On the Rayleigh-Taylor instability for the compressible non-isentropic inviscid fluids with a free interface

In this paper, we study the Rayleigh-Taylor 
instability phenomena for two compressible, immiscible, inviscid, 
ideal polytropic fluids. Such two kind of fluids always evolve 
together with a free interface due to the uniform gravitation. We 
construct the steady-state solutions for the denser fluid lying 
above the light one. With an assumption on the steady-state 
temperature function, we find some growing solutions to the related 
linearized problem, which in turn demonstrates the linearized 
problem is ill-posed in the sense of Hadamard. By such an 
ill-posedness result, we can finally prove the solutions to the 
original nonlinear problem does not have the property EE(k). 
Precisely, the $H^3$ solutions to the original nonlinear 
problem can not Lipschitz continuously depend on their initial data.

where R ± are positive constants, and e ± , θ ± , γ ± denote the inertial energy, the absolute temperature and the adiabatic exponent. Both normal velocity and pressure are assumed to be continuous across the interface ( [14]), thus it is natural to impose the following boundary conditions on the free interface Σ(t): (n · u + )| Σ(t) − (n · u − )| Σ(t) = 0, P + | Σ(t) − P − | Σ(t) = 0, (1.2) where n denotes the unit normal vector of the interface Σ(t) and f | Σ is the trace of function f on Σ(t). And the impenetrable boundary conditions are imposed on the fixed boundaries {x 3 = −m, l}: We first construct the steady-state solutions with u ± = 0 and the interface being located on {x 3 = 0} for all t ≥ 0. It is easy to find that (ρ ± , θ ± ) satisfy dR ± ρ ± θ ± dx 3 = −gρ ± , (1.4) or . (1.5) here denotes the derivative with respect to x 3 variable.
To demonstrate the Rayleigh-Taylor instability, we always assume the steady density ρ ± (x 3 ) satisfy ρ + > ρ − on the steady interface {x 3 = 0}. Once some suitable temperature function is given, it is not hard to obtain the related density function by solving (1.4) or (1.5). For example, we take θ ± = 0, which is actually the isothermal case considered in [3]. Of course, it should be guaranteed that the steady-state solutions ρ ± and θ ± are always positive in the whole interval (−m, l). In order to ensure the Rayleigh-Taylor instability phenomena indeed occur, we impose a condition (2.19) in this paper.
Without loss of generality, we assume that m = l = 1. To overcome the difficulties caused by the free interface, it is convenient to introduce the Lagrangian coordinates. We define the fixed Lagrangian domains Ω + = R 2 × (0, 1) and Ω − = R 2 × (−1, 0) and assume that there exist invertible mappings η 0 ± : Ω ± → Ω ± (0), such that Σ 0 = η 0 + ({x 3  ∂ t η ± = u ± (η ± (x, t), t) η ± (x, 0) = η 0 ± (x). (1.6) Then, we denote the Eulerian coordinates as (y, t) with y = η(x, t). In addition, the upper and lower fluids may slip across each other on the interface, and the slip operator is defined by We introduce the unknown functions in Lagrangian coordinates and denote (η ± , q ± , v ± , κ ± ) by (η, q, v, κ) without causing any confusion. Then the equations of (η, q, v, κ) can be written as where A = (( ∂η ∂x ) T ) −1 , tr(·) denotes matrix trace and Dv is the Jacobian matrix of v. The jumping conditions in Lagrangian coordinates are in turn where the unit normal vector n is defined by The fixed boundary conditions are In the remainder of this section, we shall give a description of the main results and recall some related topics. In the next section, when the steady-state solutions η = Id, q =ρ, κ =θ, v = 0 satisfy the assumptions that ρ + | x3=0 > ρ − | x3=0 and θ ± + g R ± (1 − 1 γ ± ) ≥ 0, we can prove the ill-posedness theorem of the related linearized problem of (1.8)-(1.10) around (Id,ρ, 0,θ). In section 3, based on the illposedness results for the linearized problems, we show the solutions to the original nonlinear problem (1.8)-(1.10) do not have the property EE(k). Precisely speaking, the H 3 -norms of the solutions to nonlinear problems can not Lipschitz continuously depend on their initial data in any time interval, provided that the solutions indeed exist in the Sobolev space H 3 .
The studies on the Rayleigh-Taylor instability began from the pioneering work due to Rayleigh and Taylor in [9,10,11]. From then on, many interesting physical phenomena and numerical simulations come from both physical and numerical experiments. We refer to [8] and references therein for general discussion of the physics about Rayleigh-Taylor instability. However, there are only very few analytical results from the mathematical point of view. Only very recently, Guo and Tice established the ill-posedness theory of both linearized problems and nonlinear problems for the compressible isentropic Euler equations with a free interface in [3] by the variational methods. And the dynamical Rayleigh-Taylor instability is considered for the density-dependence Euler equations for an incompressible fluid in a strip by Guo and Hwang in [4], where the background profiles are smooth. When the viscosity or the heat-conductivity is considered, the natural variational structures for the linearized problems are destroyed. Then, Guo and Tice developed a modified variational method in [2], and this method is applied for the MHD in [1,5,13] and other models [6,7,12] successfully.
Here, we deal with the compressible non-isentropic Euler equations for the ideal polytropic fluids. The effect of temperature field on the motion of fluids should be considered. We give a characterization of temperature function to ensure the Rayleigh-Taylor instability occurs. It should be remarked that the general equations of state replacing the polytropic fluids case can also be studied similarly. When the steady-state temperature is a uniform constant function, then it is reduced into the isothermal case considered in [3]. In principle, we adopt the variational methods developed in [3]. However, we need to overcome the difficulties caused by temperature field. Consequently, we propose an condition on steady temperature function θ to guarantee the Rayleigh-Taylor instability. Of course, it is more interesting to find some stabilization effect of temperature field, which is left for future study.
(2.2) Plugging (2.2) into (2.1) leads to The fixed boundary conditions are Since the coefficients in the linear equations (2.5) depend only on x 3 variable, we can adopt the horizontal Fourier transformation to (2.5) to reduce them into ordinary differential equations in term of x 3 with each spatial frequency as parameters. In the following we will denote the horizontal Fourier transformation of a function f byf or F(f ), which is the form and Then we obtain after applying the horizontal Fourier transformation to (2.5) that Assuming (φ, ζ, ψ) solves (2.8) for ξ 1 , ξ 2 and λ, then for any rotation operator R ∈ SO(2) (φ,ζ) = R(φ, ζ) solves the same equations for (ξ 1 ,ξ 2 ) = R(ξ 1 , ξ 2 ) with ψ and λ being unchanged. Hence we can choose suitable rotation operator R r ∈ SO(2) so that Therefore, it follows from (2.8) and (2.9) that ζ = 0 and Similarly, applying the horizontal Fourier transformation to (2.6) and (2.7) yields In what follows, for λ > 0, a variational principle gives a solution (φ, ψ) to (2.10)-(2.11), then using the inverse operator of the rotational operator R r , a solution (φ, ζ, ψ) to (2.8) is obtained. Then by the inverse Fourier transformation, the growing solutions to the linearized problem (2.1) along with the boundary conditions (2.6)-(2.7) can be constructed. Moreover, we show that λ depends on the frequency variable |ξ|, and λ(|ξ|) tends to +∞ as |ξ| goes to +∞. Therefore, the linearized equations are proved to be ill-posed in the sense of Hadamard.

