Noble-Abel gas diffusion at convex corners of the two-dimensional compressible magnetohydrodynamic system

1.   Introduction

A large number of fluids exist in nature. Neglecting viscosity and heat transfer, some fluids in nature can be viewed as ideal fluids ^[1,2,3,4]. The study of ideal fluid flow around a convex corner can be traced back to 1948, as proposed by Courant and Friedrichs^[5]. They discovered the existence of a solution for this problem. Subsequently, Sheng and You^[6] studied the problem of diffusion of the pseudo-steady supersonic flow of the polytropic gas into a vacuum at a convex corner and obtained the existence of a solution. Sheng and Yao^[7] studied the problem of two-dimensional pseudo-steady isentropic irrotational supersonic flow of the polytropic gas around a convex corner. They obtained the structure of non-completely centered simple wave solutions. In 2023, Chen, Shen, and Yin^[8] studied the problem of supersonic diffusion of a non-ideal gas around a convex corner into the vacuum. They proved the existence of classical solutions in the region of interaction between planar and centered simple waves. Li and Sheng^[9] studied the expansion of the Van der Waals gas into the vacuum around a convex corner. The problem is studied in terms of the interaction of a completely centered simple wave with a backward planar simple wave and an incompletely centered simple wave with a backward planar simple wave. They constructively obtained the global existence of solutions to the gas expansion problem. For other studies on the convex corner winding problem, we refer to ^{[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]}.

Since magnetic fields may affect fluids during flow, it is natural to consider compressible magnetohydrodynamic equations. There have been some results in recent years about the flow of an ideal magnetic fluid around a convex corner. In 2020, Chen, Yin, and You^[26] investigated the two-dimensional pseudo-steady compressible magnetohydrodynamic equations for the polytropic gas expansion into the vacuum at the convex corner. They obtained a classical solution for interacting with centered and planar simple waves.

The two-dimensional isentropic compressible magnetohydrodynamic equations can be written as

where $\vec{q} = (u, v)$ denotes the velocity, $\mu$ is the magnetic permeability, $\rho$ is the density, $k_0$ is the positive constant, $t$ denotes time, and $p$ is the gas pressure.

The Noble-Abel gas is

where $a > 0$ is constant and represents the compressibility limit of a gas molecule, and $\gamma$ represents the adiabatic index.

It is assumed that the supersonic incoming flow in the second quadrant travels along the horizontal solid wall to the origin with a constant velocity $\vec q_ { 0 } = (u_ { 0 }, 0)$, and the rest of the region is a vacuum (see Figure 1). The supersonic flow around a convex corner problem has the following initial value conditions:

And the boundary data

where $\rho_{0}$ denotes the incoming flow density, is a constant, and $\theta$ is the solid wall inclination.

For the sake of the discussion that follows, we propose the following notation,

where $u_ { 0 } > w_{0}$ denotes the incoming horizontal velocity as a constant and $w_{0}$ denotes the incoming magnetoacoustic velocity as a constant. $\alpha _{0}$ is the maximum characteristic inclination of the $C_{+ }$ characteristic curve of the centered simple wave at point $O$, and $\alpha _ { v }$ is the minimum characteristic inclination of the $C_{+ }$ characteristic curve of the centered simple wave at point $O$, which is determined by $\left (3.26 \right)$. The variable $\left (u, v, \rho \right)\left (\xi, \eta \right)$ is controlled by $\left (\xi, \eta \right)$, and in order to solve for the main part of the properties of the centered simple wave, we have changed the variable to a variable controlled by a single parameter $\alpha$, $\left (\hat{u }, \hat{v}, \hat{\rho}\right)\left (\alpha\right)$, and $\hat{w}$, $\hat{u}$, and $\hat{v}$ are determined by (3.25). $b$ as the Alfven velocity, $c$ as the speed of sound, and $w = \sqrt{c^{2}+b^{2}}$ as the speed of magneto-sound. $b_{ 0}$, $c_{0}$, and $w_{0} = \sqrt{c_{0}^{2}+b_{0}^{2}}$ denote the Alfven velocity of the incoming flow at t = 0, the speed of sound, and the speed of magnetoacoustic sound, respectively. All other symbols are for ease of expression and calculation and have no real physical meaning.

For the convenience of the following proof, we assume that there exists a constant $\rho _{1}$, such that the following condition holds for any $\rho < \rho _{1}$ when $0 < a\rho _{1} < 1$

In this paper, we study the Riemannian problem of a supersonic magnetic fluid with Noble-Abel gas diffusing into the vacuum around a convex corner, which is essentially the interaction of a centered simple wave with a planar simple wave and which can be solved by reducing it to a Goursat problem. The solution's hyperbolicity and a priori $C^{1}$ estimates are established using characteristic decompositions and invariant regions. In addition, pentagonal invariant regions are constructed to obtain global solutions. In addition, the generality of this gas, the sub-invariant region, is constructed, and the solved hyperbolicity is obtained based on the continuity of the sub-invariant region. Finally, the global existence of the solution to the gas expansion problem is constructively obtained. The main results of this paper are as follows:

Theorem 1.1. If $u_{0} > w_{0}$, $\theta \le \alpha _ { v }$, $\rho _{0} < \rho _{1}, \max\left (\alpha _{0}-4\bar{\delta }\left (\rho _{0} \right)+ 4\bar{\delta }_{\ast }-\frac{\pi }{2}, \alpha _{0}-\alpha _{v}-2\bar{\delta }\left (\rho _{0} \right) \right) < 0$, then the problem of a magnetoacoustic velocity flow with Noble$\mathit{\mbox{–}}$Abel gas expanding into vacuum at the convex corner (1.1)–(1.4) has a global classical solution.

