Large time behavior for a compressible two-fluid model with algebraic pressure closure and large initial data

In this paper, we consider a compressible two-fluid system with a common velocity field and algebraic pressure closure in dimension one. Existence, uniqueness and stability of global weak solutions to this system are obtained with arbitrarily large initial data. Making use of the uniform-in-time bounds for the densities from above and below, exponential decay of weak solution to the unique steady state is obtained without any smallness restriction to the size of the initial data. In particular, our results show that degeneration to single-fluid motion will not occur as long as in the initial distribution both components are present at every point.


Introduction
In multi-component flows the presence of topologically complex interphase separating the components is a great difficulty from physical as well as mathematical point of view. However, in most of engineering applications precise description of motion of each component of interphase are not rarely needed and only the averaged macroscopic description is important. We will focus on the averaged two-component model derived in the monograph of Ishii and Hibiki in its inviscid form [14]. We refer the interested reader to [4] for concise overview of various modelling and mathematical aspects related to such models. In the present paper we immediately assume that the two components of the flow share a common velocity field and that their pressures are equal (algebraic pressure closure). We obtain the following system of partial differential equations in dimension one: ∂ t (α ± ̺ ± ) + ∂ x (α ± ̺ ± u) = 0, (1.1) where α + and α − are the volumetric rates of the two fluids; ̺ + and ̺ − are the two mass densities; u is the common velocity field, and µ > 0 is the viscosity coefficient. The two internal pressures are given by for adiabatic exponents γ ± > 1. Following [6], we introduce the notation and reformulate (1.1)-(1.5) to ∂ t R + ∂ x (Ru) = 0, (1.7) ∂ t Q + ∂ x (Qu) = 0, (1.8) ∂ t ((R + Q)u) + ∂ x ((R + Q)u 2 ) + ∂ x Z γ+ = µ∂ xx u. (1.9) Due to the algebraic closure (1.4), Z is an implicit function of R and Q interrelated by (1.11) The same model, but in semi-stationary Stokes regime, has been recently investigated by Bresch, Mucha and the third author in the three-dimensional setting. They proved the global-in-time existence of weak solutions without any restriction on the initial data. Similar result for the general Navier-Stokes system, with generalized equation of state was later obtained by Novotný and Pokorný [22]. Earlier results in this spirit concern existence of weak solutions to very particular two-component models including the fluid model of atmospheric flow with transport of potential temperature [19], and the hydrodynamic limit of Vlasov-Fokker-Planck system modelling suspension of the particles in the compressible fluid [24]. For other related results in case of one-dimensional two-fluid models, including density dependent coefficients or the socalled drift-flux model, we refer to [5], to the works of Evje et. al [9,10,11,12,13] and to the recent overview paper [25]. The present paper is, as far as we know, the first attempt to provide some more information about quantitative properties of weak solutions to this system. In order to investigate the large time behavior of solutions, we furthermore rewrite (1.7)-(1.9) in Lagrangian coordinates. To do this, we assume that, for definiteness, the fluids occupy the closed interval [0, 1] with no-slip boundary conditions and make the change of variables As a consequence, where we set τ := (R + Q) −1 .
The equations (1.12)-(1.14) are supplemented with the initial and boundary conditions as follows: and we denote This paper is mainly devoted to the large time behavior of weak solutions to (1.12)-(1.16) with large initial data. Existence, uniqueness and stability of weak solutions are obtained by making full use of the specific structure of the equations. Unlike in the three-dimensional regime [6,22], we prove the existence of weak solutions by approximation based on the strong solutions. Then the stability of weak solutions is verified by adapting the arguments for singlefluid equations [1,27]. The key step in the asymptotic analysis is to show uniform-in-time bounds on the densities from above and below. Due to the complicated form of the pressure, classical methods used in [3,15,16,20,23] cannot be applied here. However, thanks to the structure of the pressure, we are able to adapt the argument from [26] so as to obtain the twosided bounds; see Lemma 4.1. Based on these bounds, we show the exponential decay of weak solution by choosing suitable test functions in the momentum equation and making another use of the structure of the pressure.
The functional spaces we use are standard. For brevity, we denote by L p the Lebesgue space L p ((0, 1)) with the norm · L p , and by H 1 the Sobolev space H 1 ((0, 1)).
