A new upper bound for the largest growth rate of linear Rayleigh–Taylor instability

We investigate the effect of (interface) surface tensor on the linear Rayleigh–Taylor (RT) instability in stratified incompressible viscous fluids. The existence of linear RT instability solutions with largest growth rate Λ is proved under the instability condition (i.e., the surface tension coefficient ϑ is less than a threshold ϑc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\vartheta _{\mathrm{c}}$\end{document}) by the modified variational method of PDEs. Moreover, we find a new upper bound for Λ. In particular, we directly observe from the upper bound that Λ decreasingly converges to zero as ϑ goes from zero to the threshold ϑc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\vartheta _{\mathrm{c}}$\end{document}.


Introduction
Considering two completely plane-parallel layers of stratified (immiscible) fluids, the heavier one on top of the lighter one and both subject to the earth's gravity, it is well known that such an equilibrium state is unstable to sustain small disturbances, and this unstable disturbance will grow and lead to a release of potential energy as the heavier fluid moves down under the gravitational force and the lighter one is displaced upwards. This phenomenon was first studied by Rayleigh [28] and then Taylor [29], and it is therefore called the Rayleigh-Taylor (RT) instability. In the last decades, this phenomenon has been extensively investigated from mathematical, physical, and numerical aspects, see [2,10,31] for instance. It has been also widely investigated how the RT instability evolves under the effects of other physical factors, such as elasticity [3,13,23,25,32], rotation [2,6], (internal) surface tension [11,16,36], magnetic fields [18-22, 24, 33, 34], and so on. In this article, we are interested in the effect of surface tension on the linear RT instability in stratified incompressible viscous fluids. To conveniently introduce relevant mathematical progress and our main results, next we mathematically formulate our problem in detail.

Motion equations in Eulerian coordinates
Let us first recall a mathematical model, which describes the horizontally periodic motion of stratified incompressible viscous fluids in an infinity layer domain [23]: div v ± = 0 on ± (t), (1.1) The momentum equations in (1.1) 1 describe the motion of both upper heavier and lower lighter viscous 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. Moreover, the fluids are incompressible due to (1.1) 2 . The two fluids interact with each other by the motion equation of a free interface (1.1) 3 and the interfacial jump conditions in (1.1) 4 . The first jump condition in (1.1) 4 represents that the velocity is continuous across the interface. The second jump in (1.1) 4 represents that the jump in the normal stress is proportional to the mean curvature of the surface multiplied by the normal to the surface. The non-slip boundary condition of the velocities on both upper and lower fixed flat boundaries are described by (1.1) 5 . (1.1) 6 and (1.1) 7 represent the initial status of the two fluids. Next we further explain the notations in (1.1) in detail.
For given t > 0, v ± (x, t) : ± (t) → R 3 are the velocities of the two fluids, and S ± are the stress tensors enjoying the following expression: In the above expression the superscript T means matrix transposition and I is the 3 × 3 identity matrix. ρ ± are the density constants, and the constants μ ± > 0 are the shear viscosity coefficients. g and ϑ represent the gravitational constant and the surface tension coefficient, resp. In addition, e 3 := (0, 0, 1) T .
For a function f defined on (t), we define tf ± u := f + | (t)f -| (t) , where f ± | (t) are the traces of the quantities f ± on (t). ν is the unit outer normal vector at boundary (t) of -(t), and C is the twice of the mean curvature of the internal surface (t), i.e., Now we further introduce the indicator function χ ± (t) and denote then model (1.1) can be rewritten as follows: where we have defined that (t) := + (t) ∪ -(t), + -:= -∪ + and omitted the subscript ± in tf ± u for simplicity.

