On effects of elasticity and magnetic fields in the linear Rayleigh–Taylor instability of stratified fluids

It is well known that there exists a threshold \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\kappa_{{c}}$\end{document}κc such that the linearized stratified viscoelastic Rayleigh–Taylor problem is unstable for the elasticity coefficient κ satisfying \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\kappa <\kappa_{{c}}$\end{document}κ<κc. In this paper, we further prove that if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\kappa <\kappa_{{c}}$\end{document}κ<κc, then there exists an unstable solution to the linearized stratified viscoelastic Rayleigh–Taylor problem with a largest growth rate. Moreover, the largest growth rate decreases from a positive constant to 0 as κ increases from 0 to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\kappa_{{c}}$\end{document}κc. In addition, we further extend the obtained results in the linearized stratified viscoelastic Rayleigh–Taylor problem to the linearized stratified magnetic Rayleigh–Taylor problem.


Introduction
Consider two completely plane-parallel layers of immiscible fluids, the heavier on top of the lighter one, and both subject to the Earth's gravity. In this case, the equilibrium state is unstable to sustain small disturbances, and this unstable disturbance grows and leads to a release of potential energy, as the heavier fluid moves down under the gravitational force, and the lighter one is displaced upward. This unstable phenomenon was first studied by Rayleigh [21] and then by Taylor [22] and is called therefore the Rayleigh-Taylor (RT) instability. In the last decades, this phenomenon has been extensively investigated from both physical and numerical aspects; see [1,4,7,9,23] for examples. It has been also widely investigated how the RT instability evolves under the effects of other physical factors, such as elasticity [6,17,18], rotation [1,3], internal surface tension [5,25], magnetic fields [1,2,8,10,12,15,16], and so on.
Recently, Jiang et al. [17] established the nonlinear stability and linear instability of the stratified viscoelastic RT (VRT) problem, which models the motion of stratified immiscible viscoelastic fluids, the heavier on top of the lighter one, in the presence of a uniform gravitational field. Their results show that the elasticity can inhibit the development of RT instability. In this paper, we further investigate the effect of elasticity on the Rayleigh-Taylor instability based on the linear instability result of Jiang et al. Before stating our main result, we briefly introduce the stratified VRT problem and the results of Jiang et al.
Equations (1.1) 1 -(1.1) 2 describe the motion of the upper heavier and lower lighter viscoelastic fluids driven by the gravitational field along the negative x 3 -direction, which occupy the two time-dependent disjoint open subsets + (t) and -(t) at time t, respectively. Equation (1.1) 3 means that the fluids are incompressible. The two fluids interact with each other by the interfacial jump conditions (1.1) 5 and the motion of a free interface (1.1) 4 , in which d := d(x 1 , x 2 , t) denotes the displacement function of the point at the interface deviating from the plane {x 3 = 0}. The nonslip boundary condition of the velocities on both upper and lower fixed flat boundaries are described by (1.1) 6 , and (1.1) 7 -(1.1) 8 represent the initial status of the two fluids.
