On the Rayleigh-Taylor instability for the two coupled fluids

1.   Introduction

In this paper we are devoted to the following Euler and magnetohydrodynamics coupled system in $\Omega$:

and

where $\Omega = \mathbb{R}^{2}\times (-1, 1)\subset \mathbb{R}^3$ is divided into $\Omega_{-}$ and $\Omega_{+}$ by a moving free surface $\Sigma(t)$. As shown in the above systems (1.1) and (1.2), the "upper fluid" is called Euler fluid, which is occupying $\Omega_{+}$, and the "lower fluid", which is occupying $\Omega_{-}$, is magnetohydrodynamics fluid. We use $(u_{\pm}, p_{\pm}, h_{-})$ to describe the fluid velocity, pressure, and magnetic field. The subscript "$\pm$"refers to "upper/lower" fluid. $I$ is the identity matrix, $\rho_{\pm}$ denotes the densities of the respective fluids, $g > 0$ is the gravitational constant, and $e_3 = (0, 0, 1)$.

The conditions on $\Sigma(t)$ are as follows:

where $\nu$ is the normal vector of $\Sigma(t)$.

At the fixed boundary ${x_3 = \pm{1}}$, we impose the conditions:

for any $t\ge 0$, $(x_1, x_2)\in\mathbb{R}^{2}$.

In order to overcome the mathematical difficulties brought about by the evolution of the free interface over time, the Lagrangian coordinates are introduced. Define the following reversible maps:

satisfying $\Sigma_0 = \varphi_{\pm}^0\{x_3 = 0\}$ and $\{x_3 = \pm 1\} = \varphi_{\pm}^{0}\{x_3 = \pm1\}$. $\varphi_{\pm}^{0}$ are continuous across $\{x_3 = 0\}$. Define invertible flow maps $\varphi_{\pm}$ which solve

In this paper, $(t, y)$ with $y = \varphi(t, x)$ and $(t, x)\in\mathbb{R}^{+}\times\Omega$ denote Eulerian coordinates and Lagrangian coordinates, respectively. Since the two-layer fluids may slip each other, the slip map must be introduced. Define $S_{\pm}:\mathbb{R}^2\times\mathbb{R}^{+}\rightarrow \mathbb{R}^2\times\{0\}\subset\mathbb{R}^2\times(-1, 1)$ by

Now, we define the corresponding unknown functions in the Lagrangian coordinate

Denote by $A_{\pm}: = ((D\varphi_{\pm})^{-1})^{T}$, where $D$ is the derivative of the coordinates $x$ and superscript $T$ is the matrix transpose. Then, the evolution equations for $v_{\pm}, b_{-}, q_{\pm}, \varphi_{\pm}$ become

In the above system, we have used the Einstein summation convention. The corresponding conditions on $\Sigma(t)$ are

where

is the unit normal vector to the interface $\Sigma(t) = \varphi_{+}(t, \{x_3 = 0\})$, and $\varphi_{+}^3$ is the third component of $\varphi_{+}$. Finally, we require the impermeability conditions

In the Lagrangian coordinates, the magnetic field $b_{-}$ can be expressed by virtue of $\varphi_{-}$ as in ^[1,2]. Applying $A_{-}^{il}$ to the seventh equation of (1.9), we achieve

Thus, we have $\partial_t(A_{-}^{il} b_{-}^{i}) = 0$, which implies $A_{-}^{il}b_{-}^i = A_{-}^{il, 0}b_{-}^{i, 0}$ and

Now, we check the last equation of (1.9). Applying the geometric identities, we have

where $J = |D\varphi|$. Utilizing $A_{-}^{ik}\partial_k$ to (1.13), one gets

The compatibility conditions for the initial value are imposed as follows:

Combining (1.14), we have

For simplicity, we assume that

By virtue of (1.13) and (1.17), we can use the forcing term by the flow map $\varphi_{-}$ to represent the Lorentz term in the fifth equation of (1.9). Thus, (1.9) becomes a two-fluids Navier-stokes system:

where the magnetic field $\bar{M}$ can be considered as a vector parameter.