Reformulation in Lagrangian coordinates
Next we adopt the transformation of Lagrangian coordinates so that the interface and the domains stay fixed in time.
We define that and assume that there exist invertible mappings We further define ζ 0 := ζ 0 + χ + (0) + ζ 0 -χ -(0) and the flow map ζ as the solution to ⎧ ⎨ where + -:= + ∪ -. We denote the Eulerian coordinates by (x, t) with x = ζ (y, t), whereas the fixed (y, t) ∈ + -× R + stand for the Lagrangian coordinates. In order 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 view of the non-slip boundary condition v| + -= 0, we have Now we set the Lagrangian unknowns then problem (1.3) can be rewritten as an initial-boundary value problem with an interface for (ζ , u) in Lagrangian coordinates: where we have defined that We shall introduce the notations involving A. The matrix A := (A ij ) 3×3 is defined via where ∂ j denotes the partial derivative with respect to the jth components of variables y. A := A -I, and I is the 3 × 3 identity matrix. The differential operator ∇ A is defined by for a vector function w := (w 1 , w 2 , w 3 ), and the differential operator div A is defined by 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.

Linearized motion
We choose a constantd ∈ (-h -, h + ). Without loss of generality, we assume thatd = 0. Then we consider an RT equilibrium state where ρ satisfies the RT (jump) condition tρu > 0 on .
The inhibition of RT instability by surface tension was first analyzed by Bellman and Phennington [1] based on a linearized two-dimensional (2D) motion equations of stratified incompressible inviscid fluids defined on the domain 2πL 1 T 1 × (-h -, h + ) (i.e., μ = 0 in the corresponding 2D case of (1.10)) in 1953. More precisely, they proved that the linear 2D stratified incompressible inviscid fluids are stable, resp. unstable for ϑ > gtρuL 2 1 , resp. ϑ < gtρuL 2 1 . The value gtρuL 2 1 is a threshold of surface tension coefficient for linear stability and linear instability. Similar result was also found in the 3D viscous case; for example, Guo and Tice proved that ϑ c := gtρu max{L 2 1 , L 2 2 } is a threshold of surface tension coefficient for stability and instability in the linearized 3D stratified compressible viscous fluids defined on [11]. Next we further review the mathematical progress for the nonlinear case.
Prüess and Simonett first proved that the RT equilibria solution of the stratified incompressible viscid fluids defined on the domain R 3 is unstable based on a Henry instability method [27]. Later Wang, Tice, and Kim verified that the RT equilibria solution of stratified incompressible viscous fluids defined on is stable, resp. unstable for ϑ > ϑ T , resp. ϑ ∈ [0, ϑ T ) [35,36]. Jang, Wang, and Tice further obtained the same results of stability and instability in the corresponding compressible case [15,16]. Recently, Wilke also proved there exists a threshold ϑ c for the stability and instability of stratified viscous fluids (with heavier fluid over lighter fluid) defined on a cylindrical domain with finite height [37]. Finally, we mention that the results of nonlinear RT instability in inhomogeneous fluid (without interface) were obtained based on the classical bootstrap instability method, see [14], resp. [17] for inviscid, resp. viscous cases.

