Global solutions to a one-dimensional non-conservative two-phase model

In this paper we investigate a basic one-dimensional viscous gas-liquid model based on the two-fluid model formulation. 
The gas is modeled as a polytropic gas whereas liquid is assumed to be incompressible. A main challenge with this model is the appearance of a non-conservative pressure term which possibly also blows up at transition to single-phase liquid flow (due to incompressible liquid). 
We investigate the model both in a finite domain (initial-boundary value problem) and in the whole space (Cauchy problem). 
We demonstrate that under appropriate smallness conditions on initial data we can 
obtain time-independent estimates which allow us to show existence and uniqueness of regular solutions as well as to gain insight into the long-time behavior of the model. These results rely strongly on the fact that we can derive appropriate upper and lower uniform bounds on the gas and liquid mass. In particular, the estimates guarantee that gas does not vanish at any point for any time when initial gas phase has a positive lower limit. The discussion of the Cauchy problem is general enough to take into account the possibility that the liquid phase may vanish at some points at initial time.

1. Introduction. A viscous version of the non-conservative equal-pressure twofluid model in one dimension can be stated as follows [18]:

STEINAR EVJE, HUANYAO WEN AND LEI YAO
This kind of two-fluid gas-liquid model plays a crucial role in the design of different flow systems applied by the oil and gas industry. When the model is used to study deepwater wellbore operations there are many challenging phenomena that can take place. Some of them are: (i) dynamic transition zones separating twophase and single-phase regions; (ii) strong expansion effects related to compressed gas which moves upwardly towards a lower pressure; (iii) complicated friction terms to take into account more realistic flow patterns; (iv) transition from one flow regime to another; (v) fluid flow between the wellbore and surrounding reservoir. A good understanding of mathematical properties of (1.1) is important both for increased insight into physical mechanisms that will dictate the behavior of the flow system as well as for the construction of reliable discrete versions of (1.1).
More generally, the application range of two-fluid based models is wide. Often, they are formulated similar to (1.1), however, without viscous terms in the momentum equations. On the other hand, it is well known that the resulting system may give rise to complex eigenvalues (loss of hyperbolicity), which has motivated researchers to include additional terms to make the model well-defined. For a discussion of different aspects of such two-fluid models and how to compute accurate solutions, we refer to [6,8,9] and references therein. We also refer to [14] for related studies of inviscid two-fluid models with incompressible fluids. Another variant of two-fluid models are obtained by assuming a pressure for each of the two phases leading to a two-pressure formulation. For a discussion of this and some results we refer to [12,19].
A different class of two-fluid based models that are commonly used in the industry is the so-called drift-flux model where the two momentum equations have been replaced by a mixture momentum equation. In order to have a closed system an algebraic equation which relates the two fluid velocities is added. This slip-relation is derived from laboratory experiments and will typically impose a stronger coupling between the two phases which is natural for flow systems where the two phases show a stronger mixing. For some mathematical results on the drift-flux model where viscous effects are included we refer to [10,20,11] and references therein.
To our knowledge few mathematical results can be found for models similar to (1.3). The most prominent is the recent work by Bresch et al [5]. In the study of the initial-boundary value problem, we employ techniques that first were used in the seminal works by Bresch and collaborators [4,5] for a two-fluid model and Li et al [15] for Navier-Stokes equations. A main difference between the one-dimensional model studied in [5] and the model we study is the pressure law. The model studied in [5] assumes two pressure-density relations of the form p = aρ γ . A challenge with the model we study is that the liquid is assumed to be incompressible. Hence, the pressure function becomes different. In particular, we must ensure that pressure remains well-defined by obtaining a strict upper bound on the liquid mass (i.e., the gas phase will not vanish at any point in the spatial domain). Another extension from [5] is that we consider the Cauchy problem.
