The risk-return relationship and volatility feedback in South Africa: a comparative analysis of the parametric and nonparametric Bayesian approach

1.   Introduction

As it is well-known that the cylindrically symmetric Navier-Stokes equations take the form

where $\rho(r, t)$ is the density, $u(r, t)$, $v(r, t)$, and $w(r, t)$ are velocities in different directions, $\theta(r, t)$ is the temperature, the pressure $P$ and the internal energy $e$ are related with the density and temperature

the specific gas constant $R$ and the specific heat at constant volume $c_v$ are positive constants, respectively; the symbol $Q$ denotes

$\mu$ and $\lambda$ are viscosity coefficients, and $\kappa$ is the heat conductivity coefficient.

Without loss of generality, we shall consider the system (1.1) with the following initial-boundary data:

Our main goal is to show the large-time behavior of global solutions to the initial-boundary value problem (1.1)–(1.8) with large initial data. For this purpose, it is convenient to transform the initial-boundary value problem (1.1)–(1.8) into Lagrangian coordinates. We introduce the Lagrangian coordinates $(t, x)$ and denote $(\tilde\rho, \tilde u, \tilde v, \tilde w, \tilde{\theta})(t, x) = (\rho, u, v, w, \theta)(t, r),$ where

and

Note that the function $f$ is invertible on $[a, b]$ provided that $\rho_0(y) > 0$ for each $y\in[a, b]$ (which will be assumed in Theorem 1.1). Due to (1.1)$_1$ and (1.8), we see

Then it is easy to check

which translates the domain $[0, T]\times[a, b]$ into $[0, T]\times[0, 1]$. Hereafter, we denote $(\tilde{\rho}, \tilde{u}, \tilde{v}, \tilde{w}, \tilde{\theta})$ by $(\rho, u, v, w, \theta)$ for simplicity. The identities (1.9) and (1.10) imply

where $\tau: = \rho^{-1}$ is the specific volume. By means of identities (1.11), system (1.1)–(1.8) is changed to

where $t > 0$, $x\in\Omega = (0, 1)$, $P = \frac{R\theta}{\tau}$, $e = \frac{R\theta}{\gamma-1}$, and $Q = \frac{\lambda(r u)_x^2}{\tau}-4 \mu uu_x+\frac{\mu r^2w_x^2}{\tau}+\mu\tau\left(\frac{r v_x}{\tau}-\frac{v}{r }\right)^2$.

Throughout this paper, we assume that $\mu, \lambda$, and $\kappa$ are power functions of absolute temperature as follows:

where constants $\tilde{\mu}, \tilde{\lambda}, \tilde{\kappa}, \alpha$, and $\beta$ are positive constants.

The objective of this paper is to study the global existence and stability of the solutions to an initial-boundary value problem of (1.12)–(1.16) with the initial data:

and the boundary conditions:

Using Navier-Stokes equations as a model for describing fluid motion has been widely accepted by the physics community. In recent years, some significant progress has been made in the study of Navier-Stokes equations with constant viscosity coefficients. When the initial value has a certain small property and vacuum state does not exist, the global existence, uniqueness, and large-time behavior of the solutions can be easily calculated ^{[1,2,3,4,5,6,7,8]}. However, solving the problem of large initial values is very challenging, and the first significant breakthrough was achieved by Lions ^[9]. Besides, by assuming that the initial value is only sufficiently small in the energy space, Hoff ^[10,11] confirmed the existence of global weak solutions. In the process of studying fluid motion, a vacuum state is often involved, which makes calculations far more complex. The results in ^[12,13] indicate the Cauchy problem of Navier-Stokes equations with constant coefficients containing vacuum state is not appropriate. This uncertainty is reflected by the fact that the solutions of the system have no continuous dependence on the initial values. Based on physical considerations, Liu-Xin-Yang ^[12] studied the Cauchy problem of the Navier-Stokes equations with density dependent viscosity, and proved its local suitability. However, only when the temperature and density change within a suitable range, real fluids can be considered as ideal fluids (viscosity coefficients are constants). In the case of large changes in temperature or density, the viscosity of the real fluid will vary greatly ^[14].

