Global existence of 3D rotating magnetohydrodynamic equations arising from Earth's fluid core

1.   Introduction

Multiple pieces of evidence indicate that the Earth's magnetic field existed at least 3.45 billion years ago, and it is constantly changing, and the way in which it changes also changes. Especially, the geomagnetic reversals happen every several hundred centuries. Shortly after fulfilling the theory of general relativity, science master A. Einstein attributed the generation and maintenance of geomagnetic field to one of the major unsolved problems in the field of physics.

From a numerical, experimental, physical, or mathematical point of view, much work has been done trying to find the underlying mechanism and rules of operation of the geomagnetic field. Among numerous theories, the self-excitation dynamo theory is universally accepted, which is to say that the outer core of Earth is often thought of as a giant dynamo that generates the Earth's magnetic field due to the motion of the conductive fluid; see ^[1,2,3]. Specifically, the liquid metal circulation that constitutes the Earth's outer core moves under the weak magnetic field to generate electric current, and then the additional magnetic field generated by the electric current will strengthen the original weak magnetic field. Under the action of electromagnetic coupling effect, the magnetic field is continuously enhanced and amplified, and finally the Earth's magnetic field is formed.

On the other hand, Coriolis forces, generated by the rotation of the Earth, cannot be ignored. Physically, Coriolis forces deflect the upwelling fluid along corkscrewlike, or helical, paths, as though it were following the spiraling wire of a loose spring. Mathematically, Coriolis forces give rise to the so-called Poincaré waves, which are dispersive waves, see ^[4]. Based on a comprehensive consideration, one possible model is the following one describing the magnetohydrodynamic phenomena with a reasonable addition of the Coriolis forces:

where $u = (u_1, u_2, u_3)$, $B = (B_1, B_2, B_3)$, and $p$ are the fluid velocity field, the magnetic field, and the fluid pressure, respectively. $\Omega\in\mathbb{R}\backslash\{0\}$ denotes the Coriolis parameter, and $\Omega e_3\times u$ represents the so-called Coriolis force with rotation axis $e_3 = (0, 0, 1)$. We refer readers to Section 10.2 of ^[4] for the derivation of system (1.1).

When $\Omega = 0$, Eq (1.1) becomes the 3D magnetohydrodynamic equations. Duvaut and Lions ^[5] established local well-posedness results of the 3D magnetohydrodynamic equations in $H^s(\mathbb{R}^n)$ with $s\geq n$ and global well-posedness results for the small initial data. Zhai et al. ^[6] established the global well-posedness of the 3D magnetohydrodynamic equations for initial data in critical Besov spaces and relaxed the smallness condition in the third components of the initial velocity field and initial magnetic field. For more relevant studies on the existence of solutions of the magnetohydrodynamic equations, we refer to ^[7,8,9,10]. However, the global well-posedness or global regularity for the 3D magnetohydrodynamic equations is still a challenging open problem.

When $\Omega\neq0$ but $B\equiv0$, Eq (1.1) becomes the 3D rotating Navier-Stokes equations. Chemin et al. ^[4,11] showed that for any given $L^2(\mathbb{R}^2) + H^{\frac{1}{2}}(\mathbb{R}^3)$-initial data, there exists a positive constant $\Omega_0$ such that the 3D rotating Navier-Stokes equations are globally well-posed provided that $|\Omega|\geq\Omega_0$. Iwabuchi and Takada ^[12] proved that the 3D rotating Navier-Stokes equations is globally well-posed for $u_0\in\dot H^s(\mathbb{R}^3)$ with $\frac{1}{2} < s < \frac{3}{4}$ satisfying

Later on, Koh et al. ^[13] and Sun et al. ^[14] relaxed the range of $s$ to $\frac{1}{2} < s < \frac{9}{10}$ and $\frac{1}{2} < s < 1$, respectively. We may refer to ^{[15,16,17,18]} for the global existence results on 3D rotating Navier-Stokes equations with uniformly small initial data. We also refer to ^{[19,20,21,22,23]} and the references therein for the global existence results on other models involving the Coriolis forces.

