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Review on mathematical analysis of some two-phase flow models
Acta Mathematica Scientia 2018,38B(5):1617–1636 http://actams.wipm.ac.cn
REVIEW ON MATHEMATICAL ANALYSIS OF SOME TWO-PHASE FLOW MODELS∗
§)
Huanyao WEN (
1
[)
Lei YAO (
2
Á )
Changjiang ZHU (
1†
1. School of Mathematics, South China University of Technology, Guangzhou 510641, China 2. School of Mathematics and Center for Nonlinear Studies, Northwest University, Xi’an 710127, China E-mail : [email protected]; [email protected]; [email protected]
In Memory of Professor Xiaqi DING Abstract The two-phase flow models are commonly used in industrial applications, such as nuclear, power, chemical-process, oil-and-gas, cryogenics, bio-medical, micro-technology and so on. This is a survey paper on the study of compressible nonconservative two-fluid model, drift-flux model and viscous liquid-gas two-phase flow model. We give the research developments of these three two-phase flow models, respectively. In the last part, we give some open problems about the above models. Key words
compressible nonconservative two-fluid model; drift-flux model; viscous liquidgas two-phase flow model; well-posedness
2010 MR Subject Classification
1 1.1
76T10; 76N10
Compressible Nonconservative Two-fluid Model Research background-1
One of the typical two-phase flow models, called compressible nonconservative two-fluid model, is ± ± ± ± ± ∂t (α ρ ) + div(α ρ u ) = 0, (1.1) ∂t (α± ρ± u± ) + div(α± ρ± u± ⊗ u± ) + α± ∇P ± (ρ± ) = div(α± τ ± ) + σ ± α± ρ± ∇△(α± ρ± ) − f ± u± + I(u+ − u− ) + α± ρ± g.
The variable α+ , α− ∈ [0, 1] are the volume fraction of liquid and gas, satisfy α+ + α− = 1; ± ρ± , u± , P ± (ρ± ) = A± (ρ± )γ¯ are, respectively, the densities, the velocities and the pressures of each phase, and the two pressure functions satisfy P + (ρ+ ) − P − (ρ− ) = f (α− ρ− ), where ∗ Received February 5, 2018. Wen was supported by the National Natural Science Foundation of China (11722104, 11671150), and by GDUPS (2016) and the Fundamental Research Funds for the Central Universities of China (D2172260). Yao was supported by the National Natural Science Foundation of China (11571280, 11331005) and FANEDD No. 201315. Zhu was supported by the National Natural Science Foundation of China (11331005, 11771150). † Corresponding author: Changjiang ZHU.
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A± > 0, γ¯ ± > 1 are constants, f (·) is a known smooth function, it can be a zero function (in this case, the two pressures are equal); τ ± := µ± (∇x u± + ∇tx u± ) + λ± divu± Id are the viscous stress tensors, µ± and λ± are shear and bulk viscosity coefficients, satisfying µ± > 0, 2µ± + dλ± ≥ 0; σ ± are the capillary coefficients; The last three terms on the right-hand side of momentum equations, denote wall frictional forces, interfacial forces and gravities, f ± , I, g are wall frictional coefficients, interfacial coefficient and gravity constant, respectively. This model is commonly used in industrial applications, such as nuclear, power, oil-and-gas, micro-technology and so on, here we can refer to [2]. The classical modeling technique consists of performing a volume average to derive a model without free surface, that is, a two-fluid model. Furthermore, internal viscous and capillary forces cannot be neglected for some applications such as, for instance, wave breaking. Here, we include a capillary pressure term, i.e., we do not assume that the two phase pressures P + and P − are equal. The assumption about non-equal pressure functions P + 6= P − is quite natural. This amounts to including capillary pressure forces and is commonly included in modeling of two-phase flow in porous media. For more information about this model and related models, see for instance [1, 36, 46] and references therein. There are many numerical results about this model and related models, see [46] and references therein. If we ignore frictional forces between wall and fluids, interfacial friction and gravity, then system (1.1) takes the simplified form ± ± ± ± ± ∂t (α ρ ) + div(α ρ u ) = 0, (1.2) ∂t (α± ρ± u± ) + div(α± ρ± u± ⊗ u± ) + α± ∇P ± (ρ± ) = div(α± τ ± ) + σ ± α± ρ± ∇△(α± ρ± ).