On the other hand, Navier-Stokes equations can be developed using the Chapman-Enskog expansion of the microscopic particle collision model Boltzman equation. Consequently, it can be determined that the viscosity depends on the temperature. However, compared to the abundant research using classical models, the studies on the physical case using the temperature-dependent viscosity model are lacking. Because the viscosity and heat conductivity are both temperature-dependent, degeneracy and strong non-linearity may appear. Pan-Zhang ^[15] and Huang-Shi ^[16] obtained global strong solutions and large-time behavior in bounded domains for one-dimensional Navier-Stokes equations, when $\alpha = 0$ and $0 < \beta < 1$. The studies of Liu-Yang-Zhao-Zhou ^[17] and Wan-Wang ^[18] also acquired global solutions of Navier-Stokes equations in one dimensional and cylindrically symmetrical cases, respectively, with the requirement that $|\gamma-1|$ was small enough. Wang-Zhao ^[19] removed the smallness condition of $|\gamma-1|$, and established global classical solutions to Navier-Stokes equations in the one-dimensional whole space when $\mu$ and $\kappa$ satisfy:

where $\alpha$ is small enough. In their calculations, the viscosity and heat-conductivity were dependent on temperature and density, and to overcome the difficulties caused by density, the following conditions could not be removed:

This means that estimate of $\|\tau_x\|_{L^2(\Omega)}$ can be directly obtained without the upper and lower bounds of density, as long as the coefficient $\mu^{-1}$ or $\kappa^{-1}$ appears. However, if $h(\tau)$ is constant, then the constants $l_1 = l_2 = 0$ and the result of this case cannot be established using the model in ^[19]. Recently, Sun-Zhang-Zhao ^[20] considered an initial-boundary value problem of the compressible Navier-Stokes equations for one-dimensional viscous and heat-conducting ideal polytropic fluids with temperature-dependent transport coefficients, and discovered the global-in-time existence of strong solutions. In that paper, the initial data could be large if $\alpha \geq 0$ is small and the growth exponent $\beta \geq 0$ is arbitrarily large. It is worth mentioning that the smallness of $\alpha > 0$ depends on the size of the initial data. However, unfortunately the study did not provide a specific relationship between $\alpha$ and the initial data in ^[20]. Our main results are concluded as follows.

Theorem 1.1. For given positive constants $M_0, V_0 > 0,$ assume that

Then there exist $\epsilon_0 > 0$ and $C_0$ which depend only on $\beta$, $M_0$, and $V_0$, such that the initial-boundary value problem (1.12)–(1.19) with $0\leq \alpha \leq \epsilon_0: = \min\{|\alpha_1|, |\alpha_2|\}$ and $\beta > 0$ admit a unique global-in-time strong solution $(\tau, u, v, w, \theta)$ on $[0, 1]\times[0, +\infty)$ satisfying

and

where $\alpha_1, \alpha_2$ defined in what follows are dependent only on $\beta$, $M_0$, and $V_0$ (see details in (3.2), (3.5), and (3.6)). Moreover, for any $t > 0$, the exponential decay rate is

where

A few remarks are in order.

Remark 1. For $k = 1, 2$ and $1\leq p\leq\infty$, we adopt the simplified nations for the standard Sobolev space as follows:

Remark 2. We remark here that the growth exponent $\beta\in(0, +\infty)$ can be arbitrarily large, and the choice of $\epsilon_0 > 0$ depends only on $\beta$, $V_0$, and the $H^2-$norm of the initial data. An outline of this paper is as follows. We devote Section 2 to a discussion of a number of a priori estimates independent of time, which are needed to extend the local solution to all time. Based on the previous estimates, the main results, Theorem 1.1 are proved in Section 3.

