Theorem 3.2. Let $1 < r < \infty$, $0 < \gamma < 1$ and (3.1) holds; then, suppose that

Then there is a unique function $\tilde{u}:[0, \infty)\rightarrow {H^{\gamma, r}}$ and $\alpha > \max \big(\beta, \dfrac{1}{2}\big)$ for all $at \in {H^{\gamma, r}}$ there exists $\Im_\ast > 0$ satisfying the following

(ⅰ) $\tilde{u}(0) = a\upsilon$ and $\tilde{u}:[0, \Im_*]\rightarrow {H^{\gamma, r}}$ is continuous,

(ⅱ) $\tilde{u}:[0, \Im_*]\rightarrow {H^{\alpha, r}}$ is continuous, and $\lim_{\upsilon\rightarrow 0}\upsilon^{\gamma(\alpha-\beta)}|\tilde{u}(\upsilon)|_{H^{\alpha, r}} = 0$,

(ⅲ) for $\upsilon \in[0, \Im_*]$, $\tilde{u}$ satisfies Definition 2.4.

Proof. Consider the space of all of the curves $X = X[T]$ and $\tilde{u}:(0, \Im]\rightarrow {H^{\gamma, r}}$; now, suppose that $\alpha = \dfrac{1+\beta}{2}$ so we have the following

(ⅰ) $\tilde{u}:[0, \Im]\rightarrow {H^{\gamma, r}}$ is continuous.

(ⅱ) $\tilde{u}:(0, \Im]\rightarrow {H^{\alpha, r}}$ is continuous and $\lim\limits_{\upsilon\rightarrow 0}\upsilon^{\gamma(\alpha-\beta)}|\tilde{u}(\upsilon)|_{H^{\alpha, r}} = 0,$

its original form is given by

Similar to the conclusion of Theorem 3.1, it is worth noting that the function $g:X \ast X\rightarrow X$ is linearly mappable and continuous. By Lemma 2.1 the function $\varphi(\upsilon)\in X$ for all $\upsilon\in(0, \Im]$. It is simple to assert that

Therefore by Theorem 2.3, we can say that

For all $\upsilon\in(0, \Im]$, using Theorem 2.2 implies that

We have

By using the above condition, we get

Now, consider a sufficiently small $\Im_\ast > 0$, such that

holds, resulting in F having a unique fixed point.

4.   Local existence in $H^{\gamma, r}$

This part is devoted to the iteration method's evaluation of a local mild solution to Problem (1.5) in $J_r$. Consider that $\alpha = \dfrac{1+\beta}{2}$.

Theorem 4.1. Suppose that $1 < r < \infty$, $0 < \gamma < 1$ and (3.1) holds. Suppose that

For $a\upsilon\in {H^{\gamma, r}}$, Problem (1.5) has a unique mild solution $\tilde{u}$ in $J_r$, $A^\alpha\tilde{u}$ shows continuity on $(0, \Im]$. Furthermore, $\upsilon^\gamma(\alpha-\beta)A^\alpha \tilde{u}(\upsilon)$ is bounded while $\upsilon \to 0$. Additionally, $\tilde{u}$ also shows continuity in $[0, \Im]$.

Proof. Step 1:

Let

and

According to the summary of Step 2 in Theorem 3.1, $\zeta (\upsilon)$ shows continuity on $[0, \Im]$ and $A^\alpha {\zeta (\upsilon)}$ is continuous on $(0, \Im]$ and also exists, then,

We have

Using the above equation, it gives the final result such that

Take into account the integral $\tilde{\varphi}(\upsilon)$. As the (3.1) holds then the inequality

is accomplished by a continuous function $\mathcal{M}(\upsilon)$. Now considering the Step 3 of Theorem 3.1, we show that $A^\alpha \tilde{\varphi}(\upsilon)$ shows continuity on $(0, \Im]$. Before discussing continuity, we signify

As $\upsilon \rightarrow 0$, we get that $\mathcal{M}(\upsilon) = 0$ and $|Pg(\upsilon)|_r = 0{(\upsilon^{-\gamma(\alpha-\beta)})}$. As $\upsilon \rightarrow 0$, we get that $|A^\alpha {\zeta (\upsilon)}|_r = 0{(\upsilon^{-\gamma(\alpha-\beta)})}$ according to (4.2). We show that the function $\tilde{\varphi}$ is continuous in $J_r$. In reality, considering $0\leq \upsilon_0 < \upsilon < \Im$, we have

