On Rayleigh–Taylor instability in Navier–Stokes–Korteweg equations

This paper focuses on the Rayleigh–Taylor instability in the two-dimensional system of equations of nonhomogeneous incompressible viscous fluids with capillarity effects in a horizontal periodic domain with infinite height. First, we use the modified variational method to construct (linear) unstable solutions for the linearized capillary Rayleigh–Taylor problem. Then, motivated by the Grenier’s idea in (Grenier in Commun. Pure Appl. Math. 53(9):1067–1091, 2000), we further construct approximate solutions with higher-order growing modes to the capillary Rayleigh–Taylor problem and derive the error estimates between both the approximate solutions and nonlinear solutions of the capillary Rayleigh–Taylor problem. Finally, we prove the existence of escape points based on the bootstrap instability method of Hwang–Guo in (Hwang and Guo in Arch. Ration. Mech. Anal. 167(3):235–253, 2003), and thus obtain the nonlinear Rayleigh–Taylor instability result. Our instability result presents that the Rayleigh–Taylor instability can occur in the fluids with capillarity effects for any capillary coefficient κ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\kappa >0$\end{document} if the critical capillary coefficient is infinite. In particular, it improves the previous Zhang’s result in (Zhang in J. Math. Fluid Mech. 24(3):70–23, 2022) with the assumption of smallness of the capillary coefficient.