Remark 3. In this paper, the positive $c$, $C$, and $C_i(i = 0, 1, \cdots, 16)$ are some positive constants which depend only on $\beta$, $M_0$, and $V_0$, but not on the time $t$. Furthermore, $c$ and $C$ are different from line to line.

2.   A priori estimates

First of all, define

where $N$, $m_1$, $m_2$, and $t_1, t_2(t_2 > t_1)$ are constants and

with

The main purpose of this section is to derive the global $t$-independent estimates of the solutions $(\tau, u, v, w, \theta)\in X(0, T; m_1, m_2, N).$

We start with the following basic energy estimate.

Lemma 2.1. Assume that the conditions listed in Theorem 1.1 hold. Then there exists a constant $0 < \epsilon_1 \leq 1$ depending only on $M_0$ and $V_0$, such that if

where

then for $T\geq 0$,

where

Proof. Multiplying (1.12)–(1.16) by $R\hat{\theta}(\bar{\tau}^{-1}-\tau^{-1})$, $u$, $v$, $w$, and $(1-\hat{\theta} \theta^{-1})$, respectively, integrating over $[0, 1]$, and adding them together, one obtains

where $Q = \frac{\lambda(r u)_x^2}{\tau}-4 \mu uu_x+\frac{\mu r^2w_x^2}{\tau}+\mu\tau\left(\frac{r v_x}{\tau}-\frac{v}{r }\right)^2$.

Apparently, by means of $\lambda = 2\mu+\lambda^\prime$, one has

Thus, one has

which combined with (2.1) and (2.3) yields

Integrating (2.4) over $(0, T)$, we can obtain (2.2) by the initial conditions $(\tau_0, u_0, v_0, \theta_0)$. □

Next, by means of Lemma 2.1, we derive the upper and lower bounds of $\tau$.

Lemma 2.2. Assume that the conditions of Lemma 2.1 hold. Then for $(x, t)\in \Omega\times[0, \infty)$,

Proof. The proof is divided into three steps.

Step 1 (Representation of the formula for $\tau$).

It follows from (1.13) that

that is

where

Integrating (2.5) over $[0, t]\times[x_1(t), x]$, we have

where $x_1(t)\in[0, 1]$ is determined by the following progresses. Next, for convenience, we define

It follows from the definitions above that

By the definition of $F$ and (1.12), one has

Due to (1.12) and (2.7), we have

Integrating (2.9) over $[0, t]\times \Omega$, one has

Hence, by virtue of the mean value theorem, there exits $x_1(t)\in[0, 1]$ such that $\varphi(x_1(t), t) = \int_0^1\varphi \tau \mathrm{d} x$. By the definition of $\varphi$, (2.8), and (2.10), one obtains

Putting (2.11) into (2.8), it follows that

Inserting (2.12) into (2.6), we derive

Let

where

It follows from (2.13) that

where

By (2.14), one has

Define that

Then, multiplying (2.15) by $J$ gives

Since $A(0) = 1$, integrating the above equality over $(0, t)$ about time, one has

Step 2 (Lower bound for $\tau$). First of all, by means of (2.1) and (2.2), one has

Next, we estimate $B$. Employing Jensen's inequality to the convex function $\phi$, we have

By (2.18) and Lemma 2.1, one obtains

which means that

Hence, by means of the definition of $B$ and (2.20), choosing $\varepsilon_1$ suitably small, there exist two constants $C_1$ and $C_2$, such that

That is,

Apparently, by means of (2.1) and (2.19), we deduce

Similarly, one also has

Thus, for $t\leq t_0 < \infty$,

For a enough large $t$, we have

So, we need the estimates of $\theta$ and $\frac{B(s)}{B(t)}$. By the mean value theorem and (2.19), there exits $x_2(t)\in[0, 1]$, such that

By Cauchy-Schwarz's inequality and (2.19), one has

which means that

By (2.16)–(2.17), (2.23)–(2.24), (2.21), Lemma 2.1, and (2.19), one has

that is

Putting (2.27) into (2.25), by (2.22), (2.28), and Lemma 2.1, for a enough large $t$, one has

Inserting (2.29) into (2.25), for a large enough time $T_0$, when $t > T_0$, it follows that

Step 3 (Upper bound for $\tau$). By (2.17), (2.22)–(2.24), and Lemma 2.1, one obtains

where we have used the results

In fact, by Hölder's inequality, for $0 < \beta\leq1$,

For $1 < \beta < \infty$,

By means of (2.26) and (2.32)–(2.33), we can obtain (2.31).