The conditions (1.10) can be expressed as

The boundary conditions are the same as (1.12).

We have known that $v_{\pm} = 0$, $\varphi_{\pm} = Id$, $q_{\pm} = const$ are steady -state solutions to the systems (1.18), (1.19), and (1.12). Then, $\nu = e_3$, $A = Id$, $S_{-} = Id_{\{x_3 = 0\}}$. The linearized equation system near the steady-state solution is

The corresponding jump and fixed boundary conditions are

where $[[\cdot]]$ denotes the interfacial jump quantity on the boundary $\{x_3 = 0\}$. Our aim is to study the Rayleigh-Taylor (RT) problem, so we suppose

RT instability is a ubiquitous phenomenon in nature, widely existing in various research fields such as astrophysics, atmospheric and oceanic science, laser fusion, and magnetic confinement fusion ^[3,4,5,6]. Before further discussion, we first review some results with regard to the RT instability problems. The studies on the RT instability can be traced back to the pioneering work due to Rayleigh ^[7] and Taylor ^[8]. From then on, many interesting physical phenomena and numerical simulations come from both physical and numerical experiments. Li and Luo ^[9] studied the effect of a vertical magnetic field on the RT instability of 2d nonideal magnetic fluids by constructing numerical solutions. We refer to ^[10] and references therein for a general research of the physics about RT instability. However, there are only very few analytical results from the mathematical point of view. Recently, Guo and Tice ^[11,12] studied the linear and nonlinear RT instability for Euler and Navier-Stokes fluids by the variational method or the modified variational method. In these papers, they discovered that the viscosity and surface tension have an impact on the RT instability. When considering the magnetic field, the RT instability appears by the Lorentz force. The theoretical discussion about the influence of magnetic fields was proposed by Kruskal and Schwarzchild in ^[13]. They found that the horizontal magnetic field can affect the development of RT instability but cannot suppress the growth of instability. Jiang et al. ^[1,14,15,16] used the similar method as ^[11,12] and employed the new techniques to discuss the RT instability for magnetohydrodynamics (MHD) fluids, as well as revealed the magnetic effect to the instability. In this paper we consider the mechanism for the effect of the magnetic field in the ideal fluid and magnetohydrodynamic coupled through the free interface.

2.   Notations and main results

We first introduce some definitions that are applicable throughout the paper. Define the horizontal Fourier transform for a function $g\in L^2(\Omega)$ as follows:

Due to the Fubini and Parseval theorems, one has that

Define the piecewise Sobolev space $H^{s}(\Omega)$ for any $s\in\mathbb{R}$ as follows:

equipped with the following norm:

and

for $I_{-} = (-1, 0)$ and $I_{+} = (0, 1)$.

Next, we will give the main theorems. The first one is concerned with the linearized systems (1.20)–(1.22).

Theorem 2.1. Give a constant vector $\bar{M} = (M, 0, 0)$, then for any $k$, the linear systems (1.20)–(1.22) are ill-posed in $H^k(\Omega)$. To be precise, for any fixed $k, j\in N$ with $j\geq k$, $T_0 > 0$, and $\alpha > 0$, (1.20)–(1.22) have the solutions $\{(\varphi_n, v_n, q_n)\}_{n = 1}^{\infty}$ which satisfy

but

Remark 2.2. The ill-posedness in the above theorem implies that the solutions to the linear systems (1.20)–(1.22) established in Theorem 3.6 depend disconstinuously on the initial conditions.

With the linear instability in hand, there is the nonlinear instability as follows:

Theorem 2.3. For any $k\geq 4$, the perturbed problem (4.2)–(4.6) does not have the property EE(k).