Recently, Ngo ^[24] studied Eq (1.1) with horizontal diffusion terms only and showed that the large Coriolis parameter implies global solvability for large initial data provided that $B_0$ is a perturbation of $e_3$. Ahn et al. ^[25] proved the existence and uniqueness of global solutions of Eq (1.1) for $u_0\in H^s(\mathbb{R}^3)$ and $B_0\in (L^2\bigcap L^q)(\mathbb{R}^3)$ with $\frac{1}{2} < s < \frac{3}{4}$ and $3 < q < \min\{\frac{6}{3-2s}, \frac{27}{6+2s}\}$, when the Coriolis parameter is sufficiently large. Kim ^[26] proved the global existence and uniqueness of smooth solutions in $H^s(\mathbb{R}^3)$ with $\frac{1}{2} < s < \frac{3}{4}$ under large Coriolis parameter and further obtained the temporal decay estimates for the solutions.

We would like to point out that there is a great difference between the 3D rotational magnetohydrodynamic equations and the 3D magnetohydrodynamic equations or rotating Navier-Stokes equations. More specifically, the 3D magnetohydrodynamic equations (i.e., Eq (1.1) with $\Omega = 0$) are a purely dissipative system, and the 3D rotating Navier-Stokes equations (i.e., Eq (1.1) with $B\equiv0$) are a system of dissipative-dispersive jointly type, whereas in Eq (1.1), the partition of flow field is of dissipative-dispersive jointly type and the partition of magnetic field is of dissipative type. The structural asymmetry of Eq (1.1) makes the problem much harder than that for the other ones.

The main aim of this paper is to investigate the global existence issue of the Cauchy problem of the 3D rotational magnetohydrodynamic equations and further verify the spatial analyticity for the obtained global solutions by adopting the famous Gevrey class approach (see ^[27,28,29]). Specifically, by establishing new balances between the regularizing effects arising from viscosity dissipation and magnetic diffusion with the dispersive effects caused by the rotation of the Earth, we shall show the existence and uniqueness of global mild solutions to Eq (1.1) for large initial data in Besov spaces under a large Coriolis parameter. Furthermore, let $\mathcal{X}$ be a Banach space and let $\Lambda_1$ be the pseudo-differential operator with symbol given by $|\xi|_1: = \sum_{i = 1}^3|\xi_i|$; we will prove that the obtained solutions $(u, B)\in \mathcal{X}$ of problem (1.1) hold $\|e^{\sqrt{t}\Lambda_1}u\|_{\mathcal{X}} < \infty,$ which implies the spatial analyticity of solutions. The main results of this paper are as follows.

Theorem 1.1. Let $p\in(\frac{3}{2}, 2)$, $r\in[1, \infty]$, $\delta\in(2, \infty)$ and $\rho\in(2, 2\delta)$ satisfy

Then there exist two positive constants, $c$ and $C$, such that for $(u_0, B_0)\in \dot B^{-1+\frac{3}{p}+\frac{2}{\delta}}_{p, r}(\mathbb{R}^3) \times\dot B^{-1+\frac{3}{p}}_{p, r}(\mathbb{R}^3)$ satisfying ${\rm{div}} u_0 = 0$, ${\rm{div}} B_0 = 0$ and

Eq (1.1) has a unique global mild solution

with $\theta\in\{0, 1\}$ and $\frac{1}{p}+\frac{1}{p'} = 1$.

The next result involves the case of $p = 2$.

Theorem 1.2. Let $q\in(2, 3)$, $r\in[1, \infty]$, $\delta\in(2, \infty)$ and $\rho\in(2, 2\delta)$ satisfy

Then there exist two positive constants, $c$ and $C$, such that for {$(u_0, B_0)\in \dot B^{\frac{1}{2}+\frac{2}{\delta}}_{2, r}(\mathbb{R}^3) \times\dot B^{-1+\frac{3}{q}}_{q, r}(\mathbb{R}^3)$} satisfying ${\rm{div}} u_0 = 0$, ${\rm{div}} B_0 = 0$, and

Eq (1.1) has a unique global mild solution

with $\theta\in\{0, 1\}$.

Remark 1.3. Theorems 1.1 and 1.2 with $\theta = 0$ indicate that for any given $u_0\in\dot B^{-1+\frac{3}{p}+\frac{2}{\delta}}_{p, r}(\mathbb{R}^3)$ and sufficiently small $B_0\in\dot B^{-1+\frac{3}{q}}_{q, r}(\mathbb{R}^3)$ with specified $p$, $r$, $\delta$, and $q$, Eq (1.1) admits a unique global mild solution if the Coriolis parameter $|\Omega|$ is large enough. Moreover, Theorems 1.1 and 1.2 with $\theta = 1$ imply these solutions possess Gevrey analyticity in the spatial variables. The main results in this paper are seen as a generalization of the global existence results on the 3D rotating Navier-Stokes equations to the 3D rotating magnetohydrodynamic equations.

