Large-time behavior of cylindrically symmetric Navier-Stokes equations with temperature-dependent viscosity and heat conductivity

1.   Introduction

The models of the differential and integral equations have been appeared in different applications (see ^{[1,2,3,4,6,7,9,12,13,14,15,16,17,18,19,20]}).

Boundary value problems involving fractional differential equations arise in physical sciences and applied mathematics. In some of these problems, subsidiary conditions are imposed locally. In some other cases, nonlocal conditions are imposed. It is sometimes better to impose nonlocal conditions since the measurements needed by a nonlocal condition may be more precise than the measurement given by a local condition. Consequently, a variety of excellent results on fractional boundary value problems (abbreviated BVPs) with resonant conditions have been achieved. For instance, Bai ^[4] studied a type of fractional differential equations with m-points boundary conditions. The existence of nontrivial solutions was established by using coincidence degree theory. Applying the same method, Kosmatov ^[17] investigated the fractional order three points BVP with resonant case.

Although the study of fractional BVPs at resonance has acquired fruitful achievements, it should be noted that such problems with Riemann-Stieltjes integrals are very scarce, so it is worthy of further explorations. Riemann-Stieltjes integral has been considered as both multipoint and integral in a single framework, which is more common, see the relevant works due to Ahmad et al. ^[1].

The boundary value problems with nonlocal, integral and infinite points boundary conditions have been studied by some authors (see, for example ^[8,10,11,12]).

Here, we discuss the boundary value problem of the nonlinear differential inclusions of arbitrary (fractional) orders

with the nonlocal boundary condition

the integral condition

and the infinite point boundary condition

We study the existence of solutions $x\in C[0, 1]$ of the problems (1.1) and (1.2), and deduce the existence of solutions of the problem of (1.1) with the conditions (1.3) and (1.4). Then the existence of the maximal and minimal solutions will be proved. The sufficient condition for the uniqueness and continuous dependence of the solution will be studied.

This paper is organised as: In Section 2, we prove the existence of continuous solutions of the problems (1.1) and (1.2), and deduce the existence of solutions of the problem of (1.1) with the conditions (1.3) and (1.4). In Section 3, the existence of the maximal and minimal solutions is proved. In Section 4, the sufficient condition for the uniqueness and continuous dependence of the solution are studied. Next, in Section 5, we extend our results to the nonlocal problems (1.3) and (2.1). Finally, some existence results is proved for the nonlocal problems (1.4) and (2.1) in Section 6.

2.   Main results

Consider the following assumptions:

(I) (i) The set $F_{1}(t, x, y)$ is nonempty, closed and convex for all $(t, x, y)\in [0, 1]\times R\times R$.

(ii) $F_{1}(t, x, y)$ is measurable in $t\in [0, 1]$ for every $x, y\in R$.

(iii) $F_{1}(t, x, y)$ is upper semicontinuous in $x$ and $y$ for every $t\in [0, 1]$.

(iv) There exist a bounded measurable function $a_{1}: [0, 1]\longrightarrow R$ and a positive constant $K_{1}$, such that

Remark 2.1. From the assumptions (i)–(iv) we can deduce that (see ^[3,6,7,13]) there exists $f_{1}\in F_{1}(t, x, y)$, such that

(v) $f_{1}:[0, 1]\times R\times R\longrightarrow R$ is measurable in $t$ for every $x, y\in R$ and continuous in $x, y$ for $t\in [0, 1]$, and there exist a bounded measurable function $a_{1}:[0, 1]\rightarrow R$ and a positive constant $K_{1} > 0$ such that

and the functional $f_{1}$ satisfies the differential equation

(II) $f_{2}:[0, 1]\times R\longrightarrow R$ is measurable in $t$ for any $x\in R$ and continuous in $x$ for $t\in [0, 1]$, and there exist a bounded measurable function $a_{2}:[0, 1]\rightarrow R$ and a positive constant $K_{2} > 0$ such that

and

(III) $2K_{1}\gamma+K_{1}K_{2}\alpha < \alpha\gamma\Gamma(2-\alpha), \; \alpha, \gamma \in (0, 1].$

Remark 2.2. From (I) and (v) we can deduce that every solution of (1.1) is a solution of (2.1). Now, we shall prove the following lemma.