Remark 2.4. We can extend the conclusions in Theorems 2.1 and 2.3 to the general horizontal magnetic field $\bar{M} = (M_1, M_2, 0)$. In practice, since the $L^2-$ norm of the velocity remains unchanged under the horizontal rotation, one may rotate the coordinates so that $\bar{M} = (M, 0, 0)$ with $M = \sqrt{M_1^2+M_2^2}$.

The paper is arranged as follows. In Section 1, we introduce the Lagrangian coordinates and linearize the nonlinear system. Some notations and main results are given in Section 2. In Section 3 we establish the growing mode solution to the linearized system and prove the uniqueness of the solution and discontinuous dependence on the initial value. In the last section, we investigate the ill-posedness of the nonlinear system.

3.   Ill-posedness of linearized problems (1.20)

When discussing the posedness of linearized Eqs (1.20)–(1.22), studying normal mode solutions is a standard practice. To this end, for some $\lambda > 0$, suppose a normal mode ansatz as follows:

Substituting the above ansatz into the systems (1.20)–(1.22) and eliminating the unknown $\tilde{\varphi}_{\pm}$ by using $(1.20)_1$ and $(1.20)_3$, we arrive at the following system:

At the same time, the jump and boundary conditions become

and

Since the coefficients in (3.2) depend only on the $x_3$ variable, we can adopt the horizontal Fourier transformation to (3.2) to reduce them into ordinary differential equations (ODEs) in terms of $x_3$ with each spatial frequency as parameters. Define

so that

and

Then, we have

where $\mathcal{F}$ means the Fourier transformation and $' = \frac{d}{dx_3}$.

Note that we only consider $\bar{M} = (M, 0, 0)$, and make the Fourier transform for (3.2), then we achieve the following system of ODEs:

subject to the jump conditions

and corresponding fixed boundary conditions

Eliminating $\pi_{\pm}$ from the Eq (3.6), one has

Equations (3.7) and (3.8) become

In what follows, we will devote ourselves to build a solution for (3.9)–(3.11) based on the variational method, which deduces a solution for the system (3.6)–(3.8). Then, we will derive an exponential growth solution of time for the system (1.20)–(1.22).

Multiply $\theta_{+}$, $\theta_{-}$ to $(3.9)_1$ and $(3.9)_2$, add the resulting equations, and integrate over $(0, 1)$ and $(-1, 0)$, respectively. After integration by parts, we get

where we used boundary and jump conditions. We would like to find a growing mode solution to the system (3.2), which requires that there exists $\lambda > 0$. One can utilize the variational method to look for the smallest value $\mu$ as follows:

Define

and

It is convenient to introduce the set $\mathcal{A}$

For any $|\xi| > 0$, let

which is equivalent to

We want to find the minimizer of $E$ on the set $\mathcal{A}$ and show the existence and negativity of the infimum.

Proposition 3.1. $E$ can obtain the infimum on $\mathcal{A}$ for any fixed $|\xi|\ge 0$. If $\theta$ is a minimizer and $-\lambda^2: = E(\theta)$, then $(\theta, \lambda^2)$ solves (3.9) with (3.10) and (3.11). Moreover, $\theta$ is smooth when limited to $(-1, 0)$ or $(0, 1)$.

Proof. For any $\theta\in\mathcal{A}$, we estimate $E(\theta)$ as follows:

Therefore, $E$ has a lower bound on $\mathcal{A}$. Take $\theta_n\in\mathcal{A}$ as a minimizing sequence, then we get the boundedness of $\theta_n$ in $H_0^1(-1, 1)$, which implies that there exists $\theta\in H_0^1(-1, 1)$ to guarantee that $\theta_n$ is weakly convergent to $\theta$ in $H_0^1(-1, 1)$ and strongly convergent in $L^2(-1, 1)$. Thus, we have

Thus, $E$ takes the infimum over $\mathcal{A}$ and $\theta$ is a minimizer.