The structure of the paper is as follows: In Section 2, the characteristic forms and characteristic decompositions related to the characteristic direction, the pseudo-flow direction, and the Mach angle are given. In Section 3, we provide the expressions for the centered simple and planar simple waves. In Section 4, the primary study is the interaction problem of centered simple and planar simple waves. The $C^{0}$, $C^{1}$, and hyperbolicity estimates of the solutions are obtained. Then, we obtain the existence of global classical solutions. In Section 5, we discuss the shortcomings of Theorem 1.1.

2.   Characteristic analysis of compressible magnetohydrodynamic equations

2.1.   The characteristic equations for $\alpha, \beta$ and $w$

By the self-similar transformation $(\xi, \eta) = (x/t, y/t)$, the system of Eq (1.1) can be written as

where $(U, V) = (u-\xi, v-\eta)$ is the pseudo-flow velocity. The initial edge value condition (1.3) and (1.4) are transformed into a boundary conditions in the $(\xi, \eta)$ plane (refer to Figure 2).

and

where $u_{0}$ and $\rho_{0}$ are two different constants for the horizontal velocity and density of the incoming flow, respectively, and $\theta$ is the solid wall inclination.

For smooth solutions, the system of Eq $\left (2.1 \right)$ can be written in the following form:

In the case of the irrotational condition $u_{\eta} = v_{\xi}$, there exist potential functions $\varphi$ such that $\varphi_{\xi} = U$ and $\varphi_{\eta} = V$. By the last two equations of (2.4), the pseudo-Bernoulli's law is obtained as follows:

We define $b = \sqrt{\mu k_{0}^{2}\rho}$ as the Alfven velocity, $c = \sqrt{\frac{A\gamma\rho^{\gamma-1}}{(1-a\rho)^{\gamma+1}}}$ as the speed of sound, and $w = \sqrt{c^{2}+b^{2}}$ as the speed of magneto-sound. Through the irrotational condition $u_{\eta} = v_{\xi}$, the system of Eq (2.4) can be transformed into

The matrix form of system (2.6) is as follows:

The eigenvalues $\Lambda$ of system (2.7) are given by

That is

which yields

(2.10) indicates that when $U^{2} + V^{2} > w^{2}$, the system (2.7) is hyperbolic, with two families of wave characteristics defined by

Taking into account Eq (2.9), it is evident that $w$ is essentially a projection of the pseudo-flow velocity in the normal direction of the $C_{\pm }$ feature line. The eigenvectors on the left side that correspond to the eigenvalues $\Lambda _{\pm }$ are

By left-multiplying (2.12) by (2.7), the system of Eq (2.7) can be transformed into the following characteristic form:

where $\partial _{\pm } = \partial _{\xi } + \Lambda _{\pm } \partial _{\eta }$.

We define the characteristic inclinations $\alpha, \beta$ of $\Lambda _{+ }$ and $\Lambda _{- }$ as follows:

Joining (2.14) and (2.10) yields

That is

The pseudo-streamline inclination is $\sigma = \frac{\alpha + \beta }{2}$ and the Mach angle is $\delta = \frac{\alpha - \beta }{2}$ as shown in Figure 3, we have

By the first equation of (2.17), we have

Thus

where

From the second equation (2.17), we can conclude that

Subsequently

According to (2.5), we have

where

(2.13) and (2.27) lead to

Substituting (2.14), (2.17), (2.26), and (2.28) into (2.29)–(2.32) reaches

where

Associating (2.21), (2.25), (2.33), and (2.34), we obtain

where

2.2.   Characteristic decomposition of the variable $w$

In this section, we compute the characteristic decomposition of $w$ using the commutator relations and the characteristic equations of $\alpha$, $\beta$, $w$, and $u$.

Lemma 2.1. The commutator relations (Li, Zhang, and Zheng ^[27]).

Lemma 2.2.

where

Proof. From Lemma 2.1

Substituting (2.33) into (2.45), we have

Simplifying and combining gives

where

Inserting (2.37) and (2.38) into (2.47) and using the commutator relation (2.41), we obtain

Thus, we have

Using a simple calculation, we have

and

Therefore, (2.42) is established. The proof of (2.43) is similar, so we will not prove it in detail here. □

3.   Centered simple waves and planar simple waves

In this section, we focus on constructing centered and planar simple waves.

3.1.   Centered simple wave

In order to construct the centered simple wave solution for isentropic irrotational pseudo-steady flow at convex corners, we first discuss the nature of the general centered simple wave principal part of the equation.

