Large time behavior of strong solution to the magnetohydrodynamics system with temperature-dependent viscosity, heat-conductivity, and resistivity

1.   Introduction

The governing equations of a planar magnetohydrodynamics (MHD) compressible flow can be written in Lagrange variable form as:

Here $t > 0$ represents the time, and $x\in\Omega = (0, 1)$ denotes the Lagrange mass coordinate. The unknown functions $v(x, t) > 0$, $u(x, t)$, ${\boldsymbol{w}} = (w_1(x, t), w_2(x, t))$, ${\boldsymbol{b}} = (b_1(x, t), b_2(x, t))$, $\theta(x, t) > 0$, $e$, and $P$ are the specific volume of the gas, longitudinal velocity, transverse velocity, transverse magnetic field, absolute temperature, internal energy, and pressure, respectively. $\mu$ and $\lambda$ are the viscosity of the flow, $\nu$ is the resistivity, and $\kappa$ is the heat conductivity.

In this paper, we consider the MHD flow of a perfect gas. Thus, $P$ and $e$ satisfy:

Here $R > 0$ denotes the specific gas constant, and $C_v > 0$ stands for the heat capacity at constant volume. It is assumed that $\mu$, $\lambda$, $\nu$, and $\kappa$ satisfy

which contain the positive constants $\tilde{\mu} > 0$, $\tilde{\lambda} > 0$, $\tilde{\nu} > 0$, $\tilde{\kappa} > 0$, $\alpha > 0$, and $\beta > 0$.

The systems (1.1)–(1.7) are supplemented with initial conditions

and boundary ones

Obviously, the initial data (1.8) should be compatible with the boundary conditions (1.9).

When ${\boldsymbol{w}} = {\boldsymbol{b}} = \textbf{0}$, the Eqs (1.1)–(1.5) are converted into the compressible Navier–Stokes equation, which can be derived from Boltzmann's equation, assuming that the space and time scales are larger than all inherent scale–lengths, such as the Debye length or the gyro-radii of the charged particles ^[1,2,3,4,5]. Also, one can deduce from the Chapman–Enskog expansion for the first level of approximation in kinetic theory that the viscosity $\mu$ and $\lambda$ may depend on the temperature or the density (see Chapman and Cowling ^[6]). Experimental results ^[7] show that the transport coefficients $\mu$ and $\kappa$ vary according to gas temperature and density at very high temperatures and densities.

The central point of magnetohydrodynamics theory is that conductive fluids can support magnetic fields. Li and Shang ^[8] proved the existence and uniqueness of the global-in-time classical solution to the initial-boundary value problem when the viscosity, resistivity, and heat conductivity depend on the specific volume $v$ and the temperature $\theta$. In that paper, the coefficients are assumed to be proportional to $h(v)\theta^\alpha$, where $h(v)$ is a non-degenerate and smooth function satisfying some natural conditions, and the absolute value of the exponent $\alpha$ is sufficiently small. It's worth noting that Li and Shang considered the planar compressible magnetohydrodynamic system for the domain $[0, 1]\times \mathbb{R}^2$. Besides, Huang et al. ^[9] proved the large-time behavior of strong solutions to equations of compressible planar magnetohydrodynamic flow with the heat conductivity is the power function of temperature. Similar results can be observed in various other reports ^{[10,11,12,13,14,15,16]}.

Recently, Sun et al. ^[17] verified the existence and uniqueness of a global-in-time non-vacuum strong solution to a one-dimensional compressible Navier–Stokes system for a viscous and heat-conducting ideal polytropic gas. It was assumed that the viscosity $\mu$ and heat conductivity $\kappa$ depend on temperature $\theta$ with $\mu (\theta) = \theta^\alpha$ and $\kappa (\theta) = \theta^\beta$ for sufficiently small $\alpha > 0$ and arbitrary $\beta\geq 0$.

Before presenting our main results, we need to provide some explanations of the symbols first. Throughout this paper, the positive general constant $C$ will be different in different lines. For $1\leq p \leq \infty$, and integer $k\geq 0$, we adopt the simplified notations for the standard Sobolev space as follows:

Without loss of generality, we assume that $\tilde{\lambda} = \tilde{\mu} = \tilde{\nu} = \tilde{\kappa} = R = c_v = 1$, and

Inspired above, we have the following main results.