Further, we assume the function $\int_{0}^{\upsilon}E_\gamma(-(\upsilon-s)^\gamma A)ads$. From Theorem 2.2, it is clear that

From Lemma 2.4,

and

Step 2:

We now build the result through successive approximation:

We know that denotes increasing and continuous functions on $[0, \Im]$. Moreover, $\mathcal{K}_n(0) = 0$. However, according to (4.1) and (4.3), $\mathcal{K}_n(\upsilon)$ satisfies the inequality

Select $\Im > 0$ so that

Then, the sequence ${K_n(\Im)}$ is bounded; following a simple description of (4.4), so we get

while (4.4) is just like the quadratic equation; we have

Here, $a = \mathcal{M}\mathcal{C}_{1}\mathcal{B}\big((\gamma{(1-\alpha)}), 1- 2\gamma{(\alpha-\beta)}\big){\mathcal{K}_n}^2(\upsilon)$, $b = -1$ and $c = \mathcal{K}_0(\upsilon)$. After applying the quadratic formula, we obtain

Considering (4.6),

Conversely, $\mathcal{K}_n(\upsilon)\leq \rho(\upsilon)$ holds for any $\upsilon\in (0, \Im]$. Correspondingly, $\rho(\upsilon) \leq 2\mathcal{K}_0(\upsilon)$. Suppose the following concept of equality:

For $\upsilon \in(0, \Im]$ and $n = 0, 1, 2, ...,$ we denote $y_n = \tilde{u}_{n+1}- \tilde{u}_n$:

Using Lemma 2.3, we get

Then, we set

We get the results from Step 2 in Theorem 3.1:

It provides

In accordance with (4.5) and (4.7), we get

Hence the series $\sum_{n = 0}^{\infty}{Y_n(\Im)}$ is convergent. For $\upsilon\in(0, \Im]$ it shows that the series $\sum_{n = 0}^{\infty}\upsilon^\gamma(\alpha-\beta)A^\alpha{y_n(\upsilon)}$ is uniformly convergent. Thus, the sequence $\{\upsilon^\gamma(\alpha-\beta)A^\alpha{\tilde{u}_n(\upsilon)}\}$ is uniformly convergent in $(0, \Im]$. This signifies that $\lim_{n\rightarrow \infty}\tilde{u}_n(\upsilon) = \tilde{u}(\upsilon) \in D{(A^\alpha)}$ and $\lim_{n\rightarrow \infty}\upsilon^{\gamma(\alpha-\beta)}A^\alpha {\tilde{u}_n(\upsilon)} = \upsilon^{\gamma(\alpha-\beta)}A^\alpha{\tilde{u}(\upsilon)}$ uniformly. From the boundedness theorem, a function $f$ continuous on a bounded and closed interval is necessarily a bounded function. Therefore, $A^{\alpha}$ is closed if $A^{-\alpha}$ is bounded. As a result, the function $\mathcal{K}(\upsilon) = \sup_{s\in (0, \upsilon]}s^{\beta(\alpha-\gamma)}|A^{\alpha} \tilde{u}_n(s)|_r$ satisfies

and

Lastly, it is necessary to confirm that $\tilde{u}$ is a mild solution of (1.5) in $[0, \Im]$. Because

we get $\varpi(\tilde{u}_n, \tilde{u}_n)(\upsilon) \rightarrow \varpi(\tilde{u}, \tilde{u})(\upsilon)$. We obtain it by taking the limits of (4.2) on both sides;

Suppose that $\tilde{u}(0) = a\upsilon$. For $\upsilon\in[0, \Im]$, we can say that (4.9) holds and $\tilde{u}\in C([0, \Im], J_r)$. Furthermore, the uniform convergence of $\upsilon^{\gamma(\alpha-\beta)}A^\alpha{u_n(\upsilon)}$ to $t^\gamma(\alpha-\beta)A^\alpha{\tilde{u}(\upsilon)}$ shows continuity of $A^\alpha \tilde{u}(\upsilon)$. By (4.8) and $\mathcal{K}_0(0) = 0$, we obtain that $|A^\alpha u(\upsilon)|_r = 0{\upsilon^{(-\gamma(\alpha-\beta))}}$ is clear.