According to the definition ^[28] of a centered simple wave, its solution $\left (u, v, \rho \right)\left (\xi, \eta \right)$ can be determined by the direct characteristic line $\eta = \xi \tan \alpha$ and the function $\left (u, v, \rho \right)\left (\xi, \eta \right) = \left (\overline{u}, \overline{v}, \overline{\rho} \right)\left (r, \alpha \right)$ defined on the rectangular region $\widetilde{\Lambda } \left (t \right) = \left \{ \left (\xi, \eta \right)\mid 0\le r \le \zeta, \tilde{\alpha } _{2}\le \alpha \le \tilde{\alpha }_{1} \right \}$, see Figure 4. It is easy to see that $\xi = r\cos \alpha$ and $\eta = r\sin \alpha$, there are

From the systems (2.13) and (2.5), it is clear that the main parts $\left (u, v, \rho \right)\left (\xi, \eta \right)$ of the $C_{+ }$ type centered simple wave satisfy

For the potential function $\varphi$, we have

Let $\xi \to 0$, then (3.2) and (3.3) become

where $\left (\overline{u }\left (0, \alpha \right), \overline{v}\left (0, \alpha \right), \overline{\varphi }\left (0, \alpha \right), \overline{\rho}\left (0, \alpha \right)\right) = \left (\hat{u}\left (\alpha \right), \hat{v}\left (\alpha \right), \hat{\varphi }\left (\alpha \right), \hat{\rho }\left (\alpha \right)\right)$.

Lemma 3.1. If the principal part $\left (\overline{u}, \overline{v}, \overline{\rho} \right)\left (r, \alpha \right) = \left (\hat{u}, \hat{v}, \hat{\rho } \right)\left (\alpha \right)$ satisfies

then $\left (u, v, \rho \right)\left (\xi, \eta \right) = \left (\hat{u}, \hat{v}, \hat{\rho } \right)\left (\alpha \right), \eta = \xi \tan \alpha$ is the $C_{+ }$ centered simple wave solution of Eq (2.13) at the origin.

Proof. This proof is divided into the following three steps:

Step 1. First: prove that for any given $\alpha$, the line $\eta = \xi \tan \alpha$ is the $C_{+ }$ characteristic curve. This only requires proving that $\eta = \xi \tan \alpha$ is circular with the speed of sound $\left (\hat{u }\left (\alpha \right) -\xi \right)^{2}+ \left (\hat{v}\left (\alpha \right)- \eta \right)^{2} = \hat{w}\left (\alpha \right)^{2}$ tangent. In fact, substituting $\eta = \xi \tan \alpha$ into the velocity circle expression of sound is obtained

Substituting (3.8) into (3.9) yields the discriminant of the (3.10) root

So, for any given line, $\eta = \xi \tan \alpha$ is a $C_{+ }$ straight eigenline.

Step 2. Prove that $\left (u, v, \rho \right)\left (\xi, \eta \right)$ satisfies Eq (2.13) in the theorem. According to the definition of $\left (u, v, \rho\right)$, it can be known that along $C_{+ }$ straight eigenline is constant, that is, $\partial _{+ }u = \partial _{+ }v = \partial _{+ }\rho = 0$ and

Under condition (3.4), it is obtained that

Step 3. Prove that $\left (u, v, \rho \right)\left (\xi, \eta \right)$ satisfies the proposed Bernoulli law in the theorem. This can be obtained from Eqs (3.4) and (3.5).

and

(3.14) and (3.15) show that Bernoulli's law holds. □

Next, we obtain the expression of the centered simple wave solution. Since $\left (u, v, \rho \right)\left (\xi, \eta \right) = \left (\hat{u}, \hat{v}, \hat{\rho } \right)\left (\alpha \right)$ is the principal part of the centered simple wave for type $C_{+ }$ and $\alpha$ is the characteristic angle of the $C_{+ }$ characteristic line $\eta = \xi \tan \alpha$. We decompose the pseudo-flow velocity $\left (U, V \right) = \left (\hat{u}\left (\alpha \right) -\xi, \hat{v}\left (\alpha \right)-\eta \right)$ along the $\left (\sin \alpha, -\cos \alpha \right)$ and $\left (\cos \alpha, \sin \alpha \right)$ directions, respectively. I obtain the velocity components $w$ and g as follows:

When $\xi \to 0$, we get

It is easy to see that $\hat{g}\left (\alpha \right)^{2}+ \hat{w}\left (\alpha \right)^{2} = \hat{u}\left (\alpha \right)^{2}+\hat{v}\left (\alpha \right)^{2}$, derivation of $\alpha$ in each of the last two equations of (3.17) yields

Substituting Eq (3.18) into Eq (3.4), we have

The derivation of $\alpha$ in Bernoulli's law (3.5) and then the union (3.19) yield

Then the derivation of $\alpha$ in (3.20) yields

Using the initial value condition $\left (\hat{u}, \hat{v}, \hat{\rho }\right)\left (\alpha _{0} \right) = \left (u_{0}, 0, \rho _{0} \right)$ to solve Eq (3.21), we have

in which

and

According to (3.17), it is not hard to obtain

Lemma 3.2. Suppose $\rho _{0} < \rho _{1}$ and $\theta \le \alpha _{v}$, then $R_{C}$ is a complete simple wave that connects the constant state with the vacuum (see Figure 5), where $\alpha _{0}$ and $\alpha _{v}$ satisfy

and $\hat{q }^{2} = \hat{u}^{2}+ \hat{v}^{2}$.

