On instability of a generic compressible two-fluid model in $\mathbb R^3$

We are concerned with the instability of a generic compressible two-fluid model in the whole space $\mathbb{R}^3$, where the capillary pressure $f(\alpha^-\rho^-)=P^+-P^-\neq 0$ is taken into account. For the case that the capillary pressure is a strictly decreasing function near the equilibrium, namely, $f'(1)<0$, Evje-Wang-Wen established global stability of the constant equilibrium state for the three-dimensional Cauchy problem under some smallness assumptions. Recently, Wu-Yao-Zhang proved global stability of the constant equilibrium state for the case $P^+=P^-$ (corresponding to $f'(1)=0$). In this work, we investigate the instability of the constant equilibrium state for the case that the capillary pressure is a strictly increasing function near the equilibrium, namely, $f'(1)>0$. First, by employing Hodge decomposition technique and making detailed analysis of the Green's function for the corresponding linearized system, we construct solutions of the linearized problem that grow exponentially in time in the Sobolev space $H^k$, thus leading to a global instability result for the linearized problem. Moreover, with the help of the global linear instability result and a local existence theorem of classical solutions to the original nonlinear system, we can then show the instability of the nonlinear problem in the sense of Hadamard by making a delicate analysis on the properties of the semigroup. Therefore, our result shows that for the case $f'(1)>0$, the constant equilibrium state of the two-fluid model is linearly globally unstable and nonlinearly locally unstable in the sense of Hadamard, which is in contrast to the cases $f'(1)<0$ and $P^+=P^-$ (corresponding to $f'(1)=0$) where the constant equilibrium state of the two--fluid model is nonlinearly globally stable.


Introduction.
1.1. Background and motivation. As is well-known, most of the flows in nature are multi-fluid flows. Such a terminology includes the flows of non-miscible fluids such as air and water; gas, oil and water. For the flows of miscible fluids, they usually form a "new" single fluid possessing its own rheological properties. One interesting example is the stable emulsion between oil and water which is a non-Newtonian fluid, but oil and water are Newtonian ones.
One of the classic examples of multi-fluid flows is small amplitude waves propagating at the interface between air and water, which is called a separated flow. In view of modeling, each fluid obeys its own equation and couples with each other through the free surface in this case. Here, the motion of the fluid is governed by the pair of compressible Euler equations with free surface: (1.2) In above equations, ρ 1 and v 1 represent the density and velocity of the upper fluid (air), and ρ 2 and v 2 denote the density and velocity of the lower fluid (water). p i denotes the pressure. −gρ i e 3 is the gravitational force with the constant g > 0 the acceleration of gravity and e 3 the vertical unit vector, and F D is the drag force. As mentioned before, the two fluids (air and water) are separated by the unknown free surface z = η(x, y,t), which is advected with the fluids according to the kinematic relation: When the wave's amplitude becomes large enough, wave breaking may happen. Then, in the region around the interface between air and water, small droplets of liquid appear in the gas, and bubbles of gas also appear in the liquid. These inclusions might be quite small. Due to the appearances of collapse and fragmentation, the topologies of the free surface become quite complicated and a wide range of length scales are involved. Therefore, we encounter the situation where two-fluid models become relevant if not inevitable. The classic approach to simplify the complexity of multi-phase flows and satisfy the engineer's need of some modeling tools is the well-known volume-averaging method (see [9,16] for details). Thus, by performing such a procedure, one can derive a model without surface: a two-fluid model. More precisely, we denote α ± by the volume fraction of the liquid (water) and gas (air), respectively. Therefore, α + + α − = 1. Applying the volume-averaging procedure to the equations (1.1) and (1.2) leads to the following generic compressible two-fluid model: ∂ t (α ± ρ ± ) + div(α ± ρ ± u ± ) = 0, ∂ t (α ± ρ ± u ± ) + div (α ± ρ ± u ± ⊗ u ± ) + α ± ∇P ± = −gα ± ρ ± e 3 ± F D . (1.4) We have already discussed the case of water waves, where a separated flow can lead to a two-fluid model from the viewpoint of practical modeling. As mentioned before, two-fluid flows are very common in nature, but also in various industry applications such as nuclear power, chemical processing, oil and gas manufacturing. According to the context, the models used for simulation may be very different. However, averaged models share the same structure as (1.4). By introducing viscosity effects and capillary pressure effects, one can generalize the above system (1.4) denote the densities, the velocities of each phase, and the two pressure functions, respectively.