Theorem 1.1. For given positive constants $M_0 > 0$ and $V_0 > 0$. Assume that

Then there exist $\epsilon_0 > 0$, $C_0 > 0$ and $C_1 > 0$ which depend only on $\beta$, $M_0$, $V_0$, such that the initial boundary value problem (1.1)–(1.9) with $0\leq\alpha\leq \varepsilon_0$ (see (3.5)) admits a unique global-in-time strong solution $(v, u, \textbf{w}, \textbf{b}, \theta)(x, t)$ on $[0, 1]\times [0, +\infty)$ satisfying

and

Furthermore, for any $t\geq 0$, it holds that

where $C, \eta_0 > 0$ are some positive constants.

Remark 1.1. From the view of physics, the resistance is a function of temperature (e.g., ^[18]). This implies that our results is physical. Dou et al.'s 2021 study^[19], published in Scientific Reports, delves into a variety of issues, including Enhanced Oil Recovery, where the technology can potentially improve the extraction of residual oil from oil fields. The equations presented in our study, especially those related to magnetic force distribution and the relationship between magnetic force and displacement, are pivotal for understanding and optimizing these applications. They assist in predicting the behavior of magnetic foams under various conditions, which is crucial for designing effective systems in the aforementioned fields.

In this paper, we try to use the framework of Li and Liang ^[20] to prove the global wellposedness of the solution. It should be emphasized that the key step is to derive the time–independent positive lower and upper bounds of specific volume and temperature. The foremost obstacles lie in the strong non-linearities caused by the temperature-dependent viscosity, resistivity, and heat-conductivity from Eq (1.7). Fortunately, these nonlinear terms are involved with $\mu_x$, $\lambda_x$, $\nu_x$, $\mu_t$, $\lambda_t$, or $\nu_t$ which can be controlled by the smallness of $\alpha$. With the help of upper and lower bounds of the specific volume, we can then estimate the higher-order derivatives of the solutions, and the upper and lower bounds of the temperature.

The rest of this paper is organized as follows. Section 2 is devoted to a discussion of a number of a priori estimates independent of time, which are required to extend the local solution to the time global. Based on the estimates given in Section 2, the main results of Theorem 1.1 are established in Section 3.

2.   A priori estimates

For constants $N, m_1, m_2$, and $T$, define

where

The main purpose of this section is to derive certain $t$-dependent a priori estimates for the variables $(v, u, \theta, {\boldsymbol{w}}, {\boldsymbol{b}})$ in the function space $X(0, T;m_1, m_2, N)$, relevant to the initial boundary value problem (1.1)–(1.9) for $T > 0$ and $0 < m_i\le 1(i = 1, 2)$, $2\le N < +\infty$. It follows from Sobolev's inequality that

Firstly, let us derive the time-independent lower and upper bounds of $v$.

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 $\beta$, $M_0$, and $V_0$, such that if

where

Then for $(x, t)\in [0, 1]\times [0, T]$,

Here (and in what follow), $C_0, C_i(i = 1, 2, \cdots, 10)$, and $C$ denote some generic positive constants depending only on $\beta$, $\|(v_0, u_0, \textbf{w}_0, \textbf{b}_0, \theta_0)\|_{H^2}$, $\inf_{x\in (0, 1)}v_0(x)$, and $\inf_{x\in (0, 1)}\theta_0(x)$.

Proof. The proof is divided into four steps.

Step 1. (Basic energy estimate)

According to (1.1), (1.5), (1.9), and (1.10), for $t > 0$, one has

In light of (2.1), it is deduced that

Simplifying (1.5) gives

Then, multiplying (1.1)–(1.4), and (2.5) by $(1-v^{-1})$, $u$, ${\boldsymbol{w}}$, ${\boldsymbol{b}}$, and $(1-\theta^{-1})$, respectively, and integrating them over $[0, 1]\times [0, T]$, together with (2.4) gives

where

and $E_0$ is the initial total entropy defined by

Step 2. (Representation formula for $v$)