Proof. Deriving and substituting $\hat{u} = \hat{q}\cos \sigma$ and $\hat{v} = \hat{q}\sin \sigma$ into the first equation of (3.4), and also deriving Eq (3.5), we have

Therefore, we have

it is easy to see that $\alpha _{v} = \lim_{\rho \to 0}\sigma$. By deflating $\alpha _{v}$ one obtains

So $\alpha _{v}$ is bounded. □

3.2.   Planar simple wave

By (1.1) and its initial value condition

where $n_{1}^{2} + n_{2}^{2} = 1$.

By coordinate transformations $\hat{x} = n_{1}x+ n_{2}y$, $\hat{y} = -n_{2}x+ n_{1}y$, let $\hat{u} = n_{1}u+ n_{2}v$, $\hat{v} = -n_{2}u+ n_{1}v$. Solving a 1-D Riemann problem, we obtain

where $\hat{\xi } = \frac{\hat{x} }{t}$, $\hat{\xi }_{1} = \lim_{\rho _{r} \to 0}u_{r }\left (\rho _{r} \right)$, $\hat{\xi }_{2} = u_{0}-w\left (\rho _{0} \right)$. The functions $u_{r}\left (\hat{\xi} \right)$ and $\rho _{r}\left (\hat{\xi } \right)$ are implicitly determined by

Therefore, the solutions of (1.1) and (3.30) are obtained as follows:

4.   Interaction of centered and planar rarefaction waves

Suppose that the centered simple wave Rc and the planar simple wave Rp start to interact at point $I\left (u_{0}-w_{0}, w_{0}\sqrt{\frac{u_{0 }-w_{0} }{u_{0}+ w_{0} } } \right)$ to form an interaction region $\Omega$. $\Omega$ is enclosed by the $C_{- }$ penetrating characteristic curve $\widetilde{IH}$ of the centered simple wave Rc, the $C_{+ }$ penetrating characteristic curve $\widetilde{IK}$ of the backward planar simple wave Rp, and the interface $\widetilde{HK}$ between the gas and the vacuum region, where $\widetilde{IH}$ is determined by the centered simple wave Rc and point Ⅰ, while $\widetilde{IK}$ is determined by the planar simple wave Rp and point Ⅰ; see Figure 5.

According to ^[29], a simple calculation yields the following parametric expression for $\widetilde{IK}$ with respect to the density $\rho$

where

From (4.1), when $\rho = 0$, the coordinates of the point K can be obtained as

The solution of the system of Eqs (2.1)–(2.3) outside the interaction region a consists of the constant state $\left (u_{0}, 0, w_{0 }\right)$, the vacuum state, the centered simple wave Rc, and the backward planar simple wave Rp; see Figure 5. Solving the solution of the system of Eq (2.1) in the interaction region $\Omega$ boils down to a Goursat problem, i.e., the system of Eq (2.1) as well as the boundary conditions

where $\left (u_{\pm}, v_{\pm}, \rho _{\pm}\right)(\xi, \eta)$ is determined by the simple wave Rp (Rc).

Lemma 4.1. For any $\rho \in \left (0, \rho_{0} \right)$, $\Omega _{w_{\rho } }$ denotes the region enclosed by the $C_{+ }$ characteristic curve $\widetilde{IK_{1}}$, the $C_{- }$ characteristic curve $\widetilde{IH_{1}}$, and the isomagnetoacoustic velocity line $w\left (\xi, \eta \right) = w_{\rho }$. Here $K_{1}$ and $H_{1}$ are the intersections of the isomagnetoacoustic velocity line $w\left (\xi, \eta \right) = w_{\rho }$ with $\widetilde{IK}$ and $\widetilde{IH}$, respectively. When $\rho$ is sufficiently close to $\rho _{0}$, then there exists a unique $C^{1}$ solution to the Goursat problem (2.1) and (4.4) on $\Omega _{w_{\rho } }$.

Proof. According to the theory of the existence of local classical solutions of hyperbolic equations ^[28], there is a local $C^{1}$ solution to the Goursat problems (2.4) and (4.4). Thus, there is a sufficiently small normal number such that $\varepsilon$ Goursat problems (2.4) and (4.4) have a $C^{1}$ solution on $\Sigma ^{\varepsilon }$, where $\Sigma ^{\varepsilon }$ is the closed region bounded by $\widetilde{IK}$, $\widetilde{IH}$, and the straight line

Let

In the planar simple wave Rp, we have $\beta = -\frac{\pi }{2}$. Combining with Eq (2.36), on the $C_{+ }$ characteristic curve $\widetilde{IK}$, we obtain

Similarly, in the centered simple wave Rc, from Eq (2.37), on the $C_{- }$ characteristic curve $\widetilde{IH}$, we have

Combining with (4.7), (4.8), (2.42), and (2.43), we have $\bar{\partial }_{\pm }w < 0$ in the region $\Sigma ^{\varepsilon }$, evidently, $l < w_{0}$. Therefore, there exists a unique $C^{1}$ solution to the Goursat problem on $\Omega _{w_{\rho } }$. □

Lemma 4.2. On the $C_{+ }$ characteristic curve $\widetilde{IK}$, there is

Meanwhile, on the $C_{- }$ characteristic curve $\widetilde{IH}$, there is

where

Proof. In view of (2.36), we have

Due to

We have

Moreover

Due to $\bar{\partial }_{+ }m\left (\rho \right) > 0$, we have the following two possible cases, see Figure 6.