Introduction
The two-dimensional (2D) motion equations of a nonhomogeneous incompressible viscous fluid with capillarity effects in the presence of a uniform gravitational field in a domain D ⊂ R 2 are given as follows: where the unknowns ρ := ρ(x, t), V := V (x, t) and P := P(x, t) denote the density, velocity and the pressure, respectively, μ > 0 stands for the coefficient of shear viscosity, g > 0 is the gravitational constant.The positive constant κ > 0 represents for the capillary coefficient, e 2 := (0, 1) T the vertical unit vector, the superscript T the transposition and -ge 2 the gravitational force.Here and in what follows x ∈ R 2 and t ≥ 0 are the spacial and temporal variables, respectively.In the system (1.1), the equation (1.1) 1 describes the law of conservation of mass and (1.1) 2 the law of conservation of momentum.We call (1.1) the inhomogeneous incompressible Navier-Stokes-Korteweg equations.We mention that the general capillary tensor K is written as where I denotes the identity matrix.However, we assume that the capillarity function κ is a positive constant for the sake of the simplicity, and thus div K = -κ∇ρ ρ.
It is well-known that the equilibrium of a heavier fluid on top of a lighter one, subject to gravity, is unstable.In this case, the equilibrium state is unstable to sustain small disturbances, and this disturbance grow and leads to the 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 [17] and then Taylor [18], and thus is called the Rayleigh-Taylor (RT) instability.In the last decades, this phenomenon has been extensively investigated from mathematical, physical, and numerical aspects, see [19][20][21][22][23][24] for examples.Moreover, the RT instability also has been investigated under other physical factors, such as internal surface tension [25][26][27][28][29], the elasticity [22,[30][31][32][33][34][35][36][37], magnetic fields [38][39][40][41][42][43][44], rotation [45,46] and so on.Next we further introduce the nonlinear RT instability results, which are closely related to our results in this paper, on the inhomogeneous incompressible fluids.
In 2003, Hwang-Guo [2] proved the existence of classical solutions of (nonlinear) RT instability in the sense of L 2 -norm for a 2D inhomogeneous incompressible inviscid pure fluid.Then Jiang-Jiang further showed the existence of strong solutions of RT instability for the nonhomogeneous incompressible viscous pure fluids in the sense of L 2 -norm [47].Similar instability result was also established in the inhomogeneous incompressible magnetohydrodynamics fluids [48].Later, Jiang-Wu-Zhong [23] also investigated the RT instability in the inhomogeneous incompressible viscoelastic fluids and surprisingly found that the elasticity can inhibit RT instability.
Recently, Zhang proved the RT instability for viscous incompressible fluids with capillarity effects for small enough capillary coefficient [3], in which the fluid domain is (2πT) 2 × R. Motivated by Zhang's result, we further investigate the RT instability for the system (1.1) with any given capillary coefficient by a new method, where the fluid domain is given as follows: (1.3) Obviously, we automatically obtain the RT instability solutions for the three-dimensional inhomogeneous incompressible Navier-Stokes-Korteweg equations in the presence of a uniform gravitational field, defined on (2πT) 2 × R, if the ones for the system (1.1) defined on (2πT) 2 × R are constructed, see (1.12) for the details.Now we consider the RT equilibrium state (ρ 0 , 0, P 0 ) of the system (1.1),where the density profile ρ 0 is independent of x 1 and satisfies Here and in what follows := d/dx 2 , l 0 >0 is a fixed constant and x 2 denotes the second component of x ∈ D. Then the corresponding equilibrium pressure can be computed out by the following relation of hydrostatics: We remark that the second condition in (1.4) prevents us from treating vacuum, while the last condition in (1.4) implies that the steady density profile has larger density with increasing height x 2 , thus may lead to the classical RT instability, as shown below.
We now consider a perturbation around the RT equilibrium state (ρ 0 , 0, P 0 ) by then, the triple (σ , v, p) satisfies the following equations: To complete the statement of the perturbed problem, we specify the initial and boundary conditions: From now on, we call the initial-boundary value problem (1.8)-(1.10) the CRT problem for the sake of simplicity.
In order to study the instability of RT equilibrium state, it seems to be convenient to start with the linearized (perturbation) equations, because the linearized perturbation equations not only enable us to understand the physical and mathematical mechanisms of capillarity, but also provide a beginning for the study of the nonlinear case.Hence, we omit the nonlinear terms in (1.8) and thus get the linearized CRT equations (1.11) The linearized system (1.11) with the initial-boundary conditions (1.9)-(1.10)constitutes the linearized CRT problem.Now we state the main result of this paper.
Theorem 1.1 For any κ > 0, the steady state (ρ 0 , 0, P 0 ) of system (1.1) is unstable under the assumption (1.4)-(1.5),that is, there is positive constant ε 0 > 0, such that for any small δ > 0 there exists a family of classical solutions (ρ δ (t, x), V δ (t, x), P δ (t, x)) to (1.1) such that The proof of Theorem 1.1 is based on the bootstrap instability method, which has its origin in Guo and Strauss's articles [49,50].Later, various versions of bootstrap instability approaches were established by many authors, see [51][52][53][54] for instance.Recently, Zhang used the bootstrap instability method with the energy inequality of Gronwall-type in [54] to prove the RT instability for the inhomogeneous incompressible viscous fluids with capillarity effects for small enough capillary coefficient [3].However, in this paper we use the other version of bootstrap instability method established by [2] to prove Theorem 1.1 for any given capillary coefficient.Such bootstrap instability method was also used to investigate the nonlinear instability of Hele-Shaw flows with smooth viscous profiles by Daripa-Hwang [55].In addition, we automatically obtain the RT instability solutions for the threedimensional inhomogeneous incompressible Navier-Stokes-Korteweg equations defined on (2πT) 2 × R. In fact, let where (ρ δ (t, x), U δ (t, x), Q δ (t, x)) is the instability solution in Theorem 1.1.It is easy to check that the solution ( δ , U δ , Q δ ) is the RT instability solution for the three-dimensional inhomogeneous incompressible Navier-Stokes-Korteweg equations in the presence of a uniform gravitational field, and such instability result presents that the smallness condition of capillary coefficient in Zhang's result can be removed.
In view of Hwang-Guo's bootstrap instability method in [2,55], the proof of Theorem 1.1 can be divided into three steps.First, we use modified variational method to construct (linear) unstable solutions for the linearized CRT problem in Sect. 2. Then we further construct approximate solutions with higher order growing modes to the CRT problem in Sect.3.1 as in Grenier's work [1] (also see [56,57]); moreover we also derive the error estimates between both the approximate solutions and nonlinear solutions of the CRT problem in Sect.3.2.Finally, we prove the existence of escape points based on the bootstrap instability approaches, and thus completes the proof of Theorem 1.1 in Sect.3.3.We end this section by introducing some abbreviations, which will be repeatedly used in the rest parts of this paper.

Linear instability
This section is devoted to constructing a family of unstable solutions to the linearized CRT problem (1.9)-(1.11),which have growing H k -norm for any k.We will construct such solutions via Fourier synthesis by first constructing a growing mode for any but fixed spatial frequency.