We are interested in both existence and uniqueness of (1.3) as well as information about the long time behavior, both on a bounded and unbounded domain. A rough summary of the results presented in this work is as follows: To obtain the basic energy equality, especially for the initial value problem (Cauchy problem), we introduce an energy function G for the pressure function which is designed to fit the model problem we consider. It takes the following form: Hereρ g = ρ lñ ρ l −m , with ρ l >m > 0 andñ > 0, represents a positive gas reference density. The role ofm andñ is clarified in Theorem 2.1. For the Cauchy problem, m = 0 andρ g =ñ.
Then we introduce the following two quantities in order to characterize the initial state where ρ g0 = ρ l n0 ρ l −m0 . Appropriate smallness conditions on E 0 and E 1 will be necessary for the result we obtain in this work as summed up by Theorem 2.1-2.4. Note that for the initial condition (1.4), is replaced by m 0 u 2 l0 (since for this case m 0 is assumed to possess a positive lower limit).
More precisely, for the initial-boundary value problem (1.3)-(1.5) we will assume that initial masses m 0 and n 0 have positive lower and upper bounds (which are related tom andñ) and then show that we obtain uniform lower and upper bounds for m and n as expressed by Theorem 2.1. Hence, for this case neither gas nor liquid mass will vanish at any point in the spatial domain. We also obtain information about the long-time behavior for this model problem as described by Theorem 2.2. For the Cauchy problem (1.3) and (1.6) we allow the initial liquid mass m 0 to vanish whereas n 0 does not. We then obtain uniform lower and upper bounds on m and n as expressed by Theorem 2.3. In particular, we must take into account the possibility that the liquid mass may vanish at some points. Finally, a characterization of the long-time behavior for the Cauchy problem is given by Theorem 2.4.
For both model problems, a special challenge is that we must ensure that the gas volume fraction α g does not vanish at any point, i.e., sup x α l < 1. This is necessary in order to ensure that the pressure function P (m, n), given by (1.2), makes sense and is defined through the gas density ρ g . As far as this point is concerned a key estimate is the so-called BD entropy estimate developed by Bresch and Desjardins for Navier-Stokes equations, see for instance [1,2,3]. This estimate hangs on the fact that the viscosity coefficients take the form g = n and l = m. Combining the BD estimate with the smallness assumption on the initial data represented by E 0 and E 1 , we can obtain the desired uniform bounds on m and n, both for the initial-boundary value problem and the Cauchy problem. However, in order to deal with the Cauchy problem where the liquid mass may vanish we must introduce an appropriate approximate system, see (4.2), where a constant viscosity term is added in the liquid momentum equation combined with additional first order terms in both the liquid and gas momentum equation. This approximate system is designed to fit our gas-liquid model problem where the liquid is incompressible.
In order to study the long-time behavior we need more regularities of the solutions, which usually requires that the masses have positive lower bounds. To achieve this, for the initial-boundary value problem we assume that the initial masses are positive and then get a positive bound for both gas and liquid mass for all time with the help of smallness assumptions. Note that the mass-dependent viscosity coefficients l = m, g = n make it difficult to get the positive lower bounds when the initial data can be large. For the Cauchy problem, we still assume that the gas mass has a positive lower bound initially but assume that the initial liquid mass is non-negative, m 0 ≥ 0. The reason why the initial assumptions on m 0 are different is that we have Ω m = Ω m 0 for the bounded interval Ω but not for the case Ω = R if inf m 0 > 0. Thus for the Cauchy problem, we assume that m 0 ≥ 0 and m 0 ∈ L 1 (R). Then we still get Ω m = Ω m 0 which, combined with the estimate of ( √ m) x (due to the BD type estimate) and Sobolev inequality implies that m is bounded. It's still unknown if m has a positive lower bound for the Cauchy problem when inf m 0 > 0 even if the initial data is small, since in this case we do not have any information about m 0 or m 0 −m in some L p for some constantm. Fortunately, we can obtain an estimate of n −ñ in L 2 thanks to the pressure function, the basic energy estimate and m 0 ∈ L 1 . Hence, we can get a positive lower bound of n by combining the estimate of ( √ n) x and smallness assumptions. We refer to Lemma 4.4 for more details.