Constrained minimization formulation.
To build the variational framework, we multiplyφ,ψ to (2.10) 1 and (2.10) 2 respectively to get after integration by parts that where we have used the boundary conditions (2.11). Note that −λ 2 ∈ R, then it follows that Re(φ), Re(ψ) are also solutions to (2.10)-(2.11). So we restrict ourselves to finding the real solutions to (2.10)-(2.11). For any |ξ| > 0, let and (2.14) Now we show the infimum value of E is negative with the constrain that J = 1. Moreover, it achieves its infimum value at some (φ, ψ). Then we prove the infimum value point (φ, ψ) is indeed a solution to (2.10)-(2.11). It is convenient to define the following functions space for any given |ξ| > 0. It is necessary to construct a function ψ ∈ H 1 0 , so that Integration by parts gives that Then ψ(0) = 0.

Proposition 2.1. E achieves its infimum in A.
Proof. For any (φ, ψ) ∈ A, it holds that
In the rest of this subsection, the growing solution to (2.1) will be obtained from the solution to (2.10) by using the inverse Fourier synthesis.
where (φ, ζ, ψ) are solutions to (2.8). (2.36) Then (η, v, q, κ) is a solution to (2.1) with the boundary conditions (2.6) and (2.7). For every k ∈ N, we have the estimates For every t > 0, we have (η(t), v(t), q(t), κ(t)) ∈ H k (Ω ± ) and (2.38) Proof. The existence of the solution (φ, ζ, ψ) to (2.8) are guaranteed by the solution to (2.10) and a suitable choice of the rotation operator SO(2). One can refer to Corollary 3.9 in [3]. And the proof of Theorem 2.2 is based on the properties of the Fourier transformation and the estimates of the eigenvalue λ(|ξ|) in Remark 2.1 and Lemma 2.3, which is similar as that of Theorem 3.10 in [3].
In addition, we can obtain the similar equality in Ω − = R 2 × (−1, 0) with the opposite sign on the right hand side. Then, summing up the two equalities leads to (2.45) The first term on the right hand side of (2.45) disappears due to jumping conditions (2.42). Then, the proof of Lemma 2.5 is completed.