Step 3:

Now, we have to suppose that $\tilde{u}$ and $\tilde{w}$ are mild solutions of (1.5) because to prove that the mild solutions are unique. Let $y = \tilde{u}-\tilde{w}$; we consider the equality

Now, we defining the following functions:

According to Theorem 3.1 and Lemma 2.4, we get

For $\upsilon \in (0, \Im]$, the Gronwall inequality demonstrates that $A^\alpha y(\upsilon) = 0$. It shows that for $\upsilon \in [0, \Im]$, $y(\upsilon) = \tilde{u}(\upsilon)-\tilde{w}(\upsilon) \equiv 0$. Hence, the mild solutions are unique.

5.   Consequences of regularity for vertical flow

In this part, we evaluate the regularity's behavior ^[22] of a solution $\tilde{u}$ which satisfies (1.5). We will assume throughout this section that $({\bf{k}}_1)$ : $Pg(\upsilon)$ has Hölder's continuity with an exponent ${\theta \in (0, \gamma(1 - \alpha))}$, which is given as

Definition 5.1. The problem (1.5) has a solution that is a classical solution of a function $\tilde{u}:[0, \Im] \to J_r$, and $\tilde{u} \in C([0, \Im], J_r)$ with $^{c}D_{\upsilon}^{\upsilon} \in C([0, \Im], J_r)$, which satisfies (1.5) for all $\upsilon \in (0, \Im]$ and accepts values in $D(A)$.

Lemma 5.1. Let Definition 5.1 be satisfied if

so $A\varphi_{1}(\upsilon) \in C^\theta([0, \Im], J_r)$ and $\varphi_{1}(\upsilon) \in D(A)$.

Proof. From Lemma 2.4 and (5.1), we fixed $\upsilon \in (0, \Im]$,

Then

we get $\varphi_{1}(\upsilon) \in D(A)$ by the closeness property of $A$. We must demonstrate that $A\varphi_{1}(\upsilon)$ has Hölder continuity. Because

it follows that

Substituting $-\mu \upsilon^\gamma = \xi$ implies that $-\upsilon^\gamma d\mu = d\xi$. So we have that $d\mu = -\dfrac{1}{\upsilon^\gamma} d\xi$:

Since

and

for every $0 < s < \upsilon \leq \Im$, we apply the mean value theorem

Now for $0 < \upsilon < \upsilon+\hbar\leq \Im$, we suppose that $\hbar > 0$, so

For convenience, we solve each term separately. For $\hbar_1(\upsilon)$, from (5.2) and (5.1) we have

For $\hbar_2(t)$, using Lemma 2.4 and (5.1), we get

Now for $\hbar_3(\upsilon)$, using Lemma 2.4 and (5.1), we have

Hence we can say that $A\varphi_1(\upsilon)$ is Hölder continuous by combining all of the above results.

Theorem 5.1. Consider that the supposition of Theorem 4.1 is satisfied. If (5.1) holds, then the classical ^[23] mild solutions of (1.5) are obtained for every $a\upsilon\in D(A)$.

Proof. Step 1:

We have that $a\upsilon\in D(A)$. So, $\int_{0}^{\upsilon}E_{\gamma}(-\upsilon^{\gamma}A)ads$ is said to be classical solution in regard to the following problem:

We also verify that

is a classical approach to the following problem:

$\tilde{\varphi}\in \mathcal{C}([0, \Im], J_r)$ follows from Theorem 4.1. We can write $\tilde{\varphi}(\upsilon) = \tilde{\varphi}_1(\upsilon) + \tilde{\varphi}_2(\upsilon)$, while

We can say that $\tilde{\varphi}_1(\upsilon)\in D(A)$ according to Lemma 5.1. In order to prove the same conclusion for $\tilde{\varphi}_2(\upsilon)$, according to Lemma 2.2(ⅲ), we realized that

It follows that (5.1):

So, for $\upsilon \in(0, \Im]$ we can say that $\tilde{\varphi}_2(\upsilon)\in D(A)$ and $\tilde{\varphi}_2(\upsilon)\in \mathcal{C}^{\nu}{((0, \Im], J_r)}$. Now we have to show that $^{c}D_{\upsilon}^{\gamma}(\tilde{\varphi})\in \mathcal{C}{((0, \Im], J_r)}$. Taking $\tilde{\varphi}(0) = 0$ and in view of Lemma 2.2(ⅳ),