Linear growing modes
We make the following ansatz of growing mode solutions to the linearized problem (1.9)- (1.11).
σ (x, t), v(x, t), p(x, t) = e λt σ (x), ṽ(x), p(x) for some λ > 0. (2.1) Substituting this ansatz into (1.11) and then eliminating σ (x) by using the first equation, one obtains the following time-independent system: with boundary-value condition We fix a spatial frequency ξ ∈ (L -1 Z), and define the new unknowns ϕ, ψ and , which depend on x 2 by the following relations Inserting the above expressions into (2.2) and (2.3), it is easy to check that ϕ, ψ, , and λ satisfy the following ODEs: Eliminating in (2.4) 2 by using (2.4) 1 and (2.4) 3 , we have the following ODE for ψ with boundary-value condition Next, we use the modified variational method to construct a solution of (2.6)-(2.7).(For more details on this idea, check out Guo and Tice's paper on compressible viscous stratified flows [26]).We now fix a non-zero vector ξ ∈ (L -1 Z) and s > 0, then we get a family of the following modified problems from (2.6)-(2.7). - (2.9) Then the standard energy functional for the problem (2.8) is given by with an associated admissible set where Thus we can find a -λ 2 (depending on ξ ) by minimizing (2.12) In order to emphasize the dependence on s ∈ (0, ∞), we will sometimes write Before constructing the growth solutions, we shall introduce some preliminary results, which will be used later.In order to get a positive λ(ξ ) in the variational problem (2.12), let the critical capillary coefficient κ c and the critical frequency constant |ξ c | by the following variational forms and for κ ∈ (0, κ c ), More precisely, we have the following conclusions: Proof Since the proof is similar to Proposition 2.1, we omit the details here.
Next we show that a minimizer of (2.12) exists and that the corresponding Euler-Lagrange equations are equivalent to (2.7)-(2.8).
Proposition 2.3 For any fixed s > 0 and ξ with ξ = 0, the following assertions hold.
(2) Let ψ 0 be a minimizer and -λ 2 := E(ψ 0 ), then the pair (ψ 0 , λ) satisfies the problem Proof (1) Noting that for any ψ ∈ A we see that E(ψ) is bounded from below on A by virtue of (1.4), then inf ψ∈A E(ψ) is well defined and finite.We choose a minimizing sequence {ψ n } ∞ n=1 ⊂ A. It is easy to check that E(ψ n ) is bounded.This fact, together with (2.10) and (2.13), implies that {ψ n } ∞ n=1 is bounded in H 2 .Therefore, there exists a weak limit ψ 0 ∈ H 2 (R) and a subsequence (still denoted by ψ n for simplicity) such that and Next, we show that ψ 0 is a minimizer and satisfies the constraint (2.11).By the lower semicontinuity and the properties of minimizing sequence, one has Suppose by contradiction that J(ψ 0 ) < 1, we could find an α > 1 such that J(αψ 0 ) = 1 by the homogeneity of J(ψ), i.e., αψ 0 ∈ A, which implies that leading to a contradiction.Thus ψ 0 is a minimizer satisfying the constraint (2.11).( 2) By the same order homogeneity of E(ψ) and J(ψ), we can find that (2.12) is equivalent to

.16)
For any τ ∈ R and ψ ∈ H 2 (R), we define that ψ(τ ) := ψ 0 + τ ψ.Then, by (2.16) we have which, together with the arbitrariness of ψ, shows that ψ 0 satisfies the equation (2.8) in the weak sense on R for the horizontal case.In order to improve the regularity of ψ 0 , we rewrite (2.18) as which together with ψ 0 ∈ H 2 (R) yields ψ 0 ∈ H 1 loc (R) and Integrating by parts, we can rewrite (2.19) as follows which, keeping in mind that Using these facts, Hölder's inequality, and integration by parts, we conclude that (2.20) Consequently, ψ 0 ∈ H 4 (R) solves (2.7) and (2.8).This immediately gives that ψ 0 ∈ H ∞ (R) by applying the bootstrap method and the classical elliptic regularity theory to the formula (2.9).