Thus, the inequality (2.30) combined with Gronwall's inequality and Lemma 2.1 yields that for any $t\geq0$,

□

However, we cannot get the time-space estimate of $v_x$ in Lemma 2.1. To obtain this estimate, we need the following result.

Lemma 2.3. Assume that the conditions listed in Lemma 2.1 hold. Then for any $p > 0$ and $T\geq0$,

Proof. By Lemma 2.1, the result of (2.34) has been established for $p = 1$. In the following steps, we do the estimate for $p > 0$ and $p\neq1$. Multiplying (1.16) by $\theta^{-p}$, integrating over $[0, 1]$, and using integration by parts gives

Apparently, there exists constant $C(p)$ depending on $p$ such that

By means of (2.35), (2.36), Lemma 2.2, (1.13), and (1.12), we deduce

Thus, employing the truth of

we can conclude from (2.37), Grönwall's inequality, and Lemma 2.2 that (2.34) is correct. In fact,

By virtue of Lemmas 2.1–2.2 and (2.19), one has

where we have used the fact that

For $\beta < 1$, it follows from Lemma 2.1 and (2.19) that

For $1\leq\beta < \infty$,

Hence, by (2.39)–(2.42) and Lemmas 2.1–2.2, we can derive (2.38). The proof of Lemma 2.3 is thus complete. □

According to Lemmas 2.1–2.3, we can conclude that the following results have been established.

Corollary 2.1. Assume that the conditions listed in Lemma 2.1 hold. Then for $-\infty < q < 1$, $0 < p < \infty$, and $T\geq0$,

Here, we have taken $p = \alpha$ in (2.34) to obtain the time-space estimates of $v$ and $v_x$.

Using the result above, we establish the following estimate about $\tau_x$.

Lemma 2.4. Assume that the conditions listed in Lemma 2.1 hold. Then for $T\geq0$,

Proof. According to the chain rule, one has

By means of (1.12), (1.13), and (2.44), we have

Multiplying (2.45) by $\frac{\lambda\tau_x}{\tau}$, integrating over $[0, 1]$ about $x$, and using (1.12) and (2.44), we obtain

By Hölder's inequality, (2.1), (1.12), and Corollary 2.1, one has

Using Corollary 2.1 and taking $p = \beta$, one has

Hence, we argue the term $I_2$ as the following

By means of integration by parts, Corollary 2.1, and (2.1), one can derive

By virtue of (2.1), we derive

By means of (2.1), Corollary 2.1, and (1.16), one can deduce

Inserting (2.47) and (2.49)–(2.52) into (2.46), and choosing $\varepsilon$ suitable small, we obtain

Integrating (2.53) over $[0, t]$, using Cauchy-Schwarz's inequality, (2.48), and Corollary 2.1, for any $t\geq0$, one has

By virtue of (2.54), we have

It follows from (2.19) and (2.48) that

Thus, it follows from (2.55)–(2.56) that

The proof of Lemma 2.4 has been completed by (2.54) and (2.57). □

Next, based on the estimate of $\tau_x$, we are devoted to derive the estimates on the first-order derivatives of $w_x$.

Lemma 2.5. Assume that the conditions listed in Lemma 2.1 hold. Then for $T\geq0$,

Proof. Multiplying (1.15) by $w_{xx}$ and integrating over $[0, 1]$ about $x$, we find from (2.1) and Lemma 2.4 that

Taking $\varepsilon$ suitably small in (2.58) finds

The proof of Lemma 2.5 is complete by integrating (2.59) over $(0, t)$ about time and choosing $\varepsilon$ suitably small. □

Based on the above result, we have the following uniform first-order derivatives estimates on the velocity $(u, v)$.