Before stating our main results, we specify the meaning of weak solutions. (1.20) A triple (R, Q, u) is said to be a weak solution of (1.12)-(1.16) on [0, 1] × [0, T ] provided that • 0 < R(y, t), Q(y, t) < ∞, a.e. in (0, 1) × (0, T ), (1.21) Clearly, the positive lower bound of Z 0 follows from (1.21) 2 and moreover R 0 < Z 0 a.e. in (0, 1) in accordance with (1.21) 1 . To get the upper bound of Z 0 we again make use of (1.21) 1 . Indeed, suppose on the contrary that Z 0 > max{2R 0 , which is a contradiction. Therefore Z 0 must be bounded from above, more precisely The main results of this paper are the following two theorems. The first one is concerned with the stability of weak solutions.
where C is a generic positive constant depending on T .
The second theorem gives the large time behavior of weak solutions. More precisely, we show the asymptotic decay of weak solutions to (R ∞ , Q ∞ , u ∞ ) -the unique steady state for problem (1.12)-(1.16) given implicitly by Here, C ⋆ is the positive constant uniquely determined by R 0 , Q 0 , γ ± and the conservation of mass (1.23) 5 .
Theorem 1.2 Let (R, Q, u) be the unique weak solution to (1.12)-(1.16) provided by Theorem 1.1. Then, for any t ≥ 0, it holds where C 1 and C 2 are generic positive constants independent of time.
Remark 1.2 Given suitably regular initial data, it can be shown, adapting the arguments from [26,17], that The rest of this paper is structured as follows. In Section 2.1 we show global existence and uniqueness of strong solutions to (1.12)-(1.16). In Section 2.2 we prove the existence of global weak solutions via approximation based on regular solutions corresponding to regularized initial data and the weak convergence method. In Section 3 we verify the stability of weak solutions. In Section 4, we obtain the exponential decay of weak solution to the unique steady state in L 2 -norm with large initial data.
Then there exists a unique global strong solution (R, Q, u) to (1.12)-(1.16). Furthermore, for any 0 < T < ∞, it holds that C −1 ≤ R(y, t), Q(y, t), Z(y, t) ≤ C, for any (y, t) Local-in-time existence and uniqueness of strong solutions to (1.12)-(1.16) is proved by the classical method based on the linearization of the problem and Banach fixed point theorem. We refer to [8,21] for similar calculations. Therefore, it remains to derive sufficient global a priori estimates so as to extend the local solution globally.
We start by giving the conservation of mass and the elementary energy inequality. To simplify the expression, we define In what follows, various positive constants are expressed by the same letter C depending on T . (2.9) Thus we can decompose the pressure as (2.10) Multiplying (1.14) by u and integrating by parts yields (2.11) The second term on the right-hand side of (2.11) can be computed, with the help of (2.10), through where we have used (2.6) for the second equality, and (2.9) in the fourth equality. Thus, combining (2.11) and (2.12) gives rise to (2.8). This finishes the proof of Lemma 2.1. By virtue of Lemma 2.1 and the specific mathematical structure of the equations, we are able to show the upper and lower bounds for R and Q. This plays a crucial role in the proof of Proposition 2.1. The idea of proof comes from [2]. Lemma 2.2 Let the assumptions of Lemma 2.1 be satisfied. Then, there exists a constant C > 0 such that (2.13) Proof. We observe first that the positiveness of R and Q follow from the method of characteristics and the regular initial data. Given positive R and Q, there exists a unique Z satisfying (1.10)-(1.11). This fact can be justified easily and we refer to Lemma 2.1 in [6] for the details.
That is, Z can be regarded as a function of R and Q. Moreover, due to (2.6) and (1.10), Z can be also seen as a function of y and τ .
In accordance with (2.7) and due to continuity of τ , for any t Integrating (1.14) with respect to time over (0, t), and then with respect to space over (a(t), y), followed by taking exponentials on both sides of the resulting equation, we arrive at where Next, in order to get the two-sided bounds from the representation formula (2.14), one needs the bounds for Y and B from above and below. Obviously, by (2.8) and Cauchy-Schwarz's inequality, it follows that It is also clear that For the purpose of obtaining the upper bound of Y , we rewrite (2.14) in another form. Noticing and integrating it with respect to time, we get Inserting the above relation back to (2.14) yields In light of (2.10), we have due to (2.8). Now we integrate both sides of (2.17) with respect to space over (0, 1), and make use of (2.7), (2.15) and (2.18) to conclude that A straightforward application of Gronwall's inequality gives And so, using (2.15) and (2.19), we deduce from (2.14) that It remains to get the upper bound of τ . To this end, we rewrite (1.10) as Differentiating both sides of (2.21) with respect to τ leads to or equivalently, The denominator is positive as we have due to (1.11), assumption about the smoothness of the solution, and Remark 1.1. Therefore, which means that the pressure is decreasing with respect to τ . From (2.14) and the two-sided bounds of Y (t), B(y, t), we see since τ ≥ τ due to (2.20) and the fact that the pressure is decreasing with respect to τ . Notice that Z(y, τ ) obeys Arguing as in Remark 1.1 one sees that Z(y, τ ) must be bounded from above with upper bound depending only on R 0 , Q 0 , τ . Thus, We are now in a position to prove that (R, Q, u) are more regular in order to finish the proof of Proposition 2.1. This process is quite standard due to Lemma 2.2. Thus the estimates are listed below and the details are omitted here and we refer to [2] for similar calculations.