Throughout the paper, we denote by $c$ and $C$ the constants, which may differ in each line. $C = C(\cdot, \cdots, \cdot)$ denotes the constant, which depends only on the quantities appearing in parentheses.

2.   Preliminaries

Let $\mathscr{S}(\mathbb{R}^3)$ be the set of Schwartz functions, $\mathscr{S}'(\mathbb{R}^3)$ be the set of tempered distributions, and $\{\psi_j\}_{j\in\mathbb{Z}}$ be a dyadic partition of unity satisfying

and

Here $\hat{f}$ represents the Fourier transform of $f$. Define the Littlewood-Paley frequency localized operator $\Delta_j$ in $\mathscr{S}'(\mathbb{R}^3)$ by:

Firstly, we give the definitions and product law of the homogeneous Besov spaces $\dot{B}^{s}_{p, r}(\mathbb{R}^3)$ and Chemin-Lerner spaces $\tilde{L}^\delta(0, \infty; \dot{B}^{s}_{p, r}(\mathbb{R}^3))$. The Chemin-Lerner spaces were first introduced by ^[30].

Definition 2.1. (Besov Space, ^[31]) Let $s\in\mathbb{R}$ and $1\leq p, r\leq \infty$, we define

● For $s < \frac{3}{p}$ $({or}\; s = \frac{3}{p}, {if}\; r = 1)$, we define $\dot{B}^{s}_{p, r}(\mathbb{R}^3): = \big\{u\in \mathscr{S}'(\mathbb{R}^3)/\mathcal{P}[\mathbb{R}^3]\mid\|u\|_{\dot{B}^{s}_{p, r}} < \infty\big\}$, where $\mathcal{P}[\mathbb{R}^3]$ is the set of all polynomials on $\mathbb{R}^3$;

● If $k\in \mathbb{N}$, $\frac{3}{p}+k\leq s < \frac{3}{p}+k+1$ $({or}\; s = \frac{3}{p}+k+1, {if}\; r = 1)$, then $\dot{B}^{s}_{p, r}(\mathbb{R}^3)$ is defined as the subset of $\mathscr{S}'(\mathbb{R}^3)/\mathcal{P}[\mathbb{R}^3]$ such that $\partial^\delta u\in \dot{B}^{s-k}_{p, r}(\mathbb{R}^3)$ with $|\delta| = k$.

Definition 2.2. (Chemin-Lerner Space, ^[30,31]) For $s\in\mathbb{R}$ and $1\leq r, \delta\leq \infty$, we define

We then define $\tilde{L}^\delta(0, \infty; \dot{B}^{s}_{p, r}(\mathbb{R}^3))$ as the set of temperate distributions $u$ on $(0, \infty)\times\mathbb{R}^3$ with $\lim\limits_{j\to-\infty} S_ju = 0$ in $\mathscr{S}'((0, \infty)\times\mathbb{R}^3)$ and $\|u\|_{\tilde{L}^\delta(0, \infty; \dot{B}^{s}_{p, r})} < \infty$.

Lemma 2.3. (Product Law, ^[14,32]) Let $\theta\in\{0, 1\}$, $r\in[1, \infty]$ and $p_0\in(1, \infty)$. Let $(p_1, p_2, \lambda_1, \lambda_2)\in [1, \infty]^4$ satisfy $\frac{1}{p_0}\leq\frac{1}{p_1}+\frac{1}{p_2}$, $p_1\leq \lambda_2$, $p_2\leq \lambda_1$, $\frac{1}{p_0}\leq\frac{1}{p_1}+\frac{1}{\lambda_1}\leq 1$ and $\frac{1}{p_0}\leq\frac{1}{p_2}+\frac{1}{\lambda_2}\leq 1$. If $s_1+s_2+3\inf\{0, 1-\frac{1}{p_1}-\frac{1}{p_2}\} > 0$, $s_1+\frac{3}{\lambda_2} < \frac{3}{p_1}$ and $s_2+\frac{3}{\lambda_1} < \frac{3}{p_2}$, then there is $C > 0$ such that

Remark 2.4. We refer to ^[32] for Lemma 2.3 with the case of $\theta = 0$ and to ^[14] for Lemma 2.3 with the case of $\theta = 1$. Moreover, Lemma 2.3 can be generalized to $\tilde{L}^\delta(0, \infty; \dot{B}^{s}_{p, r}(\mathbb{R}^3))$ with $s, p, r$ behaving just as in Lemma 2.3 and index $\delta$ behaving as the rule of Hölder's inequality, see ^[31].