γ ± ≧ 1, A ± > 0 are positive constants. In what follows, we set A + = A − = 1 without loss of any generality. As in [6], we assume that the capillary pressure f belongs to C 3 ([0, ∞)). Moreover, τ ± are the viscous stress tensors τ ± := µ ± ∇u ± + ∇ t u ± + λ ± div u ± Id, (1.6) where the constants µ ± and λ ± are shear and bulk viscosity coefficients satisfying the physical condition: µ ± > 0 and 2µ ± + 3λ ± ≧ 0, which implies that µ ± + λ ± > 0. For more information about this model, we refer to [1-3, 7, 9, 16, 17] and references therein. However, it is well-known that as far as mathematical analysis of two-fluid model is concerned, there are many technical challenges. Some of them involve, for example: • The two-fluid model is a partially dissipative system. More precisely, there is no dissipation on the mass conservation equations, whereas the momentum equations have viscosity dissipations; • The corresponding linear system of the model has zero eigenvalue, which makes mathematical analysis (well-posedness and stability) of the model become quite difficult and complicated; • Transition to single-phase regions, i.e, regions where the mass α + ρ + or α − ρ − becomes zero, may occur when the volume fractions α ± or the densities ρ ± become zero; • The system is non-conservative, since the non-conservative terms α ± ∇P ± are involved in the momentum equations. This brings various mathematical difficulties for us to employ methods used for single phase models to the two-fluid model. For the case that the capillary pressure is a strictly decreasing function near the equilibrium, namely, f ′ (1) < 0, Evje-Wang-Wen [6] obtained global stability of the constant equilibrium state for the threedimensional Cauchy problem of the two-fluid model (1.5) under the assumption that the initial perturbation is small in H 2 -norm and bounded in L 1 -norm. It should be noted that as pointed out by Evje-Wang-Wen in [6], the assumption f ′ (1) < 0 played a crucial role in their analysis and appeared to have an essential stabilization effect on the model in question. Bretsch et al. in the seminal work [2] considered a model similar to (1.5). More specifically, they made the following assumptions: • P + = P − (particularly, f ′ (1) = 0 in this case); • inclusion of viscous terms of the form (1.2) where µ ± depends on densities ρ ± and λ ± = 0; • inclusion of a third order derivative of α ± ρ ± , which are so-called internal capillary forces represented by the well-known Korteweg model on each phase.
They obtained the global weak solutions in the periodic domain with 1 < γ ± < 6. It is worth mentioning that the method of [2] doesn't work for the case without the internal capillary forces. Later, Bresch-Huang-Li [3] established the global existence of weak solutions in one space dimension without the internal capillary forces when γ ± > 1 by taking advantage of the one space dimension. However, the method of [3] relies crucially on the advantage of one space dimension, and particularly cannot be applied for high dimensional problem. Recently, Wu-Yao-Zhang [18] showed the global stability of the constant equilibrium state in three space dimension by exploiting the dissipation structure of the model (with P + = P − and without internal capillary forces) and making full use of several key observations. For the case of the special density-dependent viscosities with equal viscosity coefficients and the case of general constant viscosities, Cui-Wang-Yao-Zhu [4] and Li-Wang-Wu-Zhang [14] proved the global stability of the constant equilibrium state for the three-dimensional Cauchy problem with the internal capillary forces, respectively. To sum up, the works [6] and [18] rely essentially on the assumption f ′ (1) < 0 and P + = P − (corresponding to f ′ (1) = 0). Therefore, a natural and important problem is that what will happen for the case that the capillary pressure is a strictly increasing function near the equilibrium, namely, f ′ (1) > 0. That is to say, what about the stability of three-dimenional Cauchy problem to the two-fluid model (1.5) with f ′ (1) > 0. The main purpose of this work is to give a definite answer to this issue. More precisely, we first employ Hodge decomposition technique and make detailed analysis of the Green's function for the corresponding linearized system to construct solutions of the linearized problem that grow exponentially in time in the Sobolev space H k , thus leading to a global instability result for the linearized problem. Then, based on the global linear instability result and a local existence theorem of classical solutions to the original nonlinear system, we can prove the instability of the nonlinear problem in the sense of Hadamard by making a delicate analysis on the properties of the semigroup. Therefore, our result shows that for the case f ′ (1) > 0, the constant equilibrium state of the two-fluid model (1.5) is linearly globally unstable and nonlinearly locally unstable in the sense of Hadamard, which is in contrast to the cases f ′ (1) < 0 ( [6]) and P + = P − (corresponding to f ′ (1) = 0) ( [18]) where the constant equilibrium state of the two-fluid model (1.5) is nonlinearly globally stable.