First, (1.2) can be written as

that is,

where

Integrating (2.8) over $[0, t]\times [x_1(t), x]$, it follows

where $x_1(t)\in [0, 1]$ is determined by the following steps and $\mu_0 = \mu(\theta_0)$. Moreover, for ease of notation, define

Based on the above definitions that

It is easy to show that

With the help of (1.1) and (2.9) that

Integrating (2.11) over $[0, t] \times [0, 1]$, it follows

Hence, according to the mean value theorem, there exists $x_1(t)\in [0, 1]$ such that $\varphi(x_1(t), t) = \int_{0}^{1} v\varphi \mathrm{d} x$. By the definition of $\varphi$, (2.9) and (2.12), one obtains

Substituting (2.13) into (2.10) gives

Moreover, substituting (2.14) into (2.9),

It follows from (2.15) that

where

From (2.16), one has

Furthermore, we define

Then, multiplying (2.17) by $J$, we have

Since $A(0) = 1$, integrating the above equality over $(0, t)$ with respect to time, one obtains

Step 3. (Lower bound for $v$)

Applying Jensen's inequality to the convex function $\theta-\ln\theta$ leads to

which, together with (2.3) and (2.6), leads to

where $0 < \alpha_1 < \alpha_2$ are two roots of

The expression of $D$ and

imply that

Next, we will estimate $B$. It follows from (2.3) and (2.4) that

and

On the other hand, by the expression of $g$, there exists a sufficiently small $\epsilon > 0$ such that

Putting (2.21)–(2.23) into the expression of $B$, we will find that there exist $C_2, C_3$, such that

That means

Thus, for $0\le t < t_0$, one has

For large enough $t > t_0$, it follows

Therefore, one needs the estimates of $\theta$ and $\frac{B(s)}{B(t)}$. By the mean value theorem and (2.3), there exists $x_2(t)\in [0, 1]$, such that $C^{-1}\leq \theta(x_2(t), t)\leq C$. Based on Cauchy-Schwarz's inequality, one has

which implies

From (2.18), (2.20), (2.21), and (2.23), one has

For the large enough time $T_0$, when $t > T_0$, plugging (2.25) into (2.24) gives

Step 4. (Upper bound for $v$)

According to Holder's inequality, for $0 < \beta\leq 1$, one has

That means

For $1 < \beta < \infty$, one has

which means

Then the standard calculations give

It follows from the expression of $v$ and (2.26)–(2.29) that

By using Grönwall's inequality, one has

Up to now, the proof of Lemma 2.1 has been finished.

Lemma 2.2. Assume that the conditions listed in Lemma 2.1 hold; then for any $p > 0$, there exists some positive constant $C(p)$ such that

Proof. From (2.6), we see that (2.30) holds for $p = 1$. Then we assume $p > 0$ and $p\neq 1$. Multiplying (2.5) by $\theta^{-p}$ and integrating by parts, one can arrive at

Moreover, it follows from (2.3) and (2.19) that

For any real number $q$, it follows from (2.2) and (2.4) that

After that, for $\beta\in(0, 1)$, it follows from (2.27), (2.28), and (2.19) that

when $\beta\geq 1$, one has

Therefore, for $\beta > 0$, it follows from (2.33), (2.34), and (2.6) that

Finally, we see that for $p\in [0, 1]$, one has

And for $\beta\geq 0$, it follows from (2.33), (2.34), and (2.6) that

As a result, according to (2.33), (2.34), (2.7), (2.35), and Grönwall's inequality, we derive (2.30) from (2.31), which finishes the proof of Lemma 2.2.

Lemma 2.3. Assume that the conditions listed in Lemma 2.1 hold; then for all $T > 0$,

Proof. First, integrating (2.5) over $[0, 1]\times [0, t]$, by (2.36), one has

thus it follows from (2.2) and (2.4) that

Next, since

the momentum equation (1.2) can be rewritten as

Multiplying (2.39) by $\left(u-\frac{\theta^\alpha v_x}{v}\right)$ and integrating it over $[0, 1]\times [0, t]$ yields that for any $t\in [0, T]$,

Each $I_i(i = 1, 2, 3, 4)$ can be estimated as follows. First, based on Cauchy's inequality, we have

where it has been used

Next, by using (2.4), (2.6), and (2.30) with $p = \beta$, it follows

Combining (2.42) with Cauchy's inequality leads to

Rewriting (2.5) as

We set

where

and

Then from (2.44), one has

Each $J_i(i = 1, 2, 3)$ can be estimated as follows. First, by means of Cauchy's inequality