Define Helmholtz projection $\mathbb{P}: = (\delta_{ij}+\mathcal{R}_i\mathcal{R}_j)_{1\leq i, j\leq3}$, where $\mathcal{R}_j$ is the $j$-th Riesz transform in $\mathbb{R}^3$. By the Duhamel principle, Eq (1.1) can be equivalently written as

where $\{T_{\Omega}(t)\}_{t\geq 0}$ is the so-called Stokes-Coriolis semigroup given specifically by

Here $I$ denotes the identity operator, and $\mathcal{G}_\pm(t)$ represents the dispersive linear operator given explicitly by

and $\mathcal{R}$ is the matrix of the Riesz transforms given explicitly by

We refer to ^[16] for the deduction of $\{T_{\Omega}(t)\}_{t\geq 0}$.

The following temporal decay estimates of $\{e^{t\Delta}\}_{t\geq 0}$ and ${\{\mathcal{G}_\pm(\Omega t)\}_{t\in\mathbb{R}}}$ are the keys to studying the global existence of solutions of Eq (1.1).

Lemma 2.5. ^[33] For $-\infty < s_1\leq s_2 < +\infty$, $1\leq p_1\leq p_2\leq \infty$, and $1\leq r\leq \infty$, there is $C = C(s_1, s_2, p) > 0$ such that

for all $t > 0$ and $j\in\mathbb{Z}$. Moreover, there holds

for all $t > 0$.

Lemma 2.6. ^[13] For $s\in \mathbb{R}$, $1\leq p\leq 2$, and $1\leq r\leq \infty$, there is $C = C(p) > 0$ such that

for all $t\in \mathbb{R}$ and $j\in\mathbb{Z}$ with $\frac{1}{p}+\frac{1}{p'} = 1$. Moreover, there holds

for all $t\in \mathbb{R}$ with $\frac{1}{p'}+\frac{1}{p} = 1$.

The last two lemmas are the keys to studying the Gevrey analyticity of solutions.

Lemma 2.7. ^[28] For $1 < p < \infty$ and $a\geq0$, $E: = e^{\frac{1}{2}a\Delta+\sqrt{a}\Lambda_1}$ is a multiplier that is bounded on $L^{p}$, and the norm of the operator is bounded uniformly in regard to $a$.

Lemma 2.8. ^[28] For $0\leq s\leq t$, $E: = e^{-[\sqrt{t-s}+\sqrt{s}-\sqrt{t}]\Lambda_1}$ is either the identity operator or an $L^{1}$ kernel whose norm is bounded independently to $s$ and $t$.

3.   Linear estimates

Lemma 3.1. Let $r\in[1, \infty]$, $p\in(1, 2)$, and $\delta\in[1, \infty]$ satisfy

Then there is $C = C(p, \delta) > 0$ such that

for $\theta\in\{0, 1\}$ and $\Omega\in\mathbb{R}\backslash\{0\}$, where $\frac{1}{p}+\frac{1}{p'} = 1$.

Proof. From Definition 2.2, we see

Since the matrix $\mathcal{R}$ is bounded on $L^q(\mathbb{R}^3)(1 < q < \infty)$, by the expression of $T_\Omega(t)$, we just need to verify that

In fact, applying Lemmas 2.5–2.7, we have

Moreover, since $0 < \frac{1}{\delta} < \frac{2}{p}-1$, it is obvious that

This completes the proof.

By employing $TT^\star$ argument, we have the following estimate involving the case of $p = 2$.

Lemma 3.2. Let $r\in[1, \infty]$, $q\in(2, \infty)$, and $\delta\in[2, \infty]$ satisfy

Then there is $C = C(q, \delta) > 0$ such that

for $\theta\in\{0, 1\}$ and $\Omega\in\mathbb{R}\backslash\{0\}$.