Case 1. If the functions f and g have no intersection within $\left (0, \rho_{0} \right)$, then we have

Case 2. If the functions f and g intersect at point E in the range $\left (0, \rho_{0} \right)$, then we have $m\left (\rho \left (E \right) \right) = \tan ^{2}\delta \left (E \right)$. Clearly, $\tan ^{2}\delta > m\left (\rho \right)$ holds on IE. Then, we get $\bar{\partial }_{+ }\delta \mid _{\widetilde{IE}} < 0$ and

Furthermore, there are $\tan ^{2}\delta < m\left (\rho \right)$ holds on EK; otherwise, there exists a point $E_{1}$ on EK such that $\bar{\partial }_{+ }\delta \left (E_{1} \right) = 0$, and ${m}'\left (\rho \right) < 0$. According to (4.12), $\bar{\partial }_{+ }\delta \left (E_{1} \right) < 0$ holds, which contradicts the hypothesis. It is not difficult to obtain

Based on $\bar{\delta }_{\ast } > \bar{\delta }\left (\rho _{0} \right)$ and $\bar{\delta }\left (\rho \left (E \right) \right) > \bar{\delta }\left (\rho _{0} \right)$, it is not difficult to obtain (4.9).

Similarly, by (2.39), on the $C_{- }$ cross characteristic curve $\widetilde{IH}$, we obtain

we discuss this in the following two cases:

Case 1. If $\overline{\delta }\left (\rho \right)\le \delta$ as $\alpha _{v}\le \alpha \le \alpha _{0}$. According to Eq (4.19), we have

Case 2. If not, there must exist a point $F$ between $\delta = \overline{\delta }\left (\rho _{0} \right)$ and $\delta = \overline{\delta }_{\ast }$ such that $\delta \left (F \right) = \overline{\delta }\left (\rho \left (F \right)\right)$. Clearly, $\delta > \bar{\delta }$ holds on IF. Then, we obtain $\overline{\partial} _{- }\beta\mid _{IF} > 0$ and

Additionally, there is $\delta < \bar{\delta }$ holds on $\widetilde{FH}$; otherwise, there exists a point $F_{1}$ on $\widetilde{FH}$ such that $\bar{\partial }_{- }\beta \left (F_{1} \right) = 0$, and ${m}'\left (\rho \right) < 0$. According to (4.19), $\bar{\partial }_{- }\beta \left (F_{1} \right) < 0$ holds, which contradicts the hypothesis. It is not difficult to get

Based on $\alpha _{0}-2\bar{\delta }_{\ast } < \alpha _{0}-2\bar{\delta }\left (\rho _{0} \right)$ and $\beta \left (F \right) < \alpha _{0}-2\bar{\delta }\left (\rho _{0} \right)$, it is not difficult to obtain (4.10). Then, this lemma can be obtained. □

Lemma 4.3. If there exists a $C^{1}$ solution to the Goursat problem (2.1) and (4.4) in $\Omega _{\widetilde{\rho } }$, and $\max\left (\alpha _{0}-4\bar{\delta }\left (\rho _{0} \right)+ 4\bar{\delta }_{\ast }-\frac{\pi }{2}, \alpha _{0}-\alpha _{v}-2\bar{\delta }\left (\rho _{0} \right) \right) < 0$, then the characteristic inclination $\left (\alpha, \beta \right)$ within $\Omega _{\widetilde{\rho } }$ satisfies

where

for every $\tilde{\rho }\in \left (0, \rho _{0} \right)$ (Figure 7).

Proof. Assume

where $\varepsilon$ is any positive number close to zero and $\underline{\delta }$ is a sufficiently small constant satisfied by $0 < \underline{\delta } < \frac{\alpha _{v} }{2}-\frac{\alpha _{0} }{2}+ \bar{\delta }\left (\rho _{0} \right)$, such that

It is easy to see that when $\max\left (\alpha _{0}-4\bar{\delta }\left (\rho _{0} \right)+ 4\bar{\delta }_{\ast }-\frac{\pi }{2}, \alpha _{0}-\alpha _{v}-2\bar{\delta }\left (\rho _{0} \right) \right) < 0$, we have

Therefore, we have $0 < 2\underline{\delta } < \alpha-\beta < \pi$, which shows that $\Sigma _{1}^{\varepsilon }$ satisfies the hyperbolicity condition. Choose any point P in $\Omega _{\widetilde{\rho } }$, we have a $C_{+ }$ characteristic curve $\widetilde{PH_{2}}$ and a $C_{- }$ characteristic curve $\widetilde{PK_{2}}$, where $H_{2} \in \widetilde{IH}$ and $K_{2} \in \widetilde{IK}$. $\Omega _{P}$ is a closed region delimited by the corresponding characteristic curves $\widetilde{PK}_{2}$, $\widetilde{PH}_{2}$, $\widetilde{IH}_{2}$ and $\widetilde{IK}_{2}$.