Lemma 2.6. Assume that the conditions listed in Lemma 2.1 hold. Then for $T\geq0$,

Proof. Multiplying (1.13) and (1.14) by $u_{xx}$ and $v_{xx}$, respectively, and integrating over $\Omega$ about $x$, by integration by parts, one has

By Cauchy-Schwarz's inequality, one has

It follows from Sobolev's inequality, the boundary condition of $v$, and Corollary 2.1, that we have

Direct computation from (2.1) yields

and

Putting (2.61)–(2.65) into (2.60) and taking $\varepsilon$ suitably small gives

Integrating (2.66) over $(0, T)$ about time, and using Lemma 2.4 and Corollary 2.1, we find

For $\beta > 1$, we take $p = \beta-1$ in (2.43), and then

Substituting (2.68) into (2.67), it follows for $\beta > 1$ that

Next, we need to estimate the $L^2(\Omega\times(0, t))$-norm of $\theta_x$ for $0 < \beta\leq1$. We deduce from multiplying (1.16) by $\theta^{1-\frac{\beta}{2}}$ and integration by parts that

Notice that

and

We can conclude from (2.70)–(2.72) that

which combined with Young's inequality and Corollary 2.1 yields

By means of Lemma 2.5, (2.67), and (2.73), we find for $0 < \beta\leq 1$,

By virtue of (1.12)–(1.16), (2.1), Corollary 2.1, Lemma 2.4, (2.69), and (2.74), it follows that

□

To obtain the first-order derivative estimate of the temperature, we need to first establish the uniform upper and lower bounds of $\theta$.

Lemma 2.7. Assume that the conditions listed in Lemma 2.1 hold. Then for $T\geq0$,

Proof. First of all, multiplying (1.16) by $\theta$, and integrating over $[0, 1]$ about $x$, yields

Applying Gronwall's inequality to (2.76), we can obtain

Based on the estimate above, we can get the bound of $\int_{0}^{1}\theta^\beta\theta_x^2 \mathrm{d} x$ which will be used to obtain the upper bound of $\theta$. Multiplying (1.16) by $\theta^\beta \theta_t$ and integrating over $(0, 1)$ about $x$, it follows that

By integration by parts, one has

Inserting (2.79) into (2.78), we can deduce that

By Sobolev's inequality, Corollary 2.1, and Lemmas 2.5–2.6, one can find that

By virtue of (2.80), Grönwall's inequality, and (2.81), we can obtain

Thanks to (2.82), it follows that

That is, for $t\geq0$,

Thanks to (2.77) and (2.84), one has

Combining (2.83) and (2.84), one has

So, from (2.83), (2.85), and (2.86), one has

From (2.83), when $t\rightarrow +\infty$,

and we can obtain that there exists some time $T_0\gg1$ such that when $t > T_0$,

Fixing $T_0$ in (2.87), multiplying (1.16) by $\theta^{-p}$, $p > 2$, and integrating over $[0, 1]$ about $x$ yield

Hence,

where $C$ is a generic positive constant independent of $p$. Thus, integrating the above inequality over $(0, t)$ and letting $p\rightarrow \infty$, we arrive that

The proof of Lemma 2.7 is complete. □

Lemma 2.8. Assume that the conditions listed in Lemma 2.1 hold. Then for $T\geq0$,

Proof. Multiplying (1.16) by $\theta_{xx}$, integrating over $[0, 1]$ on $x$, and by Hölder's, Poincaré's, and Cauchy-Schwarz's inequalities, Corollary 2.1, Lemma 2.4, and Lemma 2.7, we have

Choosing $\varepsilon$ suitably small in (2.88) gives

Integrating (2.89) and using Lemmas 2.5–2.6, one has

Hence, similar to (2.75), by means of (1.16), Corollary 2.1, Lemmas 2.4–2.7, and (2.90), one can deduce that

□

Next, we derive the second-order derivatives estimates of $(\tau, u, v, w, \theta)$.