Existence of weak solutions
The main task of this subsection is to construct global-in-time weak solutions to (1.12)-(1.16) using approximation based on regular solutions. We start from regularizing the initial data Therefore, it follows from Proposition 2.1 that there exists a unique global strong solution (R ε , Q ε , u ε ) to (1.12)-(1.16) with initial data (R ε 0 , Q ε 0 , u ε 0 ). Furthermore, from Proposition 2.1 we conclude the following uniform-in-ε estimates: From (2.30)-(2.32) it follows that there exists a subsequence of {(R ε , Q ε , u ε )} ε>0 , not relabeling, such that as ε → 0, The weak convergence results (2.33)-(2.35) are not sufficient to pass to the limit in (1.12)-(1.16), in particular, in the strongly nonlinear pressure function. For the moment we only know that Z ε is the unique solution of To identify the pressure term, it suffices to verify that the pointwise limit of {Z ε } ε>0 is the unique solution of for which we need the strong convergence of the sequence {Z ε } ε>0 . In fact, since . Therefore the strong convergence of {Z ε } ε>0 will follow from that of {τ ε } ε>0 . The necessary compactness property in space is provided by the following lemma.

39)
where ∆ h F (y) := F (y + h) − F (y) is the translation in spatial variable with the step h.
Proof. Similarly to Lemma 2.2, we need a representation formula of τ ε . By setting and recalling that (R ε , Q ε , u ε ) solves (1.12)-(1.14) in the strong sense, it follows that Multiplying the above identity both sides by exp − 1 µ t 0 σ ε (y, s)ds and integrating over time, one gets the relation By definition it holds that Thanks to (2.30), we have Indeed, the lower bound of Z ε follows readily from (2.30) and the relation R ε ≤ Z ε ; the upper bound of Z ε is verified by the relation and the two-sided bounds of R ε , Q ε through the argument as in Remark 1.1. The delicate issue is to compute ∆ h τ ε Z γ+ ε . In fact, Furthermore, as pointed out before, Z ε can be regarded as a function Z ε = Z ε (Q ε 0 , R ε 0 , τ ε 0 , τ ε ). Thus by the mean value theorem Subsequent differentiations of (2.21) with respect to Q ε 0 , R ε 0 , τ ε 0 , and τ ε give rise to In view of (2.23), (2.30) and (2.42), it follows that As a consequence, we conclude from (2.30)-(2.31) and (2.41)-(2.44) that for any 0 < h < 1, 0 < s < T . This particularly implies the strong convergence of {τ ε } ε>0 to τ in L 2 (0, T ; L 2 ) and furthermore in L p (0, T ; L p ) for any 1 ≤ p < ∞. Consequently, it holds that Recalling that Z ε = Z ε (Q ε 0 , R ε 0 , τ ε 0 , τ ε ), we find Z ε converges to some limit function Z almost everywhere. Upon passing to the limit in the relations we conclude from (2.42) that {Z ε } ε>0 converges to Z strongly in L p (0, T ; L p ) for any 1 ≤ p < ∞ and Z solves exactly This finishes the proof of existence of a weak solution.