According to the definition of $\mathcal{R}_1$, one has

It follows from the definition of $\mathcal{R}_2$ that

then, one has

Inserting (2.46)–(2.48) into (2.45) gives

Putting (2.41)–(2.43), and (2.49) into (2.40), combining Grönwall's inequality gives

Note that

that means

On the other hand, it follows from (2.33) and (2.34) that

Together with (2.38), the proof of Lemma 2.3 has been completed.

Lemma 2.4. Assume that the conditions listed in Lemma 2.1 hold; then for all $T > 0$, one has

Proof. First, rewrite (1.3) as

Multiplying (2.51) by $w_{xx}$ and integrating over $[0, 1]\times [0, T]$, one obtains:

Each $I_i(i = 1, 2, 3)$ is estimated as follows. From Cauchy's inequality and (2.4), it shows

where it has been used

Next, from the a priori assumption, one has

Furthermore, according to (2.37), one has

Substituting (2.53), (2.55), and (2.56) into (2.52), one has

Combining (2.51) with (2.54) gives

Next, rewrite (1.4) as

Multiplying (2.59) by $b_{xx}$ and integrating the result over $[0, 1]\times [0, T]$ yields

Each $J_i(i = 1, 2, 3)$ is estimated as follows. From Cauchy's inequality and (2.4), one has

where it has been used

It follows from (2.2), (2.4), (2.37), and (2.62) that

From (2.2)–(2.4), one has

According to (2.37), it follows

Inserting (2.61), (2.63)–(2.65) into (2.60), which implies

From (2.59), one obtains

Therefore, it follows from (2.57), (2.58), (2.66), and (2.67) that (2.50) is correct. Then the Lemma 2.4 has been proved.

Lemma 2.5. Assume that the conditions listed in Lemma 2.1 hold; then for all $T > 0$,

Proof. For the case of $\beta > 1$, setting $p = \beta -1$ in (2.30) will give

For $0 < \beta \leq 1$, multiplying (2.5) by $\theta^{1-\frac{\beta}{2}}$ and integrating by parts, it gives

Each $I_i (i = 1, 2, 3, 4)$ can be estimated as follows. First, by (2.7), one has

According to (2.32), one obtains

It follows

Next, it follows from (2.4) that

Substituting (2.70)–(2.73) into (2.69), integrating on $[0, T]$, and combining (2.37) and Grönwall's inequality, one has when $0 < \beta \leq 1$,

Then from (2.68) and (2.74) the proof of Lemma 2.5 has ended.

Lemma 2.6. Assume that the conditions listed in Lemma 2.1 hold; then for all $T > 0$, one has

Proof. Rewrite (1.2) as

Multiplying (2.76) by $u_{xx}$ and integrating the result over $[0, 1] \times [0, T ]$, it shows

Each $J_i$ can be estimated as follows. First, according to (2.37), one has

where it has been used

Secondly, combining (2.30) with $p = \beta +1-\alpha$, one obtains

Next, from (2.2) and (2.4), one has

Furthermore, from (2.37), one has

where we have used

In fact, for $0 < \beta < 2$,

and for $\beta\geq 2$,

Finally, it follows from (2.62) that

Substituting (2.78) and (2.80)–(2.83) into (2.77) gives

On the other hand, from (1.2), one has

and

Combining this with (2.84), the proof of Lemma 2.6 has been finished.

Lemma 2.7. Assume that the conditions listed in Lemma 2.1 hold; then for all $T > 0$,

Proof. Multiplying (2.5) by $\theta$ gives

It follows from (2.32), (2.74), and (2.6) that

By combining this with (2.3), (2.37), and Grönwall's inequality, one has

Next, multiplying (2.5) by $\theta^\beta \theta_t$ and integrating it over $(0, 1)$, by (2.4), one has

Moreover, each $I_i (i = 1, 2, 3)$ can be estimated as follows. First,

Second,

Finally,

Substituting (2.88)–(2.90) into (2.87) gives

Direct calculations yield

and

From this and (2.91), and integrating over $[0, T]$, together with Grönwall's inequality, one has

Combining with (2.91), one has

On the one hand, (2.93) gives

Together with (2.4), (2.37), (2.91), and (2.92), one has

Combining (2.37), (2.93), and (2.94) leads to

Then combining (2.91) gives

Therefore, it follows from (2.19) and (2.95) that there exists some $T_0$, such that

for all $(x, t)\in [0, 1]\times [T_0, +\infty)$.