Lemma 2.9. Assume that the conditions listed in Lemma 2.1 hold. Then for $T\geq0$,

Proof. Applying $\partial_t$ to (1.13) and multiplying by $u_t$ in $L^2$, one has

Applying $\partial_t$ to (1.14) and multiplying by $v_t$ in $L^2$, one has

Applying $\partial_t$ to (1.15) and multiplying by $w_t$ in $L^2$, one has

Adding (2.91)–(2.93) together, we get

Before the computations of $III_1$ to $III_7$, we need to keep in mind the following facts:

Then, by Hölder's, Sobolev's, and Cauchy-Schwarz's inequalities, one has

and

By virtue of (1.13), one has

Thus, it follows from Hölder's, Sobolev's, and Cauchy-Schwarz's inequalities that

and

It follows from (1.14) that

Then

By virtue of (1.15), we can obtain

and

Putting (2.95)–(2.101) into (2.94) gives

Applying $\partial_t$ to (1.16) and multiplying by $\theta_t$ in $L^2$, it follows that

First of all, by means of the definition of $Q$, one has

Using (2.104) and Sobolev's inequality, we can derive from (2.103) that

Adding (2.102) to (2.105) and choosing $\varepsilon > 0$ suitably small, it follows that

By means of (2.106) and Grönwall's inequality, we deduce

According to (1.13), one has

which means that

Hence, by means of (2.107), we obtain

Similarly, use the equations (1.12)–(1.16), we also can derive

Here, we omit the details of (2.108). The proof of Lemma 2.9 is complete. □

Lemma 2.10. Assume that the conditions listed in Lemma 2.1 hold. Then for $T\geq0$,

Proof. Apply $\partial_x$ to (2.45) and multiply by $(\lambda\tau_x/\tau)_x$ in $L^2$ to get

where the following fact has been used:

By Sobolev's inequality and Lemmas 2.6–2.9, we have

and

Noting that

we can derive from Sobolev's inequality and Lemma 2.4 that

So, it follows from Cauchy-Schwarz's inequality and Lemma 2.4 that

Taking $\varepsilon$ suitably small, putting (2.110)–(2.111) into (2.109), and using Lemmas 2.4, 2.8, and 2.9, we find

By (2.113), Grönwall's inequality, Corollary 2.1, and Lemmas 2.6 and 2.8–2.9, one obtains

It follows from (2.112) and (2.114) that

Letting $\partial_x$ act on (1.13) gives

It follows from (2.115) and (2.116) that

where the following fact has been used:

Similarly, using (1.14)–(1.15), we also have

Letting $\partial_x$ act on (1.16) gives

It follows from (2.114) and (2.117) that

By the definition of $Q$, one has

Since the following estimates have been obtained:

putting (2.119) into (2.118) yields

The proof of Lemma 2.10 is complete. □

3.   The proof of Theorem 1.1

With all a priori estimates from Section $2$ at hand, we can complete the proof of Theorem $1.1$. For this purpose, it will be shown that the existence and uniqueness of local solutions to the initial-boundary value problem (1.12)–(1.19) can be obtained by using the Banach theorem and the contractivity of the operator defined by the linearization of the problem on a small time interval.

Lemma 3.1. Letting (1.20) hold, then there exists $T_0 = T_0(V_0, V_0, M_0) > 0$, depending only on $\beta$, $V_0$, and $M_0$, such that the initial boundary value problem (1.12)–(1.19) has a unique solution $(\tau, u, v, w, \theta)\in X(0, T_0;\frac{1}{2}V_0, \frac{1}{2}V_0, 2M_0)$.