The notation f + and fin (1.1) denote the values of the quantity f in the upper and lower fluids, respectively. In particular, the unknown functions v ± := v ± (x, t) ∈ R 3 , U ± (x, t), and p ± (x, t) represent the velocity, the deformation tensor (a 3 × 3 matrix-valued function), and the hydrodynamic pressure of the two fluids, and ρ ± , μ ± , and κ ± denote the constant densities, viscosity coefficients, and elasticity coefficients of the two fluids, respectively. The positive constant g, the superscript T, and the capital letter I represent the gravitational constant, the transposition, and the 3 × 3 identity matrix. The notations f 0 or f 0 denote the initial data of f . In this paper, we consider that the domain occupied by the two fluids is horizontal periodic, and thus we denote -= (x h , x 3 ) ∈ R 3 |x h := (x 1 , x 2 ) ∈ T, -l < x 3 < m with l, m > 0, (1.2) where T i := 2πL i (R/Z), T := T 1 × T 2 , and L 1 and L 2 are positive constants. Moreover, we have the following expressions: Finally, we explain the interfacial jump conditions (1.1) 5 . The notation t·u stands for where f ± | (t) are the traces of the quantities f ± on (t). For two viscous fluids meeting at a free boundary, from the physical point of view, the velocity is continuous across the interface, and the jump in the normal stress is proportional to the mean curvature of the surface multiplied by the normal to the surface (see [14,25]). Thus, we impose the jump conditions tvu = 0 on (t) and and ν is the unit normal vector on (t), ϑ is the surface tension coefficient, and C is the twice the mean curvature of the internal surface (t). Since in this paper we focus on the elasticity effect upon the RT instability, we omit the surface tension and obtain therefore the second jump condition in (1.1) 5 . In addition, since the density of the upper fluid is heavier than the lower one, we have tρ ± u > 0.
Problem (1.1) enjoys a stratified equilibrium state solution (i.e., stratified VRT equilibrium state): (v, U, d, p g ) = (0, I,d,p g ), whered ∈ (-l, m). We should point out thatp g can be uniquely computed out by hydrostatics, which depends on the variable x 3 and ρ ± and is continuous with respect to x 3 ∈ (-l, m). Without loss of generality, we assume that d = 0. Ifd is not zero, we can adjust the x 3 coordinate to maked = 0. Thus d is regarded as the displacement away from the plane To simplify the representation of problem (1.1), we introduce the indicator function χ and denote Now, we denote the perturbation quantity to the equilibrium state (0, Then we have the stratified VRT problem in a perturbation form: , and we omit the subscript ± in the jump notation t·u for simplicity. The equilibrium-state solution of (1 The movement of the free interface (t) and the subsequent change of the domains ± (t) in Eulerian coordinates result in severe mathematical difficulties. To circumvent such difficulties, we switch our analysis to Lagrangian coordinates, so that the interface and the domains stay fixed in time. To this end, we define the fixed Lagrangian domains + = T × (0, m) and -= T × (-l, 0) and assume that there exist invertible mappings and det(∇ζ 0 ) = 1. (1.6) The first condition in (1.5) means that the initial interface (0) is parameterized by the mapping ζ 0 ± restricted to , whereas the latter two conditions in (1.5) mean that ζ 0 ± map the fixed upper and lower boundaries into themselves. Define the flow maps ζ ± as the solutions to (1.7) We denote the Eulerian coordinates by (x, t) with x = ζ (y, t), whereas the fixed (y, t) ∈ × R + stand for the Lagrangian coordinates. Here we have denoted + ∪ -by .
To switch back and forth from Lagrangian to Eulerian coordinates, we assume that ζ ± (·, t) are invertible and ± (t) = ζ ± ( ± , t), and since v ± and ζ 0 ± are all continuous across , we have (t) = ζ ± ( , t). In other words, the Eulerian domains of upper and lower fluids are the images of ± under the mappings ζ ± , and the free interface is the image of under the mappings ζ ± (·, t) . In view of the nonslip boundary condition v ± | ± = 0, we have Moreover, by the incompressible condition (1.1) 3 we have as well as the initial condition (1.6), see [19,Proposition 1.4] for the derivation.