On the other hand, for $p > 2$, multiplying (2.5) by $\frac{1}{\theta^p}$, one has

That means

where the positive constant $C$ independent of $p$ and $T$. (2.97) gives

Letting $p\rightarrow +\infty$, there exists a positive constant $C_1\leq \frac{\gamma_1}{2}$ such that

for all $(x, t)\in [0, 1] \times [0, T_0]$. Combining this, (2.96), and (2.92) yields that for all $(x, t)\in [0, 1] \times [0, +\infty)$,

Together with (2.91), one has

Finally, it follows from (2.5) that

from this and (2.37), (2.97), (2.98) yields

Combining with (2.98)–(2.100), the proof of Lemma 2.7 has been finished.

Lemma 2.8. Assume that the conditions listed in Lemma 2.1 hold; then for all $T > 0$, one has

Proof. First, differentiating (1.2) with respect to $t$ shows

Multiplying the above equation by $u_t$, one obtains after integration by parts,

where in the last inequality it has been used

Next, differentiating (1.3) with respect to $t$ shows

Multiplying the above equation by $w_t$, one also gets after integration by parts,

Similarly, differentiating (1.4) with respect to $t$ shows

Multiplying the above by ${\boldsymbol{b}}_t$, one gets after integration by parts that

At the end, differentiating (2.5) with respect to $t$ shows

Multiplying the above by $\theta_t$ and integrating by parts, one obtains

According to (2.86), one has

Combining (2.102)–(2.105) and Grönwall's inequality, we deduce

Finally, we rewrite (1.2) as

It follows from (2.106), (2.85), (2.86) and (2.37) that

Similarly, rewriting (2.5) as

Using (2.5), (2.106), (2.85), (2.86), (2.37), and (2.107), one has

Next, rewriting (1.3) as

Then it follows that

At the end, rewriting (1.4) as

thus, one has

Combining (2.106)–(2.110), the proof of Lemma 2.8 has been proved.

Lemma 2.9. Assume that the conditions listed in Lemma (2.1) hold; then for all $T > 0$, one has

Proof. First, differentiating (1.2) with respect to $x$ gives

Multiplying (2.112) by $\left(\frac{v_x}{v}\right)_x$, and integrating it over $[0, 1]$, we arrive at

Together with (2.37), (2.75), (2.85), (2.86), (2.101), and Grönwall's inequality, one has

That means

Furthermore, (2.112) can be written as

Together with (2.37), (2.75), (2.85), (2.86), (2.101), and (2.113), one has

Next, differentiating (2.5) with respect to $x$, it shows

Thus, one has

Similarly, differentiating (1.4) with respect to $x$, one has

It implies that

Finally, differentiating (1.3) with respect to $x$ gives

Then, one has

Combining with (2.113)–(2.117), we obtain (2.111). The Lemma 2.9 has been proved.

3.   The proof of Theorem 1.1

With all the a priori estimates in Section 2 at hand, we will 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 (1.1)–(1.9), which 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 the (1.10) holds, 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 (1.1)–(1.9) has a unique solution $(v, u, \textbf{w}, \textbf{b}, \theta)\in X(0, T_0; \frac{1}{2}V_0, \frac{1}{2}V_0, 2M_0)$.

Proof of Theorem 1.1 First, using Lemma 3.1, the problem (1.1)–(1.9) has a unique solution $(v, u, {\boldsymbol{w}}, {\boldsymbol{b}}, \theta) \in X(0, T_1;\frac{1}{2}V_0, \frac{1}{2}V_0, 2M_0)$, where $T_1 = T_0(V_0, V_0, M_0)$.