Main result
In this paper, we investigate the effect of surface tension on the linear RT instability by the linearized motion (1.10). Wang and Tice used a discrete Fourier transformation and the modified variational method of ODEs to prove the existence of growth solutions with a largest growth rate ϑ for (1.10) with h + = 1 under the condition ϑ ∈ (0, ϑ c ) [35]. Moreover, they provided an upper bound for ϑ : In this paper, we exploit the modified variational method of PDEs and the existence theory of stratified (steady) Stokes problem to prove the existence of growth solutions with a largest growth rate ϑ for (1.10) under the instability condition ϑ ∈ [0, ϑ T ). Moreover, we find a new upper bound: It is easy to see that Therefore our upper bound is more precise than Wang and Tice's one. Moreover, we directly see from (2.1) that We mention that, in the classical Rayleigh-Taylor (RT) experiments [8,12], the phenomenon of that the instability growth is limited by surface tension during the linear stage, where the growth is exponential in time, has been shown. Obviously, the convergence behavior (2.2) mathematically verifies the phenomenon. Before stating our main results in detail, we introduce some simplified notations used throughout this article.
(1) Basic notations: where the positive constant c may depend on the domain and known parameters such as ρ ± , μ ± , g, and ϑ, and may vary from line to line.
(2) Simplified notations of Sobolev spaces: where 1 < p ≤ ∞ and i ≥ 0 is an integer. Sometimes, we denote A by A ϑ to emphasize the dependence of ϑ. In addition, to prove the existence of unstable classical solutions of the linearized RT problem, we introduce a function space In addition, we give the definition of the largest growth rate of RT instability in the linearized RT problem. Definition 2. 1 We call > 0 the largest growth rate of RT instability in the linearized RT problem (1.10) if it satisfies the following two conditions: (1) For any strong solution (η, u) (2) There exists a strong solution (η, u) of the linearized RT problem in the form Now we state the first result on the existence of the largest growth rate in the linearized RT problem. Theorem 2.1 Let g > 0, ρ > 0, and μ > 0 be given. Then, for any given there is an unstable solution to the linearized RT problem (1.10), where (w, β) ∈ H ∞ solves the boundary value problem with the largest growth rate > 0 satisfying Moreover, Next we briefly introduce how to prove Theorem 2.1 by the modified variational method of PDEs and the regularity theory of stratified (steady) Stokes problem. The detailed proof is given in Sect. 4.
We assume a growing mode ansatz to the linearized problem for some > 0. Substituting this ansatz into the linearized RT problem (1.10), we get a spectrum problem and then eliminatingη by using the first equation, we arrive at the boundary value problem (2.5) for w and β. Obviously, the linearized RT problem is unstable if there exists a solution (w, β) to the boundary value problem (2.5) with > 0.
To look for the solution, we use a modified variational method of PDEs and thus modify (2.5) as follows: where s > 0 is a parameter. To emphasize the dependence of s upon α and ϑ, we write α(s, ϑ) = α. Note that the modified problem (2.8) enjoys the following variational identity: Thus, by a standard variational approach, there exists a maximizer w ∈ A of the functional F defined on A; moreover, w is just a weak solution to (2.8) with α defined by the relation In view of the definition of α(s, ϑ) and the instability condition (2.4), we can infer that, for given ϑ, the function α(s, ·) on the variable s enjoys some good properties (see Proposition 4.3), which imply that there exists satisfying the fixed point relation = α( , ·) ∈ (0, S ϑ ). (2.10) Then we obtain a nontrivial solution (w, β) ∈ H ∞ to (2.5) with defined by (2.10), and therefore the linear instability follows. Moreover, is the largest growth rate of RT instability in the linearized RT problem (see Proposition 4.4), and thus we get Theorem 2.1.
Next we turn to introduce the second result on the properties of the largest growth rate constructed by (2.10).

Theorem 2.2
The largest growth rate ϑ := in Theorem 2.1 enjoys the estimate (2.1). Moreover, ϑ strictly decreases and is continuous with respect to ϑ ∈ [0, ϑ c ). (2.11) The proof of Theorem 2.2 is presented in Sect. 5. Here we briefly mention the idea of the proof. We find that, for fixed s, α(·, ϑ) defined by (2.9) strictly decreases and is continuous with respect to ϑ (see Proposition 5.1). Thus, by the fixed point relation (2.10) and some analysis based on the definition of continuity, we can show that ϑ := also inherits the monotonicity and continuity of α(·, ϑ). Finally, we derive (2.1) from (2.6) by some estimate techniques.

Preliminary
This section is devoted to introducing some preliminary lemmas, which will be used in the next two sections.

Lemma 3.1 Difference quotients and weak derivatives: Let D be or T.
( Proof Following the argument of [7, Theorem 3] and using the periodicity of w, we can easily get the desired conclusions.  1 in , Thus, using Poincare's inequality, we have which immediately implies assertion (3.5).
Remark 3.2 From the derivation of (3.9), we easily see that Lemma 3.6 Negative trace estimate: Lemma 3.7 Let X be a given Banach space with dual X * , and let u and w be two functions belonging to L 1 ((a, b), X). Then the following two conditions are equivalent: in the scalar distribution sense, on (a, b), where ·, · X×X * denotes the dual pair between X and X * .