In Lagrangian coordinates the deformation tensorŨ(y, t) is defined by a Jacobi matrix of ζ ± (y, t): Here and in what follows, ∂ j denotes the partial derivative with respect to the jth component of the spatial variables. When we study this deformation tensor in Eulerian coordinates, we denote it by U ± (x, t) :=Ũ(ζ -1 ± (x, t), t). Applying the chain rule, it is easy to see that U ± (x, t) satisfies the transport equation
Thus the definition of n in (1.12) is reasonable. In what follows, we call problem (1.10) the transformed stratified VRT problem. Next, we further introduce the notations involving A. The matrix A := (A ij ) 3×3 via A T = (∇ζ ) -1 := (∂ j ζ i ) -1 3×3 , and the differential operator ∇ A is defined by for vector functions w := (w 1 , w 2 , w 3 ), and the differential operator div A is defined by for vector functions It should be noted that we have used the Einstein convention of summation over repeated indices. In addition, we define A X := div A ∇ A X. Finally, we introduce some properties of A. In view of the definition of A and (1.8), we see that where A * ij is the algebraic complement minor of the (i, j)th entry of the matrix (∂ j ζ i ) 3×3 . In addition, we have (1.15) and where δ il = 1 for i = l and δ il = 0 for i = l. If (u, η) is very small, then the small terms of second order (i.e., the nonlinear terms) in (1.10) can be neglected, and we thus obtain the following linearized stratified VRT problem: where S(f ) = ∇f + ∇f T for f = η, and μu + κρη. The linearized problem is convenient to analyze in order to have an insight into the physical and mathematical mechanisms of the stratified VRT problem. In fact, using a standard energy method, Jiang et al. [17] found stability and instability criteria of the above linearized stratified VRT problem.
Before recalling the stability and instability criteria, we introduce some simplified notations: where k is a nonnegative integer, s is a real number, and the positive constant c may depend on the domain occupied by the fluids and other known physical parameters such as ρ, μ, g, and κ and varies from line to line.
Next, we introduce the stability and instability criteria of (1.17). We consider normal mode solutions of (1.17) in the form q(y, t) =q(y)e t for some constant > 0.
Substituting this ansatz into (1.17), we obtain the eigenvalue problem Eliminatingη by using the first equation, we arrive at the boundary value problem ( By the classical theory of linear RT instability, if then the linearized stratified VRT problem is unstable. Obviously, the above condition is equivalent to where we have defined Jiang et al. [17] used the discrete Fourier transformation to prove that there exist unstable solutions to the linearized stratified VRT problem under C κ > 1. Moreover, they further verified that the transformed stratified VRT problem is stable for C κ < 1. The nonlinear stability result shows that the elasticity can inhibit the development of RT instability. In this paper, we assume that κ is a constant. Then the instability criterion (1.20) reduces to Under (1.21), we prove that there exists an unstable solution of the linearized stratified VRT problem with a largest growth rate, and the largest growth rate decreases from a positive constant to 0 as κ increases from 0 to κ c . Next, we introduce the definition of largest growth rate. Definition 1. 1 We call > 0 the largest growth rate of RT instability in the linearized stratified VRT problem if it satisfies the following two conditions: (1) For any classical solution (u, η) of the linearized stratified VRT problem with an associated pressure q, we have, for any t ≥ 0, where (2) There exists a solution (u, η, q) of the linearized stratified VRT problem in the form Now we state our result on the linearized stratified VRT problem.
In addition, for given g, ρ, and μ, we can regard κ := as a function of κ ∈ [0, κ c ). It enjoys the following properties: κ strictly decreases and is continuous with respect to κ, (1.24) and In particular, we have κ → 0 as κ → κ c .
The proof of Theorem 1.1 is based on the modified variational method [5,9,13,14]. Next, we briefly introduce the proof of Theorem 1.1. Obviously, the linearized stratified VRT problem (1.17) is unstable if there exists a solution (ũ,q, ) to the boundary-value problem (1.19) with > 0. In view of the basic idea of the modified variational method, to look for the solution (ũ,q, ), we will use a modified variational approach and thus modify (1.19) as follows: where s > 0 is a given parameter, and α(s) depends on s.
Multiplying (1.26) 1 byũ and integrating the resulting identity, we get Exploiting the classical regularity theory on the stratified steady Stokes problem,ũ indeed is a classical solution of the boundary-value problem (1.26) with an associated functionq and an associated constant α(s) > 0 defined by (1.27); see Lemma 2.1 for detailed results.