The paper is structured as follows. In Section 2, we present our main results which include the well-posedness and large-time behavior for the initial-boundary value problem and Cauchy problem. In Section 3, we give some a priori estimates globally in time and prove Theorems 2.1 and 2.2. In Section 4, based on some useful estimates of Section 3 uniformly for the size of the domain and T , the proofs of Theorems 2.3 and 2.4 are completed.
2. Main results. Throughout the rest of the paper, C (C(T )) denotes a generic positive constant depending on the initial data and the known constants (T ). For an integer n ≥ 0, H n denotes the Sobolev space H n (Ω) with the usual norm · H n and L 2 = H 0 ; L q denotes the Lebesgue space L q (Ω) with the usual norm · L q with 0 ≤ q ≤ ∞. We denote For the case L < ∞, the main results are stated as follows.
The main results for the Cauchy problem are stated as follows.
Then there exists a positive constantδ 2 given by (4.33) such that the Cauchy problem (1.3) and (1.6) has a global solution (m, n, u l , u g ) for any T > 0 in the sense that and that for any T > 0 and any test function Moreover, we have the following estimates uniformly for time.
3.1. Global existence. For any given L ∈ (0, ∞), we are going to use the classical strategy, i.e., local existence in combination with global a priori estimates in time, to prove the global existence. The local existence of the solutions as in Theorem 2.1 can be done by using the classical iteration arguments (see for instance [7]). We omit it for brevity. Thus, it suffices to obtain some a priori estimates globally in time.
Let's begin with an important proposition which is about the upper bound of liquid density m.
then the following estimate holds: Proof. Proposition 3.1 is an consequence of Lemma 3.3 below.
We start with the following basic energy estimate for (m, n, u l , u g ).
Proof. Multiplying (1.3) 3 and (1.3) 4 by u l and u g , respectively, then integrating the resulting equations with respect to x over (−L, L), and adding them up, we have In order to handle I 1 we introduce a new energy function designed for the pressure function in question, which is crucial, especially for the Cauchy problem: Then we have where we have used (1.3) 2 two times, (1.3) 1 and the fact that (α g ) t = −(α l ) t . This combined with the boundary condition concludes that Integrating (3.7) with respect to t over (0, t), we complete the proof of Lemma 3.1.
Based on ideas employed in [5] and a treatment of the pressure function similar to that used in Lemma 3.1, we get the next BD type of entropy estimate. The BD entropy estimate was first proposed by Bresch and Desjardins [1,2,3] in the context of compressible Navier-Stokes equations. This provides us with some useful information on higher regularity of the liquid and gas mass (m and n).
Summing (3.9) and the equation (1.3) 3 up yields Similarly, we have Multiplying (3.10) and (3.11) by u l + mx m and u g + nx n respectively, integrating with respect to x over (−L, L), and adding the resulting equalities, we have ) and the fact α l + α g = 1, I 2 can be handled as follows: Substituting the above equality into (3.12), we have (3.14) Integrating (3.14) over (0, t), we get Here is a corollary of Lemmas 3.1 and 3.2.
Corollary 3.1. Under the assumptions of Proposition 3.1, it holds that Note that from (3.16) we directly obtain an upper bound of √ m by using that Hölder's inequality. However, this may not guarantee that the pressure P (m, n) remains bounded. Hence, a finer upper bound of m, as expressed by Proposition 3.1, must be derived.
Proof. From Corollary 3.1 and (3.1), we have This combined with Hölder inequality and the liquid mass equation deduces Using the similar arguments as that in (3.18), we have