Proof. From Definition 2.2, we have

Since the matrix $\mathcal{R}$ is bounded on $L^q(\mathbb{R}^3)\; (1 < q < \infty)$, by the expression of $T_\Omega(t)$, we just need to verify that

In fact, from Lemma 2.7, we see

We claim for $q\in(2, \infty)$ and $0 < \frac{1}{\delta} < \frac{1}{2}-\frac{1}{q}$ that there holds

Furthermore, applying the $TT^\star$ argument (see ^[34,35]), we just need to prove that

for $\phi\in C^\infty_0((0, \infty)\times\mathbb{R}^3)$ with $\frac{1}{q'}+\frac{1}{q} = 1$ and $\frac{1}{\delta}+\frac{1}{\delta'} = 1$.

In fact, we define a new operator $\widetilde{\Delta}_j$ given specifically by

It is obvious that $\widetilde{\Delta}_j\Delta_j = \Delta_j$ for all $j\in\mathbb{Z}$.

It follows from the Hölder inequality that

and

Moreover, applying Lemmas 2.5 and 2.6, we have

Substituting Eq (3.5) into Eq (3.4), applying the Hölder inequality and Young inequality yields that

Moreover, because of $0 < \frac{1}{\delta} < \frac{1}{2}-\frac{1}{q}$, it is obvious that

Therefore, we immediately obtain

Substituting Eq (3.6) into Eq (3.3) completes the proof.

Lemma 3.3. Let $r\in[1, \infty]$, $p\in (1, 2)$, and $\delta\in(2, \infty)$. Then there is $C = C(p, \delta) > 0$ such that

for $\theta\in\{0, 1\}$ and $\Omega\in\mathbb{R}\backslash\{0\}$, where $\frac{1}{p'}+\frac{1}{p} = 1$.

Proof. From Definition 2.2, we have

Since operator $\mathbb{P}$ and matrix $\mathcal{R}$ are bounded on $L^q(\mathbb{R}^3)(1 < q < \infty)$, by the expression of $T_\Omega(t)$, we just need to verify that

First, applying Lemmas 2.7 and 2.8, we deduce that

Second, it follows from Lemmas 2.5 and 2.6 and the Bernstein inequality that

Furthermore, due to $1 < \frac{\delta}{2} < \delta < \infty$, $0 < 1-\frac{1}{\delta} < 1$ and $\frac{1}{\delta} = \frac{2}{\delta}-[1-(1-\frac{1}{\delta})]$, by the Hardy-Littlewood-Sobolev inequality, we see that there is $C = C(\delta) > 0$ such that

This completes the proof.

Lemma 3.4. Let $r\in[1, \infty]$, $p\in (1, 2)$, $\delta\in(2, \infty)$, and $\rho\in(2, 2\delta)$. Then there is $C = C(p, \delta) > 0$ such that

for $\theta\in\{0, 1\}$ and $\Omega\in\mathbb{R}\backslash\{0\}$, where $\frac{1}{p}+\frac{1}{p'} = 1$.

Proof. From Definition 2.2, we have

By the expression of $T_\Omega(t)$, we just need to verify that

In fact, through the similar process of Eq (3.7), we have

Furthermore, from the Bernstein inequality and Lemmas 2.5 and 2.6, we have

Moreover, due to $1 < \frac{\rho}{2} < \delta < \infty$, $0 < 1+\frac{1}{\delta}-\frac{2}{\rho} < 1$ and $\frac{1}{\delta} = \frac{2}{\rho}-[1-(1+\frac{1}{\delta}-\frac{2}{\rho})]$, by the Hardy-Littlewood-Sobolev inequality, we see that there is $C = C(\delta) > 0$ such that

This completes the proof.

Lemma 3.5. Let $r\in[1, \infty]$, $p\in(1, \infty)$, and $\rho\in[1, \infty]$. Then there is $C = C(p, \rho) > 0$ such that

for $\theta\in\{0, 1\}$.

Proof. From Definition 2.2, we have

Applying Lemma 2.7 implies that

Moreover, it follows from Lemma 2.4 of ^[31] that there are $c > 0$ and $C > 0$ such that

Substituting Eqs (3.10) and (3.9) into Eq (3.8) completes the proof.

Lemma 3.6. Let $r\in [1, \infty]$, $p\in (1, \infty)$, $\rho\in [1, \infty]$, and $\gamma\in [1, \rho]$. Then there is $C = C(p, \rho, \gamma) > 0$ such that

for $\theta\in\{0, 1\}$.