1.2. New formulation of system (1.5) and Main Results. In this subsection, we devote ourselves to reformulating the system (1.5) and stating the main results. To begin with, noting the relation between the pressures of (1.5) 3 , one has where P ± := P ± (ρ ± ) . It is clear that Here s ± represent the sound speed of each phase respectively. Motivated by [2], we introduce the fraction densities which together with the fact that α + + α − = 1 gives By virtue of (1.7) and (1.9), we finally get Substituting (1.10) into (1.9), we deduce the following expressions: which together with (1.7) gives the pressure differential dP ± Next, by noting the fundamental relation: α + + α − = 1, we can get the following equality: Then, it holds from the pressure relation (1.5) 3 that Thus, we can employ the implicit function theorem to define ρ + . To see this, by differentiating the above equation with respect to ρ + for given R + and R − , we get which is positive for any ρ + ∈ (R + , +∞) and R ± > 0. This together with the implicit function theorem implies that ρ + = ρ + (R + , R − ) ∈ (R + , +∞) is the unique solution of the equation (1.12). By virtue of (1.8), (1.12) and the fundamental fact that α + + α − = 1, ρ − and α ± can be defined by We refer the readers to [ [3], P. 614] for more details.
Therefore, we can rewrite system (1.5) into the following equivalent form: (1. 13) In the present paper, we consider the initial value problem to (1.13) in the whole space R 3 subject to the initial condition where R ± ∞ > 0 denote the background doping profile, and for simplicity, are taken as 1 in this paper. In this work, we investigate the instability of the constant equilibrium state for the Cauchy problem (1.13)- (1.14) in the case that f ′ (1) > 0, which should be kept in mind throughout the rest of this paper. Taking , and the nonlinear terms are given by .
Taking change of variables by and setting the Cauchy problem (1.13) and (1.14) can be reformulated as subject to the initial condition where the nonlinear terms are given by and Noticing that it is clear that β + β − < 1. Before stating our main results, let us state the corresponding linearized system of (1.25) as follows: (1.28) Now, we are in a position to state our main results. The first one is concerned with the linear instability, which is stated in the following theorem.
Then for any ϑ > 0, the linearized system (1.28) admits an Moreover, the solution satisfies the following estimate: The second result is concerned with nonlinear instability, which is stated in the following theorem.
, such that for any ε ∈ (0, ε 0 ) and the initial data For the proof of Theorem 1.1, we need construct a solution to the linearized system (1.28) that has a growing H k norm for any k and the proof can be outlined as follows. First, we exclude the stabilizing part of the linearized system by employing the Hodge decomposition technique firstly introduced by Danchin [5] to split the linearized system into three systems (see (2.1) and (2.2) for details). One is a 4 × 4 system and its characteristic polynomial possesses four distinct roots, the other two systems are the heat equation. This key observation allows us to construct an unstable solution. Second, we assume a growing mode ansatz, i.e., for some λ , and submit this ansatz into the Fourier transformation of the 4 × 4 system to get a time-independent system for λ . Third, we solve the time-independent system by making careful analysis and using several key observations. Indeed, noticing that the characteristic polynomial F(λ ) defined in (2.6) is a strictly increasing function on (0, ∞), and F(θ ) > 0 for θ > 0 defined in Theorem 1.1, we show that 0 < λ 1 < θ is the unique positive root of the characteristic equation F(λ ) = 0, and θ > 0 in Theorem 1.1 is the largest possible growth rate since Re(λ i ) ≤ θ with 1 ≤ i ≤ 4. Therefore, the growing mode constructed in Theorem 1.1 actually does grow in time at the fastest possible rate.
For the proof of Theorem 1.3, we deduce the nonlinear instability. Compared to [8,[10][11][12]19] where nonlinear energy estimates and a careful bootstrap argument are employed to prove stability and instability, we need to develop new ingredients in the proof to handle with the difficulties arising from the strong interaction of two fluids, which requires some new thoughts. Indeed, since the strong coupling terms are involved in the right-hand of the system (1.25), it seems impossible to follow the energy methods of [8,[10][11][12]19] to get the lyapunov-type inequality: d dt E (t) ≤ θ E (t) to prove the largest possible growth rate. Therefore, we must pursue another route by resorting to semigroup methods to capture the largest possible growth rate, but the cost is that we need the higher regularity of the solutions. More precisely, with the help of the global linear instability result of Theorem 1.1 and a local existence theorem of classical solutions to the original nonlinear system, we can make delicate spectral analysis for the linearized system and apply Duhamel's principle to prove the nonlinear instability result stated in Theorem 1.1.

Notations and conventions.
Throughout this paper, we denote H k (R 3 ) by the usual Sobolev spaces with norm · H k and denote L p , 1 ≤ p ≤ ∞ by the usual L p (R 3 ) spaces with norm · L p . We drop the domain R 3 in integrands over R 3 . For the sake of conciseness, we do not precise in functional space names when they are concerned with scalar-valued or vector-valued functions, ( f , g) X denotes f X + g X . We will employ the notation a b to mean that a ≤ Cb for a universal constant C > 0 that only depends on the parameters coming from the problem. We denote ∇ Let Λ s be the pseudo differential operator defined by where f and F( f ) are the Fourier transform of f .