Lemma 2.1. If the solution of the problems (1.2)–(2.1) exists then it can be expressed by the integral equation

Proof. Consider the boundary value problems (1.2)–(2.1) be satisfied. Operating by $I^{1-\alpha}$ on both sides of (2.1) we can obtain

Taking

then we obtain

Putting $t = \tau$ and multiplying (2.5) by $\Sigma_{k = 1}^{m}a_{k}$, then we get

and

Then

Substituting (2.8) and (2.9) in (2.5), which completes the proof.

Theorem 2.1. Let assumptions (I)–(III) be satisfied. Then the integral equation (2.2) has at least one continuous solution.

Proof. Define a set $Q_{r}$ as

and the operator $F$ by

For $y\in Q_{r}$, then

Thus, the class of functions $\{Fy\}$ is uniformly bounded on $Q_{r}$ and $F:Q_{r}\rightarrow Q_{r}$. Let $y\in Q_{r}$ and $t_{1}, t_{2}\in [0, 1]$ such that $|t_{2}-t_{1}| < \delta$, then

Thus, the class of functions $\{Fy\}$ is equicontinuous on $Q_{r}$ and $\{Fy\}$ is compact operator by the Arzela-Ascoli Theorem ^[5].

Now we prove that $F$ is continuous operator. Let ${y_{n}}\subset Q_{r}$ be convergent sequence such that $y_{n}\rightarrow y$, then

Using Lebesgue dominated convergence Theorem ^[5] and assumptions (iv)–(II) we have

Then $F:Q_{r}\rightarrow Q_{r}$ is continuous, and by Schauder Fixed Point Theorem^[5] there exists at least one solution $y\in C[0, 1]$ of (2.2). Now

Putting $t = \tau$ and using (2.9), we obtain

then

This proves the equivalence between the problems (1.2)–(2.1) and the integral equation (2.2). Then there exists at least one solution $y\in C[0, 1]$ of the problems (1.2)–(2.1).

3.   Maximal and minimal solutions

Here, we shall study the maximal and minimal solutions for the problems (1.2) and (2.1). Let $y(t)$ be any solution of (2.2), let $u(t)$ be a solution of (2.2), then $u(t)$ is said to be a maximal solution of (2.2) if it satisfies the inequality

A minimal solution can be defined by similar way by reversing the above inequality.

Lemma 3.1. Let the assumptions of Theorem 2.1 be satisfied. Assume that $x(t)$ and $y(t)$ are two continuous functions on $[0, 1]$ satisfying

where one of them is strict.

Let functions $f_{1}$ and $f_{2}$ be monotonic nondecreasing in $y$, then

Proof. Let the conclusion (3.1) be not true, then there exists $t_{1}$ with

Since $f_{1}$ and $f_{2}$ are monotonic functions in $y,$ then we have

This contradicts the fact that $y(t_{1}) = x(t_{1})$, then $y(t) < x(t)$. This completes the proof.

For the existence of the continuous maximal and minimal solutions for (2.1), we have the following theorem.

Theorem 3.1. Let the assumptions of Theorem 2.1 be hold. Moreover, if $f_{1}$ and $f_{2}$ are monotonic nondecreasing functions in $y$ for each $t\in [0, 1]$, then Eq (2.1) has maximal and minimal solutions.