Proposition 2.5 For any
Proof Thanks to Proposition 2.4, (2.13) and (2.29), it is easy to check that λ(s) Hence, we can employ a fixed point argument to find s ∈ (0, C ξ ) such that s = λ(ξ , s), and thus obtain a solution to the original problem (2.6)-(2.7).
Proof Please refer to Theorem 3.8 in [26] (or Lemma 3.7 in [59]) Therefore, in view of the Propositions 2.5 and 2.6, one immediately gets the following conclusion.
We end this subsection by giving some properties for λ(ξ ), which shows that λ is a bounded continuous function of ξ .

Construction of a solution to the ODEs system
Next, we will find a family of unstable solutions that satisfy (2.4)-(2.5),and provide an estimate for the H k norm of the constructed solutions.
Next, we would provide an uniform estimate for the H k norm of the solutions (ϕ, ψ, ) constructed in Theorem 2.2, which would be used in the next subsection.
) be the solution constructed in Theorem 2.2.Then, for any nonnegative integer k, there exist positive constants A k , B k , and C k depending on the parameters a, b, ρ 0 , g, κ, and k, such that ) )

.43)
Proof Throughout this proof, we denote by c a generic positive constant which may vary from line to line, and may depend on a, b, ρ 0 , g, κ and k.Recalling the fact ψ(ξ , x 2 ) ∈ A, we have (2.40), and there exists a positive constant c such that (2.45) In addition, we rewrite (2.6) as, Utilizing Gagliardo-Nirenberg interpolation inequality, we obtain which, together with (2.44), (2.48), and (2.49), yields Now we proceed by induction on k.Suppose that the boundedness holds some k ≥ 1, i.e., (2.52) Then by differentiating the equation (2.46) and using (2.51), we easily derive that there exists a constant c depending on the various parameters so that It is easy to check that the bound holds for k + 1, we thus find that (2.41) holds for any nonnegative integer k.Employing the expressions (2.38) and (2.39), we can also deduce that (2.42)-(2.43)holds.

Instability of the linearized CRT problem
In this section, we will construct growing solutions to (1.9)-(1.11)by using the solutions to (2.4)-(2.5)constructed in Theorem 2.2. (2.54) where ϕ, ψ, are the solutions provided by Theorem 2.2.Let (2.57) Then, (σ , v, q) is a real-valued solution to the linearized equations (1.9)- (1.11).For every k ∈ N we have the estimate for the constant B k > 0 depending on the parameters ρ 0 , g, k, κ.Moreover, for any t > 0 we have (σ (t), v(t), q(t)) ∈ H k and satisfies where and is given by (2.30).
Proof We can easily establish the above conclusion by following the argument of Theorem 2.4 in [26], and thus the proof is omitted.

Nonlinear RT instability
This section is devoted to the proof of the nonlinear RT instability.We first construct approximate solutions to the CRT problem (1.8)-(1.10) in Sect.3.1.Then we formally derive error estimates between both the exact and approximate solutions Sect.3.2.Finally, making use of approximate solutions and the error estimates, we prove the existence of escape points, and thus complete the proof of Theorem 1.1 in Sect.3.3.

Construction of higher-order approximate solutions
In this section, we construct approximate solutions by using a similar method to [2] and make further energy estimates.Now we construct approximate solutions to (1.8), we choose and fix a ξ with λ = λ(ξ ), such that 0 < λ < . (3.1) We define T δ by where δ is an arbitrary small and positive parameter, θ is a small but fixed positive constant (independent of δ).
For s = 0, multiplying the equations (3.19) and (3.20) by φ j+1 , w j+1, respectively, and integrating over domain D, we arrive at As for the terms involving κ, using (3.19) and integrating by parts, we obtain Adding (3.26) and (3.22) together, then applying Hölder inequality and Cauchy-Schwarz's inequality to the resulting equation, coupled with the virtue of (3.17)-(3.18),we obtain Applying Gronwall's inequality, one finds that Clearly, ∂ t φ j+1 , ∂ t w j+1 and j+1 also satisfy (3.6), (3.9) and (3.10) for s = 0, we thus verify our lemma for s = 0.A similar argument works for s > 0. Now, assume that we have constructed all φ j , w j , q j for all 1 ≤ j ≤ N , we define Clearly where Compared to (3.4), we find that Noticing that R a N and S a N are the sum of all higher terms than N in the nonlinear part of the δ-expansion, which clearly satisfy (3.11).Thus the proof is complete.