Proof of Theorem 1.1: First, to prove Theorem 1.1, according to (1.20), one has

Combined with Lemma 3.1, there exists $t_1 = T_0(V_0, V_0, M_0)$ such that $(\tau, u, v, w, \theta)\in X(0, t_1;\frac{1}{2}V_0, \frac{1}{2}V_0, 2M_0).$

We find the positive constant $|\alpha| \leq \alpha_1$, where $\alpha_1$ satisfies

where $\epsilon_1$ is chosen in Lemma 2.1. That means that one can choose

One deduces from Lemmas 2.1–2.10 with $T = t_1$ that for each $t\in [0, t_1]$, the local solution $(\tau, u, v, w, \theta)$ satisfies

and

where $C_i(i = 2, \cdots, 7)$ is chosen in Section $2$ and $C_8^2: = \sum_{i = 2}^{7}C_i$. It follows from Lemma 2.9 and Lemma 2.10 that $(\tau, u, v, w, \theta)\in C([0, T); H^2).$ If one takes $(\tau, u, v, w, \theta)(\cdot, t_1)$ as the initial data and applies Lemma 3.1 again, the local solution $(\tau, u, v, w, \theta)$ can be extended to the time interval $[t_1, t_1+t_2]$ with $t_2(C_0, C_1, C_8)$ such that $(\tau, u, v, w, \theta)\in X(t_1, t_1+t_2; \frac{1}{2}C_0, \frac{1}{2}C_1, \frac{1}{2}C_8).$ Moreover, for all $(x, t)\in [0, 1]\times [0, t_1+t_2]$, one gets

and

which combined with (3.3) and (3.4) implies that for all $t\in [0, t_1+t_2],$

Take $\alpha\leq \min \{\alpha_1, \alpha_2\},$ where $\alpha_i(i = 1, 2)$ are positive constants satisfying (3.1) and

where the value of $\epsilon_1$ is chosen in Lemma 2.1. That means that we can choose

Then one can employ Lemmas 2.1–2.10 with $T = t_1+t_2$ to infer the local solution $(\tau, u, v, w, \theta)$ satisfying (3.3) and (3.4).

Choosing

and repeating the above procedure, one can extend the solution $(\tau, u, v, w, \theta)$ step-by-step to a global one provided that $|\alpha|\leq \epsilon_0$. Furthermore,

from which we derive that the solution $(\tau, u, v, w, \theta)\in X(0, +\infty; C_0, C_1, C_9).$ □

The large-time behavior (1.21) follows from Lemmas 2.4–2.10 by using a standard argument ^[21].

First, thanks to (1.15), (2.1), (2.43), (2.55), (2.62), (2.73), Corollary 2.1, and Lemmas 2.4–2.10, taking $\hat{\theta} = E_0$, one has

By Cauchy-Schwarz's inequality, one has

Hence, by means of (3.11), Poincaré's inequalities, Corollary 2.1, and Lemma 2.7, one can deduce

Multiplying (3.7)–(3.10) by $C_{14}$, $C_{15}$, and $C_{16}$, respectively, and adding them together with (3.10), one has

where we have defined

and chosen constants $C_{14} > C_{15} > C_{16} > 0$ suitably large such that

Taking $\frac{C_{14}}{2} > C_{13}$ and using Poincaré's inequality gives

where we have used the facts

By means of (3.12) and (3.13), we can derive that

By means of $\bar r$, one has

By means of (3.14) and (3.15), we have

The proof is thus complete. □

Author contributions

Dandan Song: Writing-original draft, Writing-review & editing, Supervision, Formal Analysis; Xiaokui Zhao: Writing-review & editing, Methodology, Supervision.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors are grateful to the referees for their helpful suggestions and comments on the manuscript. This work was supported by the NNSFC (Grant No. 12101200), the Doctoral Scientific Research Foundation of Henan Polytechnic University (No. B2021-53) and the China Postdoctoral Science Foundation (Grant No. 2022M721035).

Conflict of interest

The authors declare there is no conflict of interest.