Proof. First, we should demonstrate the existence of the maximal solution of (2.1). Let $\epsilon > 0$ be given. Now consider the integral equation

where

For $\epsilon_{2} > \epsilon_{1},$ we have

Also

Applying Lemma 3.1, we obtain

As shown before, the family of function $y_{\epsilon}(t)$ is equi-continuous and uniformly bounded, then by Arzela Theorem, there exist decreasing sequence $\epsilon_{n}$, such that $\epsilon_{n}\rightarrow 0$ as $n\rightarrow \infty$, and $u(t) = {\lim_{n\rightarrow \infty}}y_{\epsilon_{n}}(t)$ exists uniformly in $[0, 1]$ and denote this limit by $u(t)$. From the continuity of the functions, $f_{2, \epsilon_n}(t, y_{\epsilon_n}(t))$, we get $f_{2, \epsilon_n}(t, y_{\epsilon_n}(t))\longrightarrow f_{2}(t, y(t))$ as $n\rightarrow \infty$ and

Now we prove that $u(t)$ is the maximal solution of (2.1). To do this, let $y(t)$ be any solution of (2.1), then

and

Applying Lemma 3.1, we obtain

From the uniqueness of the maximal solution it clear that $y_{\epsilon}(t)$ tends to $u(t)$ uniformly in $[0, 1]$ as $\epsilon\rightarrow 0.$

By a similar way as done above, we can prove the existence of the minimal solution.

4.   Uniqueness of the solution

Here, we study the sufficient condition for the uniqueness of the solution $y\in C[0, 1]$ of problems (1.2) and (2.1). Consider the following assumptions:

($I^{*}$) (i) The set $F_{1}(t, x, y)$ is nonempty, closed and convex for all $(t, x, y)\in [0, 1]\times R\times R$.

(ii) $F_{1}(t, x, y)$ is measurable in $t\in [0, 1]$ for every $x, y\in R$.

(iii) $F_{1}$ satisfies the Lipschitz condition with a positive constant $K_{1}$ such that

where $H(A, B)$ is the Hausdorff metric between the two subsets $A, B\in [0, 1]\times E.\;$

Remark 4.1. From this assumptions we can deduce that there exists a function $f_{1}\in F_{1}(t, x, y)$, such that

(iv) $f_{1}:[0, 1]\times R\times R \rightarrow R$ is measurable in $t\in [0, 1]$ for every $x, y\in R$ and satisfies Lipschitz condition with a positive constant $K_{1}$ such that (see ^[3,7])

$(II^{*}) \; f_{2}:[0, 1]\times R\longrightarrow R$ is measurable in $t \in [0, T]$ and satisfies Lipschitz condition with positive constant $K_{2}$, such that

From the assumption ($I^{*}$), we have

Then

where $|a_{1}(t)| = \sup\limits_{t\in I}|f_{1}(t, 0, 0)|.$

From the assumption ($II^{*}$), we have

Then

where $|a_{2}(t)| = \sup\limits_{t\in I}|f_{1}(t, 0)|$.

Theorem 4.1. Let the assumptions $(I^{*})$ and $(II^{*})$ be satisfied. Then the solution of the problems (1.2) and (2.1) is unique.

Proof. Let $y_{1}(t)$ and $y_{2}(t)$ be solutions of the problems (1.2) and (2.1), then

Then

Since $(\alpha\gamma\Gamma(2-\alpha)-(2K_{1}\alpha+K_{1}K_{2}\gamma)) < 1$, then $y_{1}(t) = y_{2}(t)$ and the solution of (1.2) and (2.1) is unique.

4.1.   Continuous dependence of the solution

Definition 4.1. The unique solution of the problems (1.2) and (2.1) depends continuously on initial data $x_{0}$, if $\epsilon > 0$, $\exists \delta > 0$, such that

where $y^{*}$ is the unique solution of the integral equation

Theorem 4.2. Let the assumptions $(I^{*})$ and $(II^{*})$ be satisfied, then the unique solution of (1.2) and (2.1) depends continuously on $x_{0}.$

Proof. Let $y(t)$ and $y^{*}(t)$ be the solutions of problems (1.2) and (2.1), then

then we obtain

and

Definition 4.2. The unique solution of the problems (1.2) and (2.1) depends continuously on initial data $a_{k}$, if $\epsilon > 0$, $\exists \delta > 0$, such that

where $y^{*}$ is the unique solution of the integral equation

Theorem 4.3. Let the assumptions $(I^{*})$ and $(II^{*})$ be satisfied, then the unique solution of problems (1.2) and (2.1) depends continuously on $a_{k}$.