In view of the definition of α(s), we can infer that the function α(s) enjoys the three properties: α(s) ∈ C 0,1 loc (0, ∞), α(s) is strictly decreasing, and lim s→0 α(s) > 0. Obviously, √ α(s) also possesses these properties and thus has a fixed point on some internal (0, G), that is, there exists satisfying the fixed-point relation  , κ), κ , and G κ , respectively. In addition, we can further prove that, for fixed s, α(s, κ) strictly decreases and is continuous with respect to κ; moreover, there exists an estimate G κ ≤ m; see Lemma 2.4. Consequently, using the fixed-point relation (1.28), Lemma 2.4, and the definition of continuity, we can show that κ also inherits the properties of α(s, κ) with respect to κ and thus get properties (1.24)-(1.25) of κ . In the next section, we provide a detailed proof of Theorem 1.1.

Proof of Theorem 1.1
To begin with, we prove that a maximizer of (1.27) exists and that the corresponding Euler-Lagrange equations are equivalent to (1.26).

Lemma 2.1
Under the assumptions of Theorem 1.1, for any but fixed s > 0, the following assertions are valid.
(2) Letũ 0 be a maximizer, and let α := supũ ∈A E(ũ). Then there exists a pressureq 0 associated withũ 0 such that the triple (ũ 0 ,q 0 , α) satisfies the boundary problem Proof (1) Noting the estimate Using (2.1) and the Cauchy-Schwarz and Korn inequalities, we can deduce from this estimate thatũ n is bounded in H 1 . So, there existũ 0 ∈ H 1 ∩ A and a subsequence (still denoted byũ n for simplicity) such thatũ n →ũ 0 weakly in H 1 and strongly in L 2 . Further, we havẽ u n →ũ 0 strongly in L 2 ( ). Indeed, in view of (2.1), Therefore, by the lower semicontinuity we have which shows that E(ũ) achieves its supremum on A.
(2) To show the second assertion, we notice that since E(ũ) and J(ũ) are homogeneous of degree 2, (1.27) is equivalent to For any given τ ∈ R and w ∈ H 1 σ , we takew :=ũ 0 + τ w. Then (2.2) implies Now we consider the following stratified steady Stokes problem: To prove that there is a fixed point such that = √ α( ) > 0, we further give some properties of α(s) with respect to s > 0.
Thus, for any s 1 , Reversing the role of indices 1 and 2 in the derivation of this inequality, we obtain the same boundedness with the indices switched. Therefore, we deduce that which yields α(s) ∈ C 0,1 loc (0, ∞). (2) By the instability condition κ < κ c of the linearized stratified VRT problem there exists a functionũ ∈ H 1 σ such that Thus we have for two positive constants c 1 := c 1 (g, ρ, κ) and c 2 := c 2 (μ, ρ). This completes the proof of Lemma 2.2.
Next, we prove that there exists a pair of functions (ũ,q) satisfying (1.19) with a growth rate by a fixed-point argument. Thus, if s > (c 3 c 4κ min{ρ -, ρ + })/ min{μ -, μ + }, then which implies that In view of (2.5)-(2.7) and the continuity of α(s) on (0, G), we can use a fixed-point argument to deduce the following conclusion. (2.11) Multiplying (2.11) 1 with u t , integrating the resulting equation (by parts), and recalling that divu t = 0, we obtain On the other hand, by (1.17) 1 we have Inserting these equalities into (2.12), we infer that d dt Using the Newton-Leibniz formula and Cauchy-Schwarz inequality, we find that we deduce from (2.16) the differential inequality d dt Applying Gronwall's inequality [20, Lemma 1.2] to this inequality, we conclude which, together with (2.16), yields Recalling Definition 1.1, we see that is the largest growth rate of RT instability in the linearized stratified VRT problem from (2.10) and Proposition 2.2. To emphasize the dependence of α(s), , and G upon κ, we denote them by α(s, κ), κ , and G κ , respectively.