Error estimates
This section is devoted to establish the error estimates between the exact solution (σ , v) to (1.8) and the approximate solution given by Lemma 3.1.To begin with, we shall recall the local existence of the CRT problem (1.8)-(1.10).Proposition 3.1 (Local existence) Assume that ρ 0 satisfies (1.4)- (1.5).For any given initial data (σ 0 , v 0 ) ∈ H 4 × H 3 and then there exists a T > 0 and a unique solution where T denotes the maximal time of existence of the solution (σ , v).
Proof We mention that the local existence of the strong solution to the incompressible Navier-Stokes-Korteweg equations has been established, see [60][61][62][63] for examples.In particular, by a slight modification of the arguments in [63] and using the expanding domain technique in [64], we can prove that there exists a unique strong solution (σ , v) ∈ C([0, T]; H In what follows, the notation a b means that a ≤ Cb for a universal constant C > 0, which may depend on some known physical parameters.C(ε 0 ) means that the positive constant C further depends on ε 0 .We define where α = (α 1 , α 2 ) denotes a multi-index of order |α| = α 1 + α 2 .
In addition, we list some classical Sobolev embedding results, which will be repeated used in later. ) ) Let (σ (t, x), v(t, x)) ∈ C([0, T]; H 4 × H 3 ) be a local solution as constructed in Proposition 3.1 and (σ a (t, x), v a (t, x)) the approximate solution provided by Lemma 3.1.Next, we shall establish the error estimate for (σ d , v d ), which is defined as follows Noticing that (σ d , v d ) satisfies the following equations: where R a N and S a N are defined in (3.4).Next, we make standard estimates for the error terms of density and velocity.
Using (3.36), (3.37), Hölder's and Young's inequalities, G 1 can be bounded as follows: To bound G 2 and G 3 , arguing in a way similar to that used for (3.43), thus, we can deduce from (3.42) that 1 2 We proceed to derive higher-order estimates of the difference of velocity.Applying ∂ β with |β| = 3 to (3.40) 2 , and multiplying the resulting identity by ∂ β v d in L 2 , we have 1 2 Using (3.37), Hölder's and Cauchy's inequalities, I 1 can be bounded as follows: Recalling the virtue of E ≤ w/2 < 1 and putting (3.46)-(3.48)into (3.49),we get Next we continue to derive more higher-order derivatives estimates of the difference v d of perturbation velocity.Multiplying (3.40) 2 by v d t in L 2 and recalling the virtue of div v d t = 0, one gets In particular, summing up the above three estimates, we conclude that Therefore, the second term on the right-hand side of (3.45) can be bounded as follows.
Using the embedding theorem, Hölder's, Cauchy-Schwarz's inequalities, and (3.55), we have Arguing in a way similar to that used above, it is easy to verify that To estimate the term I 12 in (3.45), we shall further rewrite I 12 as follows by using the divergence-free condition and integration by parts. - Inserting all the above estimates into (3.45),we conclude that Recalling Proposition 3.2 and the definition of ρ 0 , we known the two norms • s and • H s are equivalent.By applying a similar argument to the case 0 ≤ |β| < 3, and then using (3.44), we arrive at, for sufficiently small ε 0 .
This completes the proof.

Conclusion
This paper focuses on the Rayleigh-Taylor instability in the two-dimensional system of equations of inhomogeneous incompressible viscous fluids with capillarity effects in a horizontal periodic domain with infinite height.First, we use the modified variational method to construct (linear) unstable solutions for the linearized capillary Rayleigh-Taylor problem.Then, motivated by the Grenier's idea in [1], we further construct approximate solutions with higher-order growing modes to the capillary Rayleigh-Taylor problem and derive the error estimates between both the approximate solutions and nonlinear solutions of the capillary Rayleigh-Taylor problem.Finally, we prove the existence of escape points based on the bootstrap instability method of Hwang-Guo in [2], and thus obtain the nonlinear Rayleigh-Taylor instability result.Our instability result presents that the Rayleigh-Taylor instability can occur in the fluids with capillarity effects for any capillary coefficient κ > 0 if the critical capillary coefficient is infinite.In particular, it improves the previous Zhang's result in [3] with the smallness assumption of capillary coefficient.

Proof
We first establish some estimates of the difference σ d of perturbation density.Applying ∂ α with |α| = 4 to (3.40) 1 , and multiplying the resulting identity by ∂ α σ d in L 2 , we arrive at 1 2