For $s\in \mathbb{R}$ and any $\theta_0\in H_0^1(-1, 1)$, define $\theta(s) = \theta+s\theta_0$, then

follows from (3.16). Let $L(s) = E(\theta(s))+\lambda^2 J(\theta(s))$, then there is $L(s)\geq 0$ for any $s\in \mathbb{R}$ and $L(0) = 0$. This leads to $L'(0) = 0$. By virtue of (3.14) and (3.15), we derive

By selecting $\theta_0$ with compact support in either $(-1, 0)$ or $(0, 1)$, one can get that $\theta$ solves Eq (3.9) in a weak sense. By standard bootstrap arguments, we may demonstrate that $\theta_{-}\in H^k(-1, 0)$(resp., $\theta_{-}\in H^k(0, 1)$) for all $k\geq 0$ and, hence, it is smooth when limited to the respective interval. This means that $\theta_{\pm}$ are classical solutions to the Eq (3.9). The remainder is to show that (3.10) is established. For each $\theta_0\in C_c^{\infty}(-1, 1)$, we obtain

Since $\theta_0(0)$ can be chosen arbitrary, we yield the second jump condition in (3.10). The conditions $[[\theta]] = 0$ and $\theta_{-}(-1) = \theta_{+}(1) = 0$ are satisfied trivially since $\theta\in H_0^1(-1, 1)\hookrightarrow C_0^{0, \frac{1}{2}}(-1, 1)$. □

Remark 3.2. (3.17) implies $-\lambda^2 = \mathop{\inf}\limits_{\theta\in\mathcal{A}} E(\theta)\geq -\frac{g[\rho]}{\rho_{-}}|\xi|$ and, hence,

Corollary 3.3. For any $|\xi| > 0$, system (3.6) has a solution $(\kappa_{\pm}, \psi_{\pm}, \theta_{\pm}, \pi_{\pm})$ with $\lambda = \lambda(|\xi|) > 0$. Moreover, this solution satisfies (3.7) and (3.8) and is smooth when limited to $(-1, 0)$ or $(0, 1)$.

Proof. By solving (3.6), we get

From Proposition 3.1, it is obvious that $\pi_{\pm} = \pi_{\pm}(\xi, x_3), \theta_{\pm} = \theta_{\pm}(\xi, x_3)$, and $\psi_{\pm} = \psi_{\pm}(\xi, x_3)$ are smooth over the interval $(-1, 0)$ or $(0, 1)$. Furthermore, the jump and boundary conditions (3.7) and (3.8) are satisfied. □

Lemma 3.4. Let $R_1, \xi_1$ satisfy

then the eigenvalue $\lambda = \lambda(|\xi|)$ satisfies

Proof. Denote $\bar{\theta}$ by

then

so

Since $-\lambda^2 = \inf\limits_{\theta\in H_0^1(-1, 1)} \frac{E(\theta)}{J(\theta)}$, the result follows. □

Define

Obviously, $\mathbb{D}$ is a symmetrical domain.

Lemma 3.5. Let $\xi\in \mathbb{D}$, $\kappa_{\pm}, \psi_{\pm}, \theta_{\pm}$, and $\pi_{\pm}$ be the solutions to (3.6) constructed in Corollary 3.3, then for each $k\geq 0$, the following inequalities are valid:

where

Moreover,

where $A_k, B_k, D > 0$ are constants depending on $\rho, M, R_1$, and $g$.

Proof. $\theta(\xi)\in\mathcal{A}$ implies that there are constants $A_0, A_1 > 0$ so that

By (3.9), we have

Thus,

where we used $\theta\in \mathcal{A}$. Combining (3.31) and (3.32), we arrive at

which verifies (3.28). Employing (3.23) with $|\xi|\geq R_1$, we get

for any $k\geq 0$. By virtue of the expression of $\pi$ on (3.23), (3.22), and (3.25), with $|\xi|\geq R_1$, one has

Combining (3.33) and (3.34), one can achieve (3.29).