Provide that $\left (\alpha, \beta \right)\left (\xi, \eta \right)\in \Sigma _{1}^{\varepsilon }$ for $\left (\xi, \eta \right)\in \Omega _{P}\setminus \left \{ P \right \}$, then we have

for all $\left (\xi, \eta \right)\in \Omega _{P}$, where

This lemma can be obtained. From Lemma 4.2, we obtain

Let

$\begin{matrix} \gamma _{1} = \left \{ \alpha = \underline{\alpha }-\varepsilon, \underline{\beta}_{P} -\varepsilon < \beta < \overline{\beta}+\varepsilon, \alpha -\beta > 2\underline{\delta } \right \}, \\ \gamma _{2} = \left \{ \underline{\alpha}-\varepsilon < \alpha < \overline{\alpha }_{P} +\varepsilon, \beta = \overline{\beta}+\varepsilon, \alpha -\beta > 2\underline{\delta } \right \}, \\ \gamma _{3} = \left \{ \alpha = \overline{\alpha }_{P} +\varepsilon, \underline{\beta}_{P} -\varepsilon\le \beta < \overline{\beta}+\varepsilon \right \}, \\ \gamma _{4} = \left \{ \underline{\alpha}-\varepsilon < \alpha < \overline{\alpha }_{P} +\varepsilon, \beta = \underline{\beta}_{P} -\varepsilon \right \}, \\ \gamma _{5} = \left \{ \alpha = \overline{\alpha }_{P} +\varepsilon, \beta = \overline{\beta}+\varepsilon \right \}, \\ \gamma _{6} = \left \{ \alpha = \underline{\alpha}-\varepsilon, \beta = \underline{\beta}_{P} -\varepsilon \right \}, \\ \gamma _{7} = \left \{\underline{\alpha}-\varepsilon\le \alpha \le \overline{\alpha }_{P} +\varepsilon, \underline{\beta}_{P} -\varepsilon\le \beta \le \overline{\beta}+\varepsilon, \alpha -\beta = 2\underline{\delta }\right \}. \end{matrix}$

Assuming that (4.28) does not hold, there exists a point Q in the region $\Omega _{P}$ for which $\left (\alpha, \beta \right)\left (Q \right)\in \bigcup_{i = 1}^{6}\gamma _{i}$ and $\left (\alpha, \beta \right)\left (\xi, \eta \right)\in \Sigma _{P}^{\varepsilon }$ hold for all $\left (\xi, \eta \right)\in\Omega _{Q}\setminus \left \{ Q \right \}$, where $\Omega _{Q}$ is the closed region bounded by the characteristic curves $\widetilde{QK}_{3}$, $\widetilde{QH}_{3}$, $\widetilde{IH}_{3}$, and $\widetilde{IK}_{3}$. The $C_{- }$ characteristic curve through point Q intersects $\widetilde{IK}$ at point $K_{3}$, and the $C_{+ }$ characteristic curve through point Q intersects $\widetilde{IH}$ at point $H_{3}$.

If $\left (\alpha, \beta \right)\left (Q \right)\in\gamma _{1}$, on the basis of the assumed condition that $\left (\alpha, \beta \right)\left (\xi, \eta \right)\in \Sigma _{1}^{\varepsilon }$ as $\left (\xi, \eta \right)\in\Omega _{P}\setminus \left \{ P \right \}$, we have Q = P, according to (2.36), we obtain

But based on our assumptions, we have

Thus, we obtain $\bar{\partial }_{+ }\alpha \left (Q \right)\le 0$, and that contradicts my assumptions.

If $\left (\alpha, \beta \right)\left (Q \right) \in\gamma _{2}$, we can export contradictions similarly.

If $\left (\alpha, \beta \right)\left (Q \right) \in\gamma _{3}$, it is not hard to obtain $\bar{\delta }\left (\rho \left (Q \right) \right)\le \bar{\delta }\left (\rho \left (P \right) \right) < \delta \left (\rho \left (Q \right) \right)$. Then, we obtain

But based on our assumptions, we have

Similarly, we obtain $\bar{\partial }_{+ }\alpha \left (Q \right)\ge0$. And that contradicts my assumptions.

If $\left (\alpha, \beta \right)\left (Q \right) \in\gamma _{4}$, we can export contradictions similarly.

If $\left (\alpha, \beta \right)\left (Q \right) \in\gamma _{5}$, we define $\tilde{\alpha }$ on $\widetilde{H_{3}Q }$ as follows:

Let $z = \tan \frac{\tilde{\alpha }- \overline{\alpha }_{P}- \varepsilon }{2}$; (4.35) becomes