Linear instability
In this section, we use the modified variational method to construct unstable solutions for the linearized RT problem. The modified variational method was firstly used by Guo and Tice to construct unstable solutions to a class of ordinary differential equations arising from a linearized RT instability problem [11]. This idea had been also used in [4,5].
In this paper, we directly apply Guo and Tice's modified variational method to the partial differential equations (2.5) and thus obtain a linear instability result of the RT problem by further using an existence theory of the stratified Stokes problem. Next we prove Theorem 2.1 by four subsections.

Existence of weak solutions to the modified problem
In this subsection, we consider the existence of weak solutions to the modified problem Sometimes, we denote α(s, ϑ) and F( , s) by α (or α(s)) and F( ), resp., for simplicity. Then we have the following conclusions. Proof Noting that thus, by Young's inequality and Korn's inequality (3.7), we see that {F( )} ∈A has an upper bound for any ∈ A. Hence there exists a maximizing sequence {w n } ∞ n=1 ⊂ A, which satisfies α = lim n→∞ F(w n ). Moreover, making use of (4.3), the fact √ ρw n 0 = 1, trace estimate (3.9), and Young's and Korn's inequalities, we have w n 1 + ϑ|∇ h w n 3 | 0 ≤ c 1 for some constant c 1 , which is independent of n. Thus, by the well-known Rellich-Kondrachov compactness theorem and (4.3), there exist a subsequence, still labeled by w n , and a function w ∈ A such that Exploiting the above convergence results and the lower semicontinuity of weak convergence, we have Hence w is a maximum point of the functional F( ) with respect to ∈ A. Obviously, w constructed above is also a maximum point of the functional F( )/ √ ρ 2 0 with respect to ∈ H 1 σ ,ϑ . Moreover, α = F(w)/ √ ρw 2 0 . Thus, for any given ϕ ∈ H 1 σ ,ϑ , the point t = 0 is the maximum point of the function Then, by computing out I (0) = 0, we have the weak form Note that (4.4) is equivalent to This means that w is a weak solution of the modified problem (4.1).

Improving the regularity of weak solution
By Proposition 4.1, the boundary value problem (4.1) admits a weak solution w ∈ H 1 σ ,ϑ . Next we further improve the regularity of w. Proof To begin with, we establish the following preliminary conclusion: For any i ≥ 0, we have and Obviously, by induction, the above assertion reduces to verifying the following recurrence relation: For and w satisfies Next we verify the above recurrence relation by the method of difference quotients. Now we assume that w ∈ H 1,i σ ,ϑ satisfies (4.6) for any ϕ ∈ H 1 σ ,ϑ . Noting that ∂ i h w ∈ H 1 σ ,ϑ , we can deduce from (4.6) that, for j = 1 and 2, which yield that and resp. By Korn's inequality, thus, using (4.3), Young's inequality, and the first conclusion in Lemma 3.1, we further deduce from (4.10) that Thus, using (4.3), trace estimate (3.9), and the second conclusion in Lemma 3.1, there exists a subsequence of {-h} h∈R (still denoted by -h) such that (4.11) Using the regularity of w in (4.11) and the fact w ∈ H 1,i σ ,ϑ , we have (4.7). In addition, exploiting the limit results in (4.11), we can deduce (4.8) from (4.9). This completes the proof of the recurrence relation, and thus (4.5) holds.
With (4.5) in hand, we can consider a stratified Stokes problem: where k ≥ 0 is a given integer, and we have defined that Recalling the regularity (4.5) of w, we see that ∂ k h w ∈ L 2 and ∂ k h L 1 ∈ H 1 (T). Applying the existence theory of the stratified Stokes problem (see Lemma 3.2), there exists a unique strong solution (ω k , β k ) ∈ H 2 × H 1 of the above problem (4.12).
Taking ϕ := ∂ k h wω k ∈ H 1 σ ,ϑ in the above identity and using the Korn's inequality, we find that ω k = ∂ k h w. Thus we immediately see that ∂ k h w ∈ H 2 for any k ≥ 0, (4.14) which implies ∂ k h w ∈ H 1 and ∂ k h L 1 ∈ H 2 (T) for any k ≥ 0. Thus, applying the stratified Stokes estimate (3.2) to (4.12), we have Obviously, by induction, we can easily follow the improving regularity method from (4.14) to (4.15) to deduce that w ∈ H ∞ . In addition, we have β := β 0 ∈ H ∞ ; moreover, β k in (4.12) is equal to ∂ k h β. Finally, recalling the embedding H k+2 → C 0 ( ) for any k ≥ 0, we easily see that (w, β) constructed above is indeed a classical solution to the modified problem (4.1).