Equation (3.30) follows from that for any fixed $|\xi| > 0$, $\theta(|\xi|)\in\mathcal{A}$, and (3.23). □

In Corollary 3.3, we have achieved the solution to (1.20) for the fixed spatial frequency $\xi\in\mathbb{R}^2$. In rest of this section, we will establish the solution to (1.20) by using Fourier synthesis.

Theorem 3.6. Let $1\leq R_1\leq R_2 < R_3 < \infty$ with $R_1$ satisfy (3.24). Suppose a real-valued and radial symmetric function $f\in C_0^{\infty}(\mathbb{R}^2)$ and $B(0, R_2) \subset supp(f)\subset B(0, R_3)$. For $\xi\in \mathbb{R}^2$, define

where $\kappa, \psi, \theta, \pi$ are the solutions constructed in Proposition 3.1 and Corollary 3.3 with $\lambda(\xi) > 0$.

Denote

where $x'\cdot \xi = x_1\xi_1+x_2\xi_2$, then $(\varphi, v, q)$ is a real-valued solution to the linearized problem (1.20) with the corresponding conditions. For any $k\in \mathbb{N}$, the following inequality is valid:

in which the positive constant $\tilde{C_k}$ depends on $\rho, |M|, R_1$, and $g$. Moreover, $\varphi(t), v(t), q(t)\in H^k(\Omega_{\pm})$ for every $t > 0$ satisfies the following estimates:

where $\bar{c_1} = \frac{g[\rho]}{\rho_{-}}, \bar{c_2} = \frac{g[\rho]}{4(\rho_{+}+\rho_{-})}$.

Proof. Fix $\xi\in\mathbb{R}$, and

are solutions to (1.20). Due to $B(0, R_2)\subset supp(f)\subset B(0, R_3)$, the following inequalities follow from Lemma 3.5:

and

for every $k\in \mathbb{N}$.

Meanwhile, $\lambda(\xi)\leq \sqrt{\frac{g[\rho]}{\rho_{-}}|\xi|}$. This boundedness indicates that the functions given by (3.36)–(3.38) are also a solution to (1.20).

For any $k\geq 0$, by applying Lemma 3.5, and where $f$ is compactly supported, we easily achieve the estimate (3.39). According to (3.22) and (3.25), one has

which derives the bounds (3.40). □

Now, we will study the ill-posedness for the linearized problem. Suppose that $(\varphi, v, q)$ is the solution to (1.20). Further, assume that the solution is band-limited at radius $R > 0$, that is,

Since in the lower fluid the equation in (1.20) has Lorenz force, it is appropriate to use the second-derivative of velocity. Differentiating the second equation and the fifth Eq (1.20) in time and removing $\varphi_{-}$ by using the fourth equation, one arrives at

where we used $\bar M = (M, 0, 0)$. We impose the conditions:

$\partial_t {v}(0)$ satisfies

The first result is about the estimate of energy in terms of $v$ for the evolution Eq (3.41).

Lemma 3.7. For solutions to (3.41)–(3.43), we have

Proof. Multiply $(3.41)_1$ and $(3.41)_2$ by $\partial_t v_{\pm}(t)$ and integrate over $\Omega_{\pm}$, respectively. After integration by parts and employing $(3.41)_3$, we arrive at

Adding (3.46) and (3.47), using $(3.42)_2$, we yield (3.7). □

Lemma 3.8. If $v$ satisfies that $v\in H^1(\Omega)$ is band-limited at radius $R > 0$, $div v = 0$, and $v^3(t, x_1, x_2, \pm 1) = 0$, then we arrive at

Proof. Utilizing the horizontal Fourier transform to $div v = 0$ and denoting

we have

From (2.2), (3.49), and (3.50), one has

which gives (3.48). □

We may now derive growth estimates for $v(t)$ and $\partial_t v(t)$.

Proposition 3.9. If $v$ is a solution to (3.41) and is also band-limited at radius $R > 0$, the following estimate holds:

where $c$ depends on $\rho, M, g, R$.