A simple calculation based on the method of Lemma 4.2 in ^[6] yields

Combining (2.36) and (4.35), we have

Substituting $\tilde{\alpha }\left (H_{3} \right) = \alpha \left (H_{3} \right)$ into the first equation of (4.38), we obtain $\bar{\partial }_{+ }\left (\tilde{\alpha }- \alpha \right)\left (H_{3 } \right) > 0$. We assert that $\left (\tilde{\alpha }- \alpha \right)\left (\overline{H} \right)\ge 0$, $\overline{H}\in \widetilde{H_{3}Q }$. If not, there must be a point $H_{\ast }\in \widetilde{H_{3}Q }\setminus \left \{ H_{3}, Q \right \}$ such that $\left (\tilde{\alpha }- \alpha \right)\left (H_{\ast } \right) = 0$ and $\left (\tilde{\alpha }- \alpha \right)\left (\xi, \eta \right) > 0$, $\forall \left (\xi, \eta \right)\in \widetilde{H_{3}H_{\ast } }\setminus \left \{ H_{3}, H_{\ast } \right \}$. Thus, $\bar{\partial }_{+ }\left (\tilde{\alpha }- \alpha \right)\left (H_{\ast } \right)\le 0$. On the other hand, according to the hypothesis, there are $\left (\alpha, \beta \right)\left (H_{\ast } \right)\in \Sigma _{P}^{\varepsilon }$ holds, and we obtain $\bar{\partial }_{+ }\left (\tilde{\alpha }- \alpha \right)\left (H_{\ast } \right) > 0$ according to equation (4.38), which contradicts $\bar{\partial }_{+ }\left (\tilde{\alpha }- \alpha \right)\left (H_{\ast } \right)\le 0$. Therefore, we have $\alpha \left (Q \right) \le \tilde{\alpha }\left (Q \right) < 2\bar{\delta }_{P}+ \overline{\alpha }_{P} +\varepsilon$, and that contradicts our assumptions.

If $\left (\alpha, \beta \right)\left (Q \right) \in\gamma _{6}$, we can export contradictions similarly.

If $\left (\alpha, \beta \right)\left (Q \right) \in\gamma _{7}$, we have $\bar{\partial }_{-}\delta \left (Q \right)\le 0$. According to (2.38), (2.39), and (4.26), it is obtained that

and that contradicts our assumptions. Based on the above discussion, when $\varepsilon$ tends to zero, it is easy to get this lemma. □

Lemma 4.4. If there exists a $C^{1}$ solution to the Goursat problem (2.1) and (4.4) in $\Omega _{\widetilde{\rho } }$, then there exists a constant $M_{0} > 0$ independent of $\widetilde{\rho }$ such that

where $\tilde{\rho }\in \left (0, \rho _{0} \right)$.

Proof. According to Lemma 4.3 and Eqs (2.5) and (2.17), we have

in the domain $\Omega _{\widetilde{\rho } }$, then the lemma is easy to proved. □

Lemma 4.5. If there exists a $C^{1}$ solution to the Goursat problem (2.1) and (4.4) in $\Omega _{\widetilde{\rho } }$, then we have

in the domain $\Omega _{\widetilde{\rho } }$, where

Proof. From Eq (4.7) and $\mu ^{2}\left (\rho \right) > 0$, we obtain

Moreover, according to (3.22), (1.12), and (3.1), they lead to

Based on the characteristic decompositions (2.42) and (2.43), we find by calculation that

Then, we obtain

Next, using the converse, we prove the consistent boundedness of $\bar{\partial }_{\pm }w$. Suppose that the conclusion of Lemma 4.5 does not hold, and that there exists a point $P_{0}$ inside $\Omega _{\widetilde{\rho } }$ such that $\bar{\partial }_{- }w\left (P_{0} \right) = -M_{1}$, $\bar{\partial }_{+ }w\left (P_{0} \right)\in \left [ -M_{1}, 0 \right)$ and $\left (\bar{\partial }_{+ }w, \bar{\partial }_{- }w \right)\in \left (-M_{1}, 0 \right)\times \left (- M_{1}, 0 \right)$ in $\Omega _{P_{0} }\setminus \left \{ P_{0} \right \}$, where $\Omega _{P_{0} }$ is the region bounded by the $C_{+ }$ characteristic curve $\widetilde{P_{0}H_{0}}$ over the point $P_{0}$ and the $C_{- }$ characteristic curve $\widetilde{P_{0}K_{0}}$ over the point $P_{0}$ and $\widetilde{IK}$, $\widetilde{IH}$. According to (2.42), we have

However, this would contradict the hypothesis, so $\bar{\partial }_{- }w\left (P_{0} \right)\in \left (-M_{1}, 0 \right)$. Similarly, $\bar{\partial }_{+ }w\left (P_{0} \right)\in \left (-M_{1}, 0 \right)$ can be proved, and applying the continuity method yields $\left (\bar{\partial }_{+ }w, \bar{\partial }_{- }w \right)\in \left (-M_{1}, 0 \right)\times \left (- M_{1}, 0 \right)$ in the domain $\Omega _{\widetilde{\rho } }$. □

Lemma 4.6. If there exists a $C^{1}$ solution to Goursat problem (2.1) and (4.4) in $\Omega _{\widetilde{\rho } }$, then there exists a constant $M_{2} > 0$ independent of $\widetilde{\rho }$ such that

where $\tilde{\rho }\in \left (0, \rho _{0} \right)$.

Proof. The derivation of both sides of $w^{2} = c^{2}+ b^{2}$ in the direction of the $C_{\pm }$ characteristic curve can be obtained

Observe

From Lemma 4.5, (2.33), and (2.34), it is known that there exists a normal number $M_{2}$ independent of $\widetilde{\rho }$. □

Lemma 4.7. There is a global classical solution for the boundary problem (2.1) and (4.4) in the region $\Omega$, where the $\Omega$ region is bounded by the $C_{+}$ characteristic curve $\widetilde{IK}$, the $C_{-}$ characteristic curve $\widetilde{IH}$, and the vacuum boundary $\widetilde{HK}$.