STEINAR EVJE, HUANYAO WEN AND LEI YAO
We are going to obtain H 1 estimates of (u l , u g ).
In the meantime (in view of (3.34), (3.41), (3.38), and (3.32)), we also obtain Similarly, we have Conclusion. Based on the a priori estimates globally in time as obtained in Section 3.1, it can be concluded that the maximal existence time of the solution T * = ∞. Thus, the proof of the global existence part of Theorem 2.1 is complete.

3.2.
Uniqueness. Now we are in a position to prove the uniqueness of the solutions as stated in Theorem 2.1. Let (m, n, u l , u g ) and (m,n,û l ,û g ) be the two solutions to (1.3)-(1.5) with the same initial data and the same boundary conditions. At first, using the two liquid mass equations Then, using the two liquid momentum equations we can obtain and On the other hand, multiplying (3.48) and (3.50) by (u l −û l ) and (u g −ū g ), respectively, then integrating the resulting equations with respect to x over (−L, L), and adding them up, we can obtain (3.52)

Combining (3.51) and (3.52) yields
which belongs to L 1 (0, T ) for any T < ∞. In view of Gronwall's inequality and the fact that the initial data of the two solutions is the same, we have

3.3.
Large-time behavior. In this subsection, we will explore the large time asymptotic behavior of the solution (m, n, u l , u g ) as defined in Theorem 2.1. The precise result is stated in Theorem 2.2.
Proof of Theorem 2.2. The equation (1.3) 4 can be rewritten as follows: Multiplying (3.53) byu g , and integrating the resulting equality with respect to x over (−L, L), we have Claim. g(t) → 0, as t → ∞.
(3.58) In fact, from (3.56) and the Poincaré-Sobolev inequality, we have (3.59) On the other hand, 4. Cauchy problem. In this section we focus on the Cauchy problem. Note that we will allow for the possibility that the initial liquid mass m 0 (x) may vanish at some points, as described by the assumptions stated in (2.6). In order to deal with this new situation we define an appropriate approximate system on the interval [−L, L], see (4.2)-(4.4). This is defined to fit the model problem we study where the pressure P (m, n) reflects that liquid is incompressible. In particular, since the initial gas n 0 has a positive lower limit, no additional constant viscosity term is added to the gas momentum equation, in contrast to what is done for the liquid momentum equation. However, an approximate first order term of the form 1/4 [n 2γ ] x is still included (in the gas momentum equation) and seems necessary in order to derive a BD type of estimate, see Lemma 4.2. Note that, clearly, an additional challenge with the Cauchy problem is to derive estimates that are independent of both L, , and T .

Existence.
4.1.1. An auxiliary theorem. In order to apply the conclusion of Theorem 2.1, we need to mollify the initial data as follows.
For sufficiently large L ∈ (1, ∞) and any fixed (L) ∈ (0, 1) We consider an approximate system of the Cauchy problem of (1.3) and (1.6): where α l = m ρ l , ρ g = ρ l n ρ l −m , and α g = ρ l −m ρ l , Along the lines of the proof of Theorem 2.1, one can get the following theorem about the global well posedness of (4.2)-(4.4) (the global solution (m L , n L , u L l , u L g ) is still denoted by (m, n, u l , u g )): Theorem 4.1. Under the assumptions of Theorem 2.3, there exists a positive constantδ 2 depending only on the initial data and some other known constants but independent of L and T , such that for given L, (4.2)-(4.4) has a global solution (m, n, u l , u g ) satisfying  7) and (4.9). Moreover, we have the following estimates uniformly for time.
Proof of Theorem 4.1. The local existence can be obtained by using similar arguments as in [7], we omit it here for brevity. Then we will obtain some a priori estimates (global in time) of the solution to the approximate system uniformly for L. In fact, some of them can be obtained by using the same arguments as in Section 3.1. Based on the local existence and these a priori estimates global in time, we can prove Theorem 4.1.
More precisely, we begin with an important Proposition which is used to get the upper bound of m. provided that E L 0 + E L 3,0 ≤δ 1 . The proof of this proposition is based on the next two lemmas.  Hereρ g L = ρ lñ ρ l − (L) .
Proof. Multiplying (4.2) 3 and (4.2) 4 by u l and u g , respectively, integrating the resulting equations with respect to x over (−L, L), and then using some similar arguments as in Lemma 3.1, we have Thus, we have l,x + nu 2 g,x ] = 0.  Proof. From (4.2) 1 , we obtain Summing (4.10) and (4.2) 3 up yields Similarly, we have Multiplying (4.11) and (4.12) by u l + mx m + mx m 2 and u g + nx n respectively, integrating the result over (−L, L), and using arguments similar to those of Lemmas 3.2 and 4.1, we have (4.13) For the second term on the right hand side of (4.13), we have (4.14) Let > 0 be sufficiently small such that Then substituting (4.14) into (4.13), we have (4.16) Integrating (4.16) over (0, t), we get (4.9). provided that E L 0 + E L 3,0 ≤δ 1 .
Proof. Following the calculations of Lemma 3.3 with m 0 replaced by m L 0 , we have provided that E L 0 + E L 3,0 ≤δ 1 .
The proof of Proposition 4.1 is complete. Based on it, we have the following corollary.  Then for any t ∈ [0, T ], there exist x 1 (t), x 2 ∈ (−L, L) such that This combined with Hölder inequality deduces This implies that Step 2. upper bound of n. Since n = ρg(ρ l −m) ρ l and G(n, ρ g ,ρ g L ) one can easily get ρ γ g ≤ C + CG(n, ρ g ,ρ g L ), (4.26) where C = C(ρ g , ρ l ,m, γ) and we have used Young inequality.
The next estimate is concerned about H 1 estimates of (u l , u g ). Proof. Multiplying (1.3) 4 by u g,t , integrating by parts over (−L, L), and using arguments similar to those of Lemma 3.5, we have (4.36) The next step is to handle the last term on the right hand side of (4.36).
Similar to Lemmas 3.6 and 3.7, we get the following estimates for the approximate system  Note that some of the estimates related to the liquid phase depend on . This is due to the fact that the lower bound on m has been used.
Conclusion. Based on these global a priori estimates in time in Section 4, it concludes that the maximal existence time T * = ∞. Thus, the proof of Theorem 4.1 is complete.