Proof. Integrate (3.7) with regard to time from $0$ to $t$ to achieve

Thus, we have

where

We apply (3.48) to (3.52) to get the inequality

which implies

By virtue of the Cauchy-Schwartz inequality, one can show that

where $\langle \cdot, \cdot\rangle$ denotes the $L^2$ inner product. Applying the Gronwall inequality to (3.55), we derive

Combining (3.55) and (3.56), we obtain

where $c$ depends $\rho_{\pm}, g, R, M$. □

We have shown the existence of the solutions to the Eq (1.20). To investigate the ill-posedness, we turn to verify the uniqueness and discontinuous dependence on the initial conditions of the solutions. To do this, we first build a projection operator related to the horizontal spatial frequency. Let $\Phi$ be a function that is infinitely differentiable and a compact support in $\mathbb{R}^2$, satisfies $\Phi\in [0, 1], supp(\Phi)\subset B(0, 1)$, and $\Phi(x) = 1$ for $x\in B(0, \frac{1}{2})$, then define $\Phi_{R}(x) = \Phi(\frac{x}{R})$ for $R > 0$. For $f\in L^2(\Omega)$, the projection operator $P_{R}$ is defined by

It is easy to show that $P_R$ verifies the following properties ^[11]:

(1) $P_R f$ is band-limited at radius $R$;

(2) $P_R$ is a bounded linear operator on $H^k(\Omega)$ for all $k\geq 0$;

(3)$P_R$ commutes with partial differential and multiplication by functions depending only on $x_3$;

(4) $P_R f = 0$ for all $R > 0$ if, and only if, $f = 0$.

Theorem 3.10. Solutions to (1.20) are unique.

Proof. We only need to prove that when the initial data is zero, the solutions to (1.20) are also zero. Assume that $\eta_{\pm}, v_{\pm}, q_{\pm}$ solve (1.20) with zero initial conditions. For any fixed $R > 0$, define $\eta_R = P_R\eta, v_R = P_R v, q_R = P_R q$, then $\eta_R, v_R$, and $q_R$ also solve (1.20). Moreover, $v_R$ also solves (3.41) with zero initial value. By virtue of (3.51), for any $t\geq 0$, we derive

Thus, there is $\varphi_R(t) = q_R(t) = v_R(t) = 0$ for all $t\geq 0$. Due to the arbitrariness of $R$, we have that $\varphi(t) = v(t) = q(t) = 0$ for all $t\geq 0$. □

Lastly, we will show that the solution to the problem (1.20) is discontinuously dependent on the initial data.

Now, let us complete the proof of Theorem 2.1.

Proof. Fix $j\geq k\geq 0, \alpha > 0, T_0 > 0$. Let positive constants $\tilde{C}_k, R_1, D$ come from Lemma 3.5 and Theorem 3.6. For every $n\in \mathbb{N}$, take $R(n)$ large enough such that $R(n) > R_1, \sqrt{\frac{g[\rho]}{4(\rho_{+}+\rho_{-})}R(n)} > 1$, and

Choose a family of real-valued, radial, and compact supported functions $f_n$ as $f$ in (3.36)–(3.38) so that $B(0, R(n))\subset supp(f_n)\subset B(0, R(n)+1)$ and

Take $R_2 = R(n)$ and $R_3 = R(n)+1$ in Theorem 3.6 to get $\varphi_n(t), v_n(t), q_n(t)\in H^j(\Omega)$ that solves (1.20) for all $t\geq 0$. By virtue of (3.39) and (3.60), we have that (2.4) holds for all $n$. Due to the definition (2.2), there is

where we used (3.25), (3.40), and (3.30).