Some properties of the function α(s)
In this subsection, we derive some properties of the function α(s), which ensure the existence of a fixed point of √ α(s) in R + .

Largest growth rate
Next we prove that constructed in the previous section is the largest growth rate of RT instability in the linearized RT problem and thus complete the proof of Theorem 2.1.

Proposition 4.4
Under the assumptions of Theorem 2.1, > 0 constructed by (4.22) is the largest growth rate of RT instability in the linearized RT problem.
Proof Recalling the definition of largest growth rate, it suffices to prove that enjoys the first condition in Definition 2.1. Let u be a strong solution to the linearized RT problem. Then we derive that, for a.e. t ∈ I T and all w ∈ H 1 σ , Using the regularity of (η, u), we can show that the right-hand side of (4.25) is bounded above by A(t)( w 1 + |w| 1 ) for some positive function A(t) ∈ L 2 (I T ). Then there exists f ∈ L 2 (I T , H -1 σ ) such that, for a.e. t ∈ I T , Hence it follows from Lemma 3.7 that In addition, by a classical regularization method (referring to Theorem 3 in Chap. 5.9 in [7] and Lemma 6.5 in [26]), we have Therefore, we can derive from (4.26) and the above two identities that d dt In addition, by (2.6), we have Thus, we infer from (4.27)-(4.29) that Recalling that d dt we further deduce from (4.30) the differential inequality d dt Multiplying (1.10) 2 by u t in L 2 and using the integral by parts, we get ρ|u t | 2 dy = tqu∂ t u 3 dy h + μ u · u t dy. (4.33) Exploiting (3.17), we can estimate that tqu∂ t u 3 dy h tqu 1/2 |∂ t u 3 | -1/2 tqu 1/2 u t 0 .
In addition, using (1.10) 5 and trace estimate (3.9), we have Using the above two estimates, we can derive from (4.33) that which implies that By the above estimate and Korn's inequality, we derive from (4.31) and (4.32) that Finally, from (1.10) 1 we get By the two estimates above, we see that satisfies the first condition in Definition 2.1. The proof is completed.

Properties of α(s, ϑ) with respect to ϑ
To emphasize the dependence of and G upon ϑ, we denote them by ϑ and G ϑ , respectively. To prove Theorem 2.2, we further derive relations (2.1) and (2.11) of the surface tension coefficient and the largest growth rate. To this end, we need the following auxiliary conclusions.

Conclusion
We investigate the effect of (interface) surface tension on the linear Rayleigh-Taylor (RT) instability in stratified incompressible viscous fluids. The existence of linear RT instability solutions with largest growth rate is proved under the instability condition (i.e., the surface tension coefficient ϑ is less than a threshold ϑ c ) by the modified variational method of PDEs. Moreover, we find a new upper bound for . In particular, we directly observe from the upper bound that decreasingly converges to zero as ϑ goes from zero to the threshold ϑ c . However, we have no idea to provide a numerical example, since it is very difficult for us to do the numerical experiment for the motion of stratified fluids in three dimensions. We will further investigate this topic in the future.