Linear instability.
To construct a solution to the linearized system (1.28) that has growing H k -norm for any positive integer k, by using a real method as in [13], one need to make a detailed analysis on the properties of the semigroup. To exclude the stabilizing part, we will employ the Hodge decomposition technique firstly introduced by Danchin [5] to split the linear system into three systems. One only has four equations and its characteristic polynomial possesses four distinct roots, the other two systems are the heat equation. This key observation allows us to construct a unstable solution. To see this, let ϕ ± = Λ −1 divũ ± be the "compressible part" of the velocitiesũ ± , and denote φ ± = Λ −1 curlũ ± (with (curlz) j i = ∂ x j z i − ∂ x i z j ) by the "incompressible part" of the velocitiesũ ± . Setting ν ± = ν ± 1 + ν ± 2 , the system (1.28) can be decomposed into the following three systems: To construct a solution to the linearized equations (2.3) that has growing H k -norm for any k, we shall make a growing normal mode ansatz of solutions, i.e., After a series of tedious but direct calculations, we can conclude from (2.4) that (2.5) Therefore, the system (2.4) has non-zero solutions if the characteristic equation has a real characteristic root.
Moreover, the following estimate holds Proof. Employing the similar argument of Taylor series expansion as in [15], then (2.7) follows from some tedious but direct calculations. It is noticed that F(λ ) is a strictly monotonically increasing function if λ > 0. Furthermore, (2.8) holds and the proof of lemma is completed.
Proof. Set φ ± ≡ 0. As the definition of ϕ ± and φ ± , and the relatioñ it is easy to prove that (ñ + ,ũ + ,ñ − ,ũ − ) is a solution of (1.29). Moreover, in virtue of Plancherel theorem, we have if η is large enough. Performing the similar procedures, we can prove ũ ± (t) L 2 ≤ e θt ũ ± 0 L 2 and The proof of proposition is complete.

SPECTRAL ANALYSIS AND LINEAR L 2 -ESTIMATES
In this section, we are devoted to deriving the linear L 2 -estimates, by using a real method as in [15], one need to make a detailed analysis on the properties of the semigroup.
3.1. Spectral analysis for system (2.1). We consider the Cauchy problem of (2.1) with the initial data In terms of the semigroup theory, we may represent the IVP (2.1) and (3.1) for U = (ñ + , ϕ + ,ñ − , ϕ − ) t as where the operator B 1 is defined by Taking the Fourier transform to the system (3.2), we obtain where U (ξ ,t) = F(U (x,t)) and A 1 (ξ ) is given by We compute the eigenvalues of the matrix A 1 (ξ ) from the determinant which is the same as characteristic equation (2.6) and implies that the matrix A 1 (ξ ) possesses four different eigenvalues: Consequently, the semigroup e tA 1 can be decomposed into where the projector P i (ξ ) is defined by Thus, the solution of IVP (3.3) can be expressed as To derive long time properties of the semigroup e tA 1 in L 2 -framework, one need to analyze the asymptotical expansions of λ i , P i (i = 1, 2, 3, 4) and e tA 1 (ξ ) . Employing the similar argument of Taylor series expansion as in [15], we have the following lemmas from tedious calculations.
3.2. Spectral analysis for system (2.2). We consider the Cauchy problem of (2.2) with the initial data for any t ≥ 0.
We consider the Cauchy problem of (1.28) with the initial data By virtue of the definition of ϕ ± and φ ± , and the fact that the relations involve pseudo-differential operators of degree zero, the estimates in space H k (R 3 ) for the original functionũ ± will be the same as for (ϕ ± , φ ± ). Combining Propositions 3.4 and 3.5, we have the following result concerning long time properties for the solution semigroup e tA .

NONLINEAR INSTABILITY
We mention that the local existence of strong solutions to a generic compressible two-fluid model can be established by using the standard iteration arguments as in [20] whose details are omitted. We can arrive at the following conclusion: Proposition 4.1. Assume that the notations and hypotheses in Theorem 1.3 are in force. For any given initial data n + 0 , u + 0 , n − 0 , u − 0 ∈ H 4 (R 3 ) satisfying inf x∈R 3 {n ± 0 + 1} > 0, there exist a T > 0 and a unique strong solution (n + , u + , n − , u − ) ∈ C 0 ([0, T ]; H 4 (R 3 )) to the Cauchy problem (1.26)-(1.27). Moreover, the strong solution satisfies where E (t) = (n + , u + , n − , u − ) (t) H 4 .