Proof. Suppose that the Goursat problem has a $C^{1}$ solution on the area $\Omega _{\widetilde{\rho } }$, where $\widetilde{\rho }\in \left (0, \rho _{0} \right)$. Similarly to the proof of Theorem 4.12 in ^[6], we have that the isomagnetic sound velocity line $\rho \left (\xi, \eta \right) = \widetilde{\rho }$ is Lipschitz continuous. The isomagnetic sound velocity line $\rho \left (\xi, \eta \right) = \widetilde{\rho }$ is solvable to length. Let $P_{0}, P_{1}, P_{2}, \dots, P_{n}$ be $\rho \left (\xi, \eta \right) = \widetilde{\rho }$ on sequentially different n + 1 points; $P_{0}$ is located in $\widetilde{IH}$, and $P_{n}$ is located in $\widetilde{IK}$. The $C_{+}$ eigencurve through point $P_{i}$ intersects the $C_{-}$ characteristic curve through the point $P_{i-1}$ at point $I_{i}$, where $i = 0, 1, \dots, n-1$. The contour $\rho \left (\xi, \eta \right) = \widetilde{\rho }$ is a non-characteristic curve, for any $i = 0, 1, \dots, n-1$, both $P_{i}\ne I_{i}$ and $P_{i-1}\ne I_{i}$. For any $i = 0, 1, \dots, n-1$, the system of Eq (2.1) is characterized by the characteristic curves $\widetilde{P_{i}I_{i} }$ and $\widetilde{P_{i-1}I_{i} }$. The Goursat problem is in the quadrilateral region $\Omega _{\widetilde{\rho } }$ bounded by $\widetilde{P_{i}I_{i} }$, $\widetilde{P_{i-1}I_{i} }$, $\widetilde{P_{i-1}T_{i} }$, and $\widetilde{P_{i}T_{i} }$. There is a global smooth classical solution on $\Omega _{\widetilde{\rho } }$, where $\widetilde{P_{i}T_{i} }$ is the $C_{-}$ characteristic curve of the point $P_{i}$, and $\widetilde{P_{i-1}T_{i} }$ is the $C_{+}$ characteristic curve of the point $P_{i-1}$.

Let $B = T_{0}$ and $A = T_{n+ 1}$. For each $i = 0, 1, \dots, n$, there is a $0 < \widetilde{\rho }_{i} < \widetilde{\rho }$ such that there is a $C^{1}$ solution on the Goursat problem $\rho \left (\xi, \eta \right) = \widetilde{\rho _{i} }$, $\widetilde{P_{i}T_{i} }$ and $\widetilde{P_{i}T_{i+1} }$ bounded by the characteristic curves $\widetilde{P_{i}T_{i} }$ and $\widetilde{P_{i}T_{i+1} }$. Let $\widetilde{\rho } _{\varepsilon } = \max \left \{ \widetilde{\rho }_{0}, \widetilde{\rho }_{1}, \dots, \widetilde{\rho }_{n}, \rho \left (T_{1} \right), \rho \left (T_{2} \right), \dots, \rho \left (T_{n} \right) \right \}$. We obviously have a relationship $\widetilde{\rho }_{\varepsilon } < \widetilde{\rho }$ established. Then, we obtain the solution to the Goursat problem (2.1) and (4.4) in the region $\Omega _{\widetilde{\rho } }$. By repeating the above process, we can solve the problem at the regional $\Omega$. So we obtain this lemma. □

Thus, we have completed all the discussions and obtained Theorem 1.1. The ideal magnetohydrodynamic system described in Theorem 1.1 can be used to model phenomena in astrophysics, laboratory plasmas, solar physics, etc. It represents an ideal plasma flow interacting with a magnetic field and is described by partial differential equations, including conservation of mass, momentum, total energy, and magnetic field. The conclusion of Theorem 1.1 shows that we only need to control the density $\rho _{0}$ and velocity $u_{0}$ of the incoming flow and the solid$\mbox{–}$wall inclination $\theta$ so that they satisfy the conditions. We can guarantee that the problem of expanding a magnetic fluid with a Noble$\mbox{–}$Abel gas into the vacuum at a convex corner has a global classical solution.

5.   Conclusions and discussions

We study the Riemann problem of a supersonic magnetic fluid with Noble-Abel gas diffusing into the vacuum around a convex corner, which is solved by reducing it to a Goursat problem. The solution's hyperbolicity and a priori $C^{0}$, $C^{1}$ estimates are established using characteristic decompositions and invariant regions. In addition, pentagonal invariant areas are constructed to obtain a global solution. In addition, sub-invariant regions are built, and the hyperbolicity of the solution is obtained based on the continuity of the sub-invariant regions. Finally, the global existence of the solution to the gas expansion problem is constructively received.

This paper deals with this problem under the special assumption of the solid wall angle $\theta$. For more general cases, the problem requires further attempts and discussions, for example, when the incoming flow velocity is not supersonic but sonic or subsonic. The authors will continue to study these issues later. In addition, we study the complete expansion problem in this work. How about incomplete expansion problems? These problems are interesting. Follow-up studies are needed.

Use of AI tools declaration

The author declares she have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The author would like to thank the anonymous referee and editor very much for their valuable comments and suggestions, which greatly help us improve the presentation of this article.

Conflict of interest

The author declares no conflicts of interest in this paper.