Since $\lambda(|\xi|)\geq \sqrt{\frac{g[\rho]}{4(\rho_{+}+\rho{-})}R(n)}\geq 1$ on the support of $f_n$, we also arrive at

We finish the proof of Theorem 2.1. □

4.   Ill-posedness for the nonlinear problem

We focus on showing the ill-posedness of the nonlinear system. Since $A = Id, S_{-} = Id_{\{x_3 = 0\}}$, $v_{\pm} = 0, \varphi_{\pm} = Id, q_{\pm} = const$ are the steady-state solutions to (1.18), one can rewrite (1.18) using the perturbation equations near the steady-state solutions. Let

where $G_{\pm}^T = \sum_{n = 1}^{\infty}(-1)^{n-1}(D\tilde{\varphi}_{\pm})^n$.

Substituting (4.1) into (1.18) with $\bar{M} = (M, 0, 0)$, we yield the following equations about $\tilde{\varphi}_{\pm}, v_{\pm}, \sigma_{\pm}$

where $tr(\cdot)$ is the matrix trace. The following compatibility conditions are required:

We impose the corresponding jump conditions as follows:

where

We collect the equation, jump, and boundary Eqs (4.2)–(4.6) as "the perturbed problem". For $k\geq 0$, we use the following abbreviation:

Before proving it, we give an importance definition.

Definition 4.1. (Property $EE(k)$) For any $\delta, t_0, C > 0$, and the initial data $\tilde{\varphi}_0, v_0, \sigma_0$ meeting

there exists $(\tilde{\varphi}, v, \sigma)\in L^{\infty}((0, t_0);H^3(\Omega))$, which solves the perturbed problems (4.2)–(4.6) on $\Omega\times (0, t_0)$ and satisfies:

(1) $\varphi(t) = Id+\tilde{\varphi}(t)$ is reversible and $\varphi^{-1}(t) = Id-\zeta(t)$ for $0\leq t < t_0$, and

(2)

where $Q:[0, \delta)\to \mathbb{R}^{+}$ and $Q(y)\leq Cy$ for $z\in [0, \delta)$. We say the perturbed problems (4.2)–(4.6) has property $EE(k)$.

Next, we will use the proof by contradiction to prove Theorem 2.3.

Proof. For some $k\geq 4$, we assume that the problems (4.2)–(4.6) has the property $EE(k)$ of the above definition. For $n\in\mathbb{N}$, let $T = \frac{t_0}{2}, k\geq 4$, and $\alpha = 1$ in Theorem 2.1. Then, $\bar{\varphi}, \bar{v}, \bar{\sigma}$ solves (1.20) with $\bar{M} = (M, 0, 0)$ and the initial data satisfying

but

For any $\varepsilon > 0$, denote

then we have

Select $n$ such that $n > C, \frac{\varepsilon}{n} < \delta$, where $C, \delta$ are the constants in the above property $EE(k)$.

Due to $EE(k)$, the perturbed problem exists a solution $(\tilde{\varphi}^{\varepsilon}, v^{\varepsilon}, \sigma^{\varepsilon})\in L^{\infty}((0, t_0);H^4(\Omega))$ with $(\bar{\varphi}_0^{\varepsilon}, \bar{v}_0^{\varepsilon}, \bar{\sigma}_0^{\varepsilon})$ as the initial data. In addition,

Defining

and inputting them into (4.11), we derive

and

We next demonstrate that

where $(\bar{\varphi}, \bar{v}, \bar{\sigma})$ solves the linearized system (1.20) with $\bar{M} = (M, 0, 0)$. Substitute (4.12) into (4.2), then we have

where

then $\bar{G}^{\varepsilon}_{\pm}$ is well-defined. Thus,

where the positive constant $K_1$ is the optimal constant in the inequality $\|FH\|_{H^2}\leq K_1\|F\|_{H^2}\|H\|_{H^2}$. Take $\varepsilon$ small enough so that $\varepsilon < \frac{1}{2K_1}$, then $\bar{G}_{\pm}^{\varepsilon}$ is uniform boundness in $L^{\infty}(0, t_0;H^2(\Omega))$.