Proof. From Definition 2.2, we see

Through the similar process of Eq (3.7), we have

Moreover, it follows from Lemma 2.4 of ^[31], Bernstein's inequality, and Young's inequality that

where $\frac{1}{m} = 1+\frac{1}{\rho}-\frac{1}{\gamma}$. Substituting Eqs (3.12) and (3.13) into Eq (3.11) completes the proof.

4.   Proofs of main results

Proof of Theorem 1.1. Because of $0 < \frac{1}{\delta} < \frac{2}{p}-1$, by Lemmas 3.1 and 3.5, we see that there has $C_0 > 0$ and $C_1 > 0$ such that

and

Let

and

Now, we define the mapping $\mathscr{B}$ by

where

and

And we define the solution space $Z$ by

with $\|(u, B)\|_Z: = \|u\|_X+\|B\|_Y$,

Since $\delta\in(2, \infty)$ and $\rho\in(\delta, 2\delta)$, employing Lemmas 3.3, 3.4, and 3.6, we see that there are $C_i\; > \; 0\; (i = 2, 3, 4)$ such that

and

with $\frac{1}{\gamma} = \frac{1}{\delta}+\frac{1}{\rho}$.

Moreover, because of $\frac{3}{2} < p < 2$ and $\frac{1}{\delta} < 2-\frac{3}{p}$, by taking $s_1 = s_2 = -1+\frac{3}{p'}+\frac{2}{\delta}$, $p_0 = p$, $p_1 = p_2 = p'$ and $\lambda_1 = \lambda_2 = \frac{p'}{p'-2}$, from Lemma 2.3 and Remark 2.4, we see that there is $C_5 > 0$ such that

Because of $\rho > 2$ and $p < 2$, by taking $s_1 = s_2 = -1+\frac{3}{p}+\frac{2}{\rho}$, $p_0 = p_2 = p_1 = p$, $\lambda_1 = \lambda_2 = \infty$, from Lemma 2.3 and Remark 2.4, we see that there is $C_6 > 0$ such that

Due to $p < 2$, $\delta > 2$, $\frac{1}{\rho} < 2-\frac{3}{p}$ and $\frac{1}{\gamma}: = \frac{1}{\rho}+\frac{1}{\delta}$, by taking $s_1 = -1+\frac{3}{p'}+\frac{2}{\delta}$, $s_2 = -1+\frac{3}{p}+\frac{2}{\rho}$, $p_0 = p_2 = p$, $p_1 = p'$, $\lambda_1 = \frac{p'}{p'-2}$ and $\lambda_2 = \infty$, from Lemma 2.3 and Remark 2.4, we see that there has $C_7 > 0$ such that

Therefore, combining Eqs (4.1)–(4.8) implies

and

for every $(u, B)\in Z$.

Moreover, it follows from the similar argument that

for every $(u_1, B_1)$ and $(u_2, B_2)$ in $Z$.

Hence, if $(u_0, B_0)\in\dot B^{-1+\frac{3}{p}+\frac{2}{\delta}}_{p, r}(\mathbb{R}^3)\times\dot B^{-1+\frac{3}{p}}_{p, r}(\mathbb{R}^3)$ satisfies

and

Eqs (4.9)–(4.11) imply that

and

for all $(u_1, B_1)$ and $(u_2, B_2)$ in $Z$. Then, applying the contraction mapping principle implies that there is a unique global mild solution $(u, B)\in Z$ to problem (1.1).

Proof of Theorem 1.2. The proof of Theorem 1.2 is identical to that of Theorem 1.1. We omit the proof.

Author contributions

J. Sun handled the review and supervision. W. Wang was responsible for writing the original draft. D. Zhao worked on validating.

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 would like to thank the anonymous reviewers for the careful reading and several valuable comments to revise the paper. This paper is supported by the National Natural Science Foundation of China (Grant No. 12361050), the Outstanding Youth Fund Project of Gansu Province (Grant No. 24JRRA121), the Funds for Innovative Fundamental Research Group Project of Gansu Province (Grant No. 24JRRA778), and the University Teachers Innovation Fund Project of Gansu Province (Grant No. 2023A-002).

Conflict of interest

The authors declare there is no conflict of interest.