Stability of weak solutions
In the present section, we show Lipschitz continuous dependence on the initial data of weak solutions, i.e., we prove our first main Theorem 1.1. We remark that the proof relies on the structure of the equations. As a preliminary step, we state the following lemma, the proof of which is omitted as it is similar to relevant results from [17,27]. where σ(y, t) := µ ∂ y u τ − Z γ+ (y, t), To verify the stability estimate (1.22) from Theorem 1.1, we follow the arguments in [1,17].
Let us start from introducing the following notation: Recalling that Q = Q 0 τ 0 τ −1 , R = R 0 τ 0 τ −1 , one has in light of uniform bounds for R, Q from below and from above, i.e., (2.36), that Consequently, in order to estimate L ∞ (0, T ; L ∞ )-norm of ∆R and ∆Q, it suffices to control ∆τ L ∞ (0,T ;L ∞ ) . This is the key step in proving stability of weak solutions. We follow the idea in [1,27] to accomplish this goal.
for any t ∈ (0, T ], where C denotes generic positive constant depending on T . Proof. It follows from (3.1) that Similarly to (2.42), it holds that Indeed, the upper bound of Z and Z is derived by the same argument as (2.42). Based on (2.36) and (3.6), we observe that where we used a version of (2.44) for the limit functions. Therefore, we deduce from (3.5)-(3.7) that (3.8) The rest of the proof follows the same lines as [17], and we write down the details only for the convenience of the reader. First, from the identity (3.2) and Hölder's inequality we obtain (3.9) It remains to bound ∆σ. Notice that we have and so, as for (3.7) we obtain It follows that ∆σ L 2 (0,t;L 2 ) ≤ C ∂ y (∆u) L 2 (0,t;L 2 ) + ∆R 0 L ∞ + ∆Q 0 L ∞ + t 0 ( (∂ y u)(·, s) L 2 + 1) ∆τ (·, s) L ∞ ds . (3.10) Since from (2.37) we deduce that T 0 ∂ y u 2 L 2 ds ≤ C, and therefore we can put together (3.9)-(3.10), and apply Gronwall's inequality to (3.8) to deduce (3.4). The proof of Lemma 3.2 is thus finished.
In order to use (3.4) to conclude (1.22), we need the estimates for ∆u. In fact, standard energy estimate for parabolic equation [7] gives Lemma 3.3 For any t ∈ (0, T ], it holds that (3.11) Having this, (1.22) follows by suitable combination of Lemmas 3.2-3.3. For the sake of brevity, we omit the details and refer the reader to [17] for similar steps. Clearly, (1.22) implies the uniqueness of weak solutions and so the proof of Theorem 1.1 is complete.

Large time behavior of weak solution
In this section, we show the exponential decay of weak solution in L 2 -norm. The classical methods to handle the large time behavior of the one-dimensional single-phase Navier-Stokes equations [16,20,23] are not readily applicable to our two-fluid model system. In [26], the author developed a new technique to treat one-dimensional viscous barotropic gas with nonmonotone pressure. Of great importance in [26] is to obtain the uniform-in-time bounds of the density from above and below. It turns out that the idea can be adapted to our two-fluid model. As a matter of fact, it has already been successfully adapted before to the case of one-dimensional non-resistive magnetohydrodynamic equations [17].

Two-sided bounds for R and Q
To begin with, we notice that the estimates in Lemma 2.1 are uniform-in-time. Then we have the following lemma, which is essential for the proof of Theorem 1.2. Throughout this section we use C and C i to denote generic positive constants independent of time. Proof. From (2.46) and the assumptions on the initial data (1.18)-(1.19) one sees that verification of (4.1) requires only to show the two-sided bounds for τ . By adapting the arguments in [26] (see also [17]), this follows from Lemma 2.1 and the three items below.
As a consequence, it remains to check that the three items above are satisfied. By the identity of pressure decomposition (2.10) and (2.46), it holds that where we have used the energy estimate (2.8). Clearly, we conclude from the definition of α, i.e., (2.5), and Jensen's inequality that Suppose now that τ is small, i.e., R + Q is large and we consider two possible cases. If R is large, then Z γ+ is also large due to R ≤ Z. If, on the other hand, Q is large, then also Z is large. Indeed, otherwise, we would arrive at a contradiction in the relation The third item is verified by using similar observation as above. We refer to [26] and Lemma 5.3 in [17] for the remaining details.

Remark 4.1
The key observations in Lemma 4.1 are as follows. Firstly, the pressure term may be seen as a function with variables y and τ by virtue of (1.10)-(1.11), tending to infinity as τ goes to zero and tending to zero as τ goes to infinity. Secondly, the two internal pressures satisfy γ-laws. This leads to a positive lower bound of the integral 1 0 τ Z γ+ dy; while the upper bound is obtained by the energy inequality. In this way, the arguments in [17,26] are naturally adapted.

Exponential decay
In this subsection, we prove the exponential decay of weak solution in L 2 -norm by adapting the ideas from [26,17]. It should be emphasized that the structure of pressure function is crucial for a modification of these arguments to work.
Step 5. Finally, the exponential decay of R − R ∞ L 2 and Q − Q ∞ L 2 is a direct consequence of (4.15) and the relations Q = Q 0 τ 0 τ −1 , R = R 0 τ 0 τ −1 ; The proof of Theorem 1.2 is complete.

Remark 4.2
We observe that the exponential decay of Z follows from that of τ . Indeed, in light of (4.6), (4.7) and (4.15).

Remark 4.3
The strategy adopted in this paper is strong enough to show existence, stability and exponential decay of global weak solution to two-fluid models with more general form of pressure considered for example in [22]. In particular, the two-fluid model with pressure satisfying γ-laws [18] could be included.