Now we will show some convergence results. From $(4.15)_1$, one gets

Expanding $(4.15)_2$, we have

whence

and

By virtue of $(4.15)_{3}$, we achieve

which implies

Thus, we have

$(4.15)_{4}$ implies

The convergence results about the jump conditions are as follows. Due to the invertibility of $Id+\varepsilon\bar{\varphi}^{\varepsilon}$, denote $\bar{\zeta}^{\varepsilon}$ by

which means

then $S_{-}^{\varepsilon}:\mathbb{R}^2 \times \mathbb{R}^{+}\rightarrow \mathbb{R}^2\times\{0\}$ can be expressed by

Hence,

where the positive constant $K_2$ is the Sobolev embedding constant in the trace mapping $H^4(\Omega)\hookrightarrow L^{\infty}(\mathbb{R}^2\times\{0\})$. Define $\bar{S}_{-}^{\varepsilon} = \frac{S_{-}^{\varepsilon}-Id_{\mathbb{R}^2}}{\varepsilon}$, then $\bar{S}_{-}^{\varepsilon}$ is uniform boundness in $L^{\infty}((0, t_0);L^{\infty}(\mathbb{R}^2\times\{0\}))$ by (4.27). Denote the normal at the interface by $\nu^{\varepsilon} = \frac{N^{\varepsilon}}{|N^{\varepsilon}|}$ with

As $\varepsilon\to 0$, one gets $|N^{\varepsilon}| > 0$. The jump condition (4.3) can be rewritten as follows:

It is obvious that $\sup\limits_{0\leq t < t_0}\|\bar{N}^{\varepsilon}(t)\|_{L^{\infty}}$ is uniformly bounded since

Therefore,

For the jump condition (4.4), we can rewrite it as

where

Obviously, $\sup\limits_{0\leq t < t_0}\|\bar{Q}^{\varepsilon}(t)\|_{L^{\infty}}$ is uniformly bounded. Thus, we achieve

Collecting (4.13), (4.18), (4.21), and (4.24), there exists $(\bar{\varphi}^0, \bar{v}^0, \bar{\sigma}^0, \partial_t \bar{\sigma}^0, \partial_t\bar{\varphi}_0) \in L^{\infty}(0, t_0;H^4(\Omega))$ so that

and

By virtue of the lower semi-continuity, we derive

Combining (4.13), (4.18), (4.21), and (4.24), one has

By virtue of a conclusion in ^[17], $(\bar{\varphi}^{\varepsilon}, \bar{v}^{\varepsilon}, \bar{\sigma}^{\varepsilon})$ is strongly pre-compact in $L^{\infty}(0, t_0; H^{\frac{11}{4}}(\Omega))$. So, we have

Following from the above strong convergence, the convergence results (4.20) and (4.23), and the equation $\partial_t\bar{\varphi}^{\varepsilon} = \bar{v}^{\varepsilon}$, we arrive at

and

Taking the limit for (3.44), there is

Combining (4.30) and (4.31), we infer

and

(4.35)–(4.38) imply that $(\bar{\varphi}^0, \bar{v}^0, \bar{\sigma}^0)$ solves (1.20)–(1.23) with $\bar{M} = (M, 0, 0)$ and meets the initial conditions (4.24). Thereby,

follows from Theorem 3.10. Furthermore, collect the inequality (4.33) and (4.10) to yield

which is a contradiction. Therefore, for any $k\geq 4$, the property $EE(k)$ is not valid for the perturbed problem. Theorem 2.3 has been proven. □

5.   Conclusions

We investigated the RT instability problem of the two-phase flow coupled with ideal fluid and magnetohydrodynamic. We obtained the RT instability of linear problems by establishing a growth mode solution to the linearization problem near the steady-state solution. By virtue of the instability of linearization problems, we ultimately obtained the RT instability of nonlinear problems.

Acknowledgments

The author would like to thank the editor and the referees for their careful reading and helpful comments. This work was partially supported by National Natural Science Foundation of China (Grant No. 11901249).

Conflict of interest

The author declares that there are no conflicts of interest regarding the publication of this paper.