To complete the proof of Theorem 1.1, we further derive relations (1.24) and (1.25) of elasticity coefficient and the largest growth rate. To this end, we need the following auxiliary conclusions.
(2) Let s > 0 be fixed. We  6 ], there is a functionw κ ∈ A satisfying α(s, κ) =Ê(w κ )κ (w κ ). Thus, in view of the monotonicity of α(s, κ), we know that Thus, for any κ 1 , Reversing the role of indices 1 and 2 in the derivation of this inequality, we obtain the same boundedness with the indices switched. Therefore, we deduce that Recalling the definition of κ c , we see that Hence, by the first assertion of Lemma 2.1, for any given κ, there existsw κ ∈ A such that α(s, κ) = gtρu w κ Thus, if α(s, κ) ≥ 0, then s ≤ m (refer to (1.25) for the definition of m). This means that G κ ≤ m by the definition of G κ in (2.19). This completes the proof. First, we verify the monotonicity of κ in the variable κ. For given two constants κ 1 and κ 2 satisfying 0 ≤ κ 1 < κ 2 , there exist two associated curve functions α(s, κ 1 ) and α(s, κ 2 ) defined in (0, κ c ). By the first assertion in Lemma 2.4, α(s, κ 1 ) > α(s, κ 2 ).
On the one hand, the fixed point κ i satisfying κ i = α( κ i ) can be obtained as the intersection point of the two curves y = s and y = α(s, κ i ) for i = 1 and 2. Thus we can immediately observe the monotonicity: (2.20) Second, we show the continuity. To this end, we choose a constant κ 0 > 0 and an associated function α(s, κ 0 ). Noting that α( κ 0 , κ 0 ) = 2 κ 0 > 0 and α(s, κ 0 ) ∈ C 0,1 loc [0, κ c ) are strictly decreasing with respect to κ, we have that, for any given ε > 0, there exists a constant δ > 0 such that In particular, we have By the monotonicity of κ with regard to κ we get κ 0 -δ > κ 0 > κ 0 +δ .

Extension to MHD fluids
Recently, Wang [24] established the nonlinear stability and linear instability of the stratified magnetic Rayleigh-Taylor (MRT) problem. His result shows that magnetic fields can also inhibit the development of RT instability. In this section, we extend the obtained result in linearized stratified VRT problem to the linearized stratified MRT problem. More precisely, we establish qualitative relations between magnetic fields and the largest growth rates in the linear RT instability of stratified MRT problem. Next, we introduce the stratified MRT problem investigated by Wang. In what follows, we continue to use the mathematical notations of Sect. 1, unless specified otherwise. The three-dimensional motion equations of incompressible MHD fluids without resistivity under a uniform gravitational field (along the negative x 3 -direction) can be described as follows [10,12,15]: where M denotes the magnetic field, and stress tension S(p g , v, M) is defined as follows: Here λ is the permeability of vacuum divided by 4π .