It is still necessary to demonstrate that $(E_{\gamma}(-\upsilon^{\gamma}A)*Pg)$ is differentiability continuous in $J_r$. Suppose that $0 < h \leq \Im-\upsilon$; therefore, we have

Notice that

In view of Lemma 2.4,

the dominated convergence theorem applies; then, we get

From Lemma 2.2(ⅲ), we have

Therefore,

Conversely,

Let $s^\ast = \upsilon+\hbar-s$ so $ds^\ast = -ds$ and after setting the conditions [$s = \upsilon$ implies $s^\ast = \hbar$] and [$s = \upsilon+\hbar$ implies $s^\ast = 0$], we have

By replacing $s^\ast\rightarrow s$, we get

From Lemmas 2.1–2.4 and (5.1), we have

From Lemma 2.1(ⅰ),

we deduce that $E_{\gamma}(\upsilon^\gamma A)\ast Pg$ holds differentiability $a\upsilon$ and $\dfrac{d}{d\upsilon}(E_{\gamma}(\upsilon^\gamma A)\ast Pg)_+ = A\tilde{\varphi}(\upsilon) + Pg(\upsilon)$. Similarly $E_{\gamma}(\upsilon^\gamma A)*Pg$ holds differentiability at $\upsilon$ and $\dfrac{d}{d\upsilon}(E_{\gamma}(\upsilon^\gamma A)\ast Pg)_- = A\tilde{\varphi}(\upsilon) + Pg(\upsilon)$. We can prove that $A\tilde{\varphi} = A\tilde{\varphi}_1+A\tilde{\varphi}_2 \in \mathcal{C}((0, \Im], J_r)$. Clearly, we can say that according to Lemma 2.1, $\tilde{\varphi}_2(\upsilon)$ shows continuity and $\tilde{\varphi}_2(\upsilon) = Pg(\upsilon)-E_{\gamma}(\upsilon^\gamma A)Pg(\upsilon)$ because of Lemma 2.2(ⅲ). Consequently, $A\tilde{\varphi}_1(\upsilon)$ is continuous according to Lemma 5.1. Thus, $^{c}D^{\gamma}_{\upsilon}\tilde{\varphi} \in \mathcal{C}((0, \Im], J_r)$.

Step 2:

Consider $\tilde{u}$ as the mild solution of (1.5). In order to draw the conclusion that $F(\tilde{u}, \tilde{u})\in C^{\theta}((0, \Im], J_r)$, we have to show that $A^\alpha \tilde{u}$ holds Hölder continuity in $J_r$ according to Theorem 3.1. Now for $0 < \upsilon < \upsilon+\hbar$, let $\hbar > 0$. We know that $\psi(\upsilon) = \int_{0}^{\upsilon}E_{\gamma}{(-(\upsilon-s)^\gamma)}ads$. Denoting ${\phi}(\upsilon): = \int_{0}^{\upsilon} E_{\gamma}(-\upsilon^\gamma A)ads$, and by Lemma 2.2(ⅲ) we have

Integrating on both sides, we get

After applying the limits, we have

Using the above results and Lemma 2.4, then

Thus $A^\alpha \phi\in \mathcal{C}^\theta((0, \Im], J_r)$. For every small $\varepsilon > 0$ we take $\hbar$ as $\varepsilon\leq \upsilon < \upsilon+\hbar\leq \Im$. We have

Using Lemma 2.4 and (5.1), we have

To estimate $\phi_2$, we have

which yields that

Now, we apply the mean value theorem:

Thus,

This shows that $A^\alpha \phi\in \mathcal{C}^\theta([\varepsilon, \Im], J_r)$. Therefore, $A^\alpha \phi\in C^\theta((0, \Im], J_r)$ due to an arbitrary $\varepsilon$. Try to remember that $\zeta(\upsilon) = \int_{0}^{\upsilon}(\upsilon-s)^{\gamma-1}E_{\gamma, \gamma}{(-(\upsilon-s)^\gamma A)}F(\tilde{u}(s), \tilde{u}(s))ds$. We know that $|F(\tilde{u}(s), \tilde{u}(s)|_r \leq \mathcal{M}\mathcal{K}^{2}(\upsilon)s^{-2\gamma(\alpha-\beta)}$, while $\mathcal{K}(\upsilon) = \sup\limits_{s\in (0, \upsilon]}s^{\gamma(\alpha-\beta)}|A^{\alpha} u(s)|_r$ is bounded and continuous in $(0, \Im]$. A simple reasoning allows us to take the Hölder continuity of $A^\zeta\in \mathcal{C}^\theta((0, \Im], J_r)$. Therefore we have that $A^\alpha \tilde{u}(\upsilon)$ = $A^\alpha \phi(\upsilon)+A^\alpha \varphi(\upsilon)+A^\alpha \zeta(\upsilon)\in \mathcal{C}^\theta((0, \Im], J_r)$. We know that $F(\tilde{u}, \tilde{u})\in \mathcal{C}^\theta((0, \Im], J_r)$ is verified; similarly following Step 2, this yields that $^{c}D^{\gamma}_{\upsilon}\zeta\in \mathcal{C}^\theta((0, \Im], J_r)$, $A\zeta\in \mathcal{C}^\theta((0, \Im], J_r)$ and $^{c}D^{\gamma}_{\upsilon}\zeta = -A\zeta + F(\tilde{u}, \tilde{u})$. So we get that $^{c}D^{\gamma}_{\upsilon}\zeta\in \mathcal{C}^\theta((0, \Im], J_r)$, $A\tilde{u}\in \mathcal{C}^\theta((0, \Im], J_r)$ and $^{c}D^{\gamma}_{\upsilon}\tilde{u} = -A\tilde{u} + F(\tilde{u}, \tilde{u}) +Pg$. Hence, we can say that $\tilde{u}$ has a classical solution.

Theorem 5.2. Suppose (5.1) holds if $\tilde{u}$ is a classical solution of (1.5); thus, $^{c}D^{\gamma}_{\upsilon}\in \mathcal{C}^\nu ((0, \Im], J_r)$ and $A\tilde{u} \in\mathcal{C}^\nu ((0, \Im], J_r)$.

Proof. Let $\tilde{u}$ be the classical solution of Problem (1.5) and also suppose that(5.1) holds; so, $\tilde{u}(\upsilon) = \tilde{\varphi}(\upsilon) + \phi(\upsilon) + \zeta(\upsilon)$ shows that $A\tilde{\varphi} \in \mathcal{C}^{\gamma(1-\beta)} ((0, \Im], J_r)$. It is also sufficient to demonstrate that, for every $\varepsilon > 0$ there is $A\tilde{\varphi} \in \mathcal{C}^{\gamma(1-\beta)} ((\varepsilon, \Im], J_r)$. Now by using Lemma 2.2(ⅲ) we take $\hbar$ as $\varepsilon\leq \upsilon < \upsilon+\hbar\leq \Im$;

Using (6) and rewriting $\tilde{\varphi}(\upsilon)$ in the form of

as $\upsilon \in (0, \Im]$, we can say that $A\tilde{\varphi}_1 (\upsilon) \in \mathcal{C}^\nu) ((0, \Im], J_r)$ and $A\tilde{\varphi}_2 (\upsilon) \in \mathcal{C}^\theta ((0, \Im], J_r)$. These results come from Lemma 5.1 and (5.8). As we know that $F(\tilde{u}, \tilde{u})\in \mathcal{C}^\theta ((0, \Im], J_r)$ and $\zeta(\upsilon)$ is verified analogously, so we can conclude that $A\zeta (\upsilon) \in C^\nu ((0, \Im], J_r)$. Consequently, $^{c}D^{\gamma}_{\upsilon}\tilde{u} = A\tilde{u} + F(\tilde{u}, \tilde{u}) + Pg \in \mathcal{C}^\nu ((0, \Im], J_r)$ as $A\tilde{u} \in \mathcal{C}^\nu ((0, \Im], J_r)$. This proof is now complete.

6.   Conclusions

This study demonstrates the existence-uniqueness of local and global mild solutions. Meanwhile, we offer a local reasonable solution in ${\bf{S_\wp}}$. The NS equations with time-fractional derivatives of order $\gamma \in (0, 1)$ were used to simulate anomaly diffusion in fractal media. We also demonstrate the existence of regular, classical solutions to these equations in ${\bf{S_\wp}}$. The concept could be expanded upon by including MHD effects, the concept put forth in this mwork could be developed further, observability could be added and other activities could be generalized in future work. This is an interesting area with a lot of study going on that could lead to a lot of different applications and theories.

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Author's Contributions

All authors contributed equally to the writing of this paper. All authors have read and agreed to the published version of the manuscript.

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments and suggestions.