For the positive constants $\alpha\leq \alpha_1$ with $\alpha_1$ being small enough such that

where $\epsilon_1$ is chosen in Lemma 2.1, one deduces from Lemmas 2.1–2.9 with $T = T_1$ that the solution $(v, u, {\boldsymbol{w}}, {\boldsymbol{b}}, \theta)$ satisfies

and

where $C_i(i = 2, \cdots, 10)$ is chosen in Section 2, and $C_{11}: = \sum_{i = 2}^{10}C_i$. It follows from Lemmas 2.8 and 2.9 that $(v, u, {\boldsymbol{w}}, {\boldsymbol{b}}, \theta)\in C([0, T_1);H^2)$. If one takes $(v, u, {\boldsymbol{w}}, {\boldsymbol{b}}, \theta)(\cdot, T_1)$ as the initial data and applies Lemma 3.1 again, the local solution $(v, u, {\boldsymbol{w}}, {\boldsymbol{b}}, \theta)$ can be extended to the time interval $[T_1, T_1+T_2]$ with $T_2(C_0, C_1, C_{11})$. Moreover, one obtains

and

Combining with (3.2), (3.3) implies

and

Taking $\alpha \leq \min\{\alpha_1, \alpha_2\}$, where $\alpha_1$ is chosen in (3.1) and $\alpha_2$ is chosen to be such that

where $\epsilon_1$ is chosen in Lemma 2.1. Then one can employ Lemmas 2.1–2.9 with $T = T_1+T_2$ to infer the local solution $(v, u, {\boldsymbol{w}}, {\boldsymbol{b}}, \theta)$ satisfies (3.2) and (3.3).

Thus, choosing

and repeating the above procedure, one can then extend the solution $(v, u, \textbf{w}, {\boldsymbol{b}}, \theta)$ step by step to a global one provided that $0 \leq \alpha \leq \epsilon_0$. Furthermore, one derives the initial boundary value problem (1.1)–(1.9) has a unique global solution $(v, u, {\boldsymbol{w}}, {\boldsymbol{b}}, \theta)$ satisfying (3.2) and (3.3). Moreover, $(v, u, {\boldsymbol{w}}, {\boldsymbol{b}}, \theta) \in X(0, +\infty; C_0, C_1, C_{11})$.

The large-time behavior (1.11) follows from Lemmas 2.3–2.9 by using a standard argument (see Reference ^[21]).

First, similar to (2.6), multiplying (1.1) by $(1-v^{-1})$, (1.2) by $u$, (1.3) by ${\boldsymbol{w}}$, (1.4) by ${\boldsymbol{b}}$, (2.5) by $1-\theta^{-1}$ and adding them altogether, integrating the resultant equality over $(0, 1)$, one has after using (2.2) and (2.85) that

where (and in what follows) $C_i, i = 12, \cdots, 18$ and $C, C^*$ denote some generic positive constants depending only on $\beta$ and $M_0, V_0$.

By means of (2.87), (2.2), (2.85), (2.86), (2.101) and Sobolev's inequality, one obtains

Next, multiplying (2.39) by $(u-\theta^\alpha\frac{v_x}{v})$ and integrating the resultant equality over $(0, 1)$, using (2.2), (2.85), (2.50), (2.75), and Poincare's inequality yields that

By the virtue of (2.50), (2.75), and (2.54), one obtains that

Furthermore, adding (3.5) multiplied by $\frac{2(C_{17}+1)}{C_{15}}$, (2.7) multiplied by $\frac{1}{C_{13}}\left(\frac{2C_{16}(C_{17}+1)}{C_{15}} +1\right)$, and (3.4) multiplied by $C^*$, which satisfy

From (3.6) and choosing $\epsilon$ suitably small, it follows

where

By using Cauchy–Schwarz's inequality, one obtains

which along with Poincare's inequality, yields that

Finally, using Poincare's inequality and (3.7) implies that

where it has been used the conservation of energy implied by (1.5), (2.30), and the following fact:

By means of (3.6) and (3.8), one obtains that

Thus, the proof of Theorem 1.1 has been completed.

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 anonymous referees for their helpful comments and valuable suggestions, which have improved the presentation of the manuscript. This work was supported by the Jiangxi Provincial Natural Science Foundation (Grant Nos. 20232BAB211001, 20242BAB25006).

Conflict of interest

The authors declare there are no conflicts of interest.