Referring to (1.1), we can easily use these motion equations to establish the mathematical model of the motion of the stratified incompressible MHD fluids without resistivity under a uniform gravitational field [16]: To investigate the RT instability in (3.2), we assume that tρu > 0. Then we call this model the stratified MRT problem. Since the upper fluid is heavier than the lower fluid, the stratified MRT problem may be unstable due to the RT instability. The initial-boundary problem (3.2) admits an equilibrium-state solution with v = 0, p = const, M =M, and d =d, wherē M is a uniform magnetic field. Without loss of generality, we assume thatd = 0. Let ζ and η be defined by (1.9) and η := ζy. Similarly to (1.10), using the additional definition the stratified MRT problem (3.2) in Lagrangian coordinates reads as follows: where we have defined The initial-boundary problem (3.3) admits an equilibrium-state solution with ζ = y, u = 0, q = const, and B =M. To investigate the stability and instability of the initial-boundary problem (3.3) around the equilibrium-state solution, as in Wang [24], we further assume that ζ 0 = y, i.e., η 0 = 0; (3.5) then we have det(∇ζ ± ) = 1. Next, we recall some conservation relations involving the magnetic field. Applying A T to the magnetic induction equation (3.3) 3 and using (1.15), we obtain This implies that It then follows from (1.16), div A B = 0, and (3.7) that and ∂ t B ± · (A ± e 3 ) = ∂ t e T 3 A T ± B ± e 3 = 0 on . (3.9) Hence by (3.5) we can derive from (3.7)-(3.9) that The conservation analysis above reveals that the magnetic field B should have certain relations with the flow map ζ . In turn, this motivates us to eliminate the magnetic field B in the initial-boundary values problem (3.3). Indeed, since B =M · ∇ζ by the first identity in (3.10), we may rewrite the generalized Lorentz force term: Consequently, the initial-boundary values problem (3.3) can be reformulated as follows: (3.12) We call this problem the transformed stratified MRT problem. The corresponding linearized problem of the transformed stratified MRT problem reads as follows: .
Wang [24] has proved that the linearized stratified MRT problem defined on 1 is stable forM 3 > M c and unstable forM 3 < M c . Moreover, he also verified that the nonlinear stratified MRT problem defined on 1 is also stable forM 3 > M c . The nonlinear stability result shows that the magnetic fields can inhibit the development of RT instability. Of course, Wang's results also hold for the domain .

Main result
Before introducing our result, let us recall the derivation of the instability criterion. We consider the following growing mode solutions to (3.13): for some > 0. Substituting the above ansatz into (3.13), we get an eigenvalue problem (3.15) By using (3.15) 1 we can eliminateη and arrive at the following boundary-value problem for blue(ũ,q): refer to [11,Lemma 4.5]. Finally, we introduce the largest growth rate of RT instability in the linearized stratified MRT problem. Definition 3. 1 We call > 0 the largest growth rate of RT instability in the linearized stratified MRT problem if it satisfies the following two conditions: (1) For any classical solution (u, η) of the linearized stratified MRT problem with an associated pressure q, we have, for any t ≥ 0, In this paper, forM := (0, 0, M), we further prove that the largest growth rate in the linearized stratified MRT problem decreases from a positive constant to 0 as |M 3 | increases from 0 to M c . More precisely, we have the following conclusion.
We can follow the argument of Theorem 1.1 to derive Theorem 3.1. The detailed proof of Theorem 3.1 is provided in the next subsection.

Proof of Theorem 3.1
Next, we follow the argument of Theorem 1.1 to prove Theorem 3.1. To begin with, we modify (3.16) as follows: (2) Letũ 0 be a maximizer, and let β := supũ ∈A E(ũ). Then there exists a pressure functionq 0 associated withũ 0 such that the triple (ũ 0 ,q 0 , β) satisfies the boundary problem (3.26). Moreover, Proof (1) Following the argument of the first assertion of Lemma 2.1, we can easily get the first conclusion.
Consequently, following the above argument and a standard bootstrap method of improving the regularity, we can easily see that (ũ 0 ,q 0 ) ∈ (H 1 σ ∩ H ∞ ) × H ∞ . This completes the proof.
To prove that there is a fixed point such that = β( ) > 0, we further give some properties of β(s) with respect to s > 0. (1) β(s) ∈ C 0,1 loc (0, ∞) is strictly decreasing in the variable s. Since divũ = 0 andũ| + -= 0, we immediately getũ 2 1 +ũ 2 2 = 0. In addition, similarly to Proposition 2.1, we can also verify that > 0 is the largest growth rate of RT instability in the linearized stratified MRT problem. Thus we have the following conclusion. To emphasize the dependence of β(s), , and G upon M, we will denote them by β (s, M), M , and G M , respectively. Obviously, we have M = -M . To complete the proof of Theorem 3.1, we further derive (3.24)- (3.25). To this end, we need the following auxiliary conclusions.