Proof. Let $y(t)$ and $y^{*}(t)$ be the solutions of problems (1.2) and (2.1) and $(\sum\nolimits^{m}_{k = 1}a^{*}_{k})^{-1} = A^{*}$,

then we obtain

and

5.   Riemann-Stieltjes integral condition

Let $y\in C[0, 1]$ be the solution of the nonlocal boundary value problems (1.2) and (2.1). Let $a_{k} = (g(t_{k})-g(t_{k-1})$, $g$ is increasing function, $\tau_{k}\in(t_{k-1}-t_{k})$, $0 = t_{0} < t_{1} < t_{2}, ... < t_{m} = 1$, then, as $m\longrightarrow \infty$ the nonlocal condition (1.2) will be

As the limit $m\longrightarrow \infty$, we obtain

Theorem 5.1. Let the assumptions (I)–(III) be satisfied. If $\sum\nolimits^{m}_{k = 1}a_{k}$ be convergent, then the nonlocal boundary value problems of (1.3) and (2.1) have at least one solution given by

Proof. As $m\rightarrow \infty$, the solution of the nonlocal boundary value problem (2.1) will be

6.   Infinite-point boundary condition

Theorem 6.1. Let the assumptions (I)–(III) be satisfied, then the nonlocal boundary value problems of (1.4) and (2.1) have at least one solution given by

Proof. Let the assumptions of Theorem 2.1 be satisfied. Let $\sum\nolimits^{m}_{k = 1} a_{k}$ be convergent, then take the limit to (1.4), we have

Now

and by the comparison test $(\sum\nolimits^{m}_{k = 1}a_{k}\int_{0}^{\tau_{k}}\frac{(\tau_{k}-\theta)^{\alpha-1}}{\Gamma(\alpha)}y(\theta)d\theta)$ is convergent,

Furthermore, from (2.9) we have

Example 6.1. Consider the following nonlinear integro-differential equation

with boundary condition

Let

then

and also

It is clear that the assumptions (I) and (II) of Theorem 2.1 are satisfied with $a_{1}(t) = t^{4}e^{-t}\in L^{1}[0, 1]$, $a_{2}(t) = \frac{4}{3}I^{\gamma}|(\cos(5t+1)|\in L^{1}[0, 1]$, and let $\alpha = \frac{1}{3}$, $\gamma = \frac{2}{3}$, then $2K_{1}\gamma+K_{1}K_{2}\alpha = \frac{1}{2} < \alpha\gamma\Gamma(2-\alpha) = \frac{1}{2}\Gamma(2-\alpha)$. Therefore, by applying Theorem 2.1, the nonlocal problems (6.1) and (6.2) has a continuous solution.

7.   Conclusions

In this paper, we have studied a boundary value problem of fractional order differential inclusion with nonlocal, integral and infinite points boundary conditions. We have prove some existence results for that a single nonlocal boundary value problem, in of proving some existence results for a boundary value problem of fractional order differential inclusion with nonlocal, integral and infinite points boundary conditions. Next, we have proved the existence of maximal and minimal solutions. Then we have established the sufficient conditions for the uniqueness of solutions and continuous dependence of solution on some initial data and on the coefficients $a_k$ are studied. Finally, we have proved the existence of a nonlocal boundary value problem with Riemann-Stieltjes integral condition and with infinite-point boundary condition. An example is given to illustrate our results.

Conflict of interest

The authors declare no conflict of interest.