Many models are related to system (1.2) such as compressible Navier-Stokes equations (α+ ≡ 0 or α− ≡ 0 and σ ± = 0), Navier-Stokes-Korteweg equations (α+ ≡ 0 or α− ≡ 0), etc. There is a huge literature on the investigation of global existence and large time behavior of smooth solutions in relation to these models. Here we mention several of the most relevant papers. Matsumura and Nishida [44, 45] first considered the global existence of smooth solutions to the compressible Navier-Stokes equations in multidimensional whole space and obtained that the global solutions tended to its equilibrium state, they also obtained the decay rate; Hoff and Zumbrun [33, 34] considered the Green’s function of the compressible isentropic NavierStokes equations with the artificial viscosity and showed the convergence of global solutions to diffusion waves. Later, Liu and Wang [43] investigated the properties of Green’s function for the isentropic Navier-Stokes system in odd dimension. They showed an interesting pointwise convergence of global solution to the diffusion waves with the optimal time decay rate, where the important phenomena of the weaker Huygens’principle was also justified due to the dispersion effects of compressible viscous fluids in multidimensional odd space. Recently, Li and Zhang [42] obtained the optimal Lp time-decay rate for isentropic Navier-Stokes equations in three −s dimensions when initial data belonged to some space H l ∩ B˙ 1,∞ (l ≥ 4, s ∈ [0, 1]). The same decay property also appears in the half space and exterior domain, see [37–40]. Near the constant equilibrium states, Hattori and Li [31, 32] considered the local and global existence of the smooth solution of the compressible Navier-Stokes-Korteweg system for multidimensional model in Sobolev space; Danchin and Desjardins [6] proved existence and uniqueness results of suitably smooth solution in the Besov space frame; Next, Wang and Tan
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[49] obtained the optimal time-decay of the system, which is the same as that of the NavierStokes system. Recently, Based on the weighted L2 -method and some delicate L1 estimates on solutions to the linearized problem, Chen and Zhao [4] studied the existence and uniqueness of stationary solutions of the compressible nonisentropic fluid model of Korteweg type by the contraction mapping principle. Furthermore, they also obtained the stability of the stationary solution. In recent years, some efforts were made on the existence and large time behavior of global solution to the nonconservative compressible two-fluid model (1.2). For the equal pressures, Bresch, Desjardins, Ghidaglia and Grenier [2] considered the existence of global weak solution in the periodic domain when 1 < γ¯ ± < 6; Bresch, Huang and Li [3] showed the existence of global weak solution in one-dimensional case without capillary effect (i.e., σ ± =0), when γ¯ ± > 1. Recently, Lai, Wen and Yao [41] obtained the vanishing capillary coefficient limit to the nonconservative compressible two-fluid model (1.2), where two pressure functions are unequal. We refer also to the recent work [25] which was in a gas-liquid context where a polytropic gas law was used for the gas phase whereas the liquid was assumed to be incompressible, and the global existence of strong solutions and large time behavior were obtained for both the initial boundary value problem and initial value problem. For more results about the compressible nonconservative two-fluid model (1.2), see [19, 21]. Next, we give the main well-posedness results about the model (1.2). By using the detailed analysis of the Green’s function to the linearized system and elaborate energy estimates to the nonlinear system, the time-decay estimates of classical or strong solutions of nonconservative viscous compressible generic two-fluid model with or without capillary effect in R3 were obtained in [5, 18], the equal and unequal pressures were considered respectively. 1.2
Well-posedness Results We consider the following compressible nonconservative two-fluid model in R3 : α+ + α− = 1, ∂ (α± ρ± ) + div(α± ρ± u± ) = 0, t
∂t (α± ρ± u± ) + div(α± ρ± u± ⊗ u± ) + α± ∇P ± (ρ± ) = div(α± τ ± ), + + P (ρ ) − P − (ρ− ) = f (α− ρ− ), ±
(1.3)
±
) ± where dP ± = s2± dρ± , s2± := dP ¯ ± P ρ(ρ , and s± denote the sound speed of each ± dρ± (ρ ) = γ ± ± ± phase separately. Introduce the variable R = α ρ , then ±
dρ+ =
1 (dR+ − ρ+ dα+ ), α+
dρ− =
1 (dR− − ρ− dα+ ). α−
(1.4)
Furthermore, from (1.4) and dP + − dP − = df (α− ρ− ), we obtain α− s2 α+ α− dα = − + 2 + + − 2 dR+ − − + 2 α ρ s+ + α ρ s− α ρ s+ + α+ ρ− s2− +
s2− ′ + f dR− . α−
Substitute (1.5) into (1.4), we get the following expressions: − ′ ρ+ ρ− s2− − + + +α f − dρ+ = − + 2 2 ρ dR + ρ + ρ dR , R (ρ ) s+ + R+ (ρ− )2 s2− s2−
(1.5)
(1.6)
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ρ+ ρ− s2+ dρ = − + 2 2 R (ρ ) s+ + R+ (ρ− )2 s2− −
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+ ′ − + + −α f − , ρ dR + ρ − ρ dR s2+
(1.7)
which give dP
−
dP
+
and
where we denote C 2 := Define
+ ′ − + + −α f − , = C ρ dR + ρ − ρ dR s2+
(1.8)
− ′ − + + +α f − = C ρ dR + ρ + ρ , dR s2−
(1.9)
2
2
s2− s2+ . α− ρ+ s2+ +α+ ρ− s2−
ϕ(ρ+ ) = P + (ρ+ ) − P −
R − ρ+ + ρ − R+
− f (R− ) = 0.
It is easy to see that in (R+ , +∞), ϕ′ (ρ+ ) = s2+ + s2−
R− R+ > 0, (ρ+ − R+ )2
and ϕ: (R+ , +∞) 7→ (−∞, +∞). Applying the implicit function theorem to (1.3)4 , there exists ρ+ = ρ+ (R+ , R− ) such that + ϕ(ρ ) = 0. Furthermore, we can define R− ρ+ (R+ , R− ) , − R+ R+ α+ (R+ , R− ) = + + − , ρ (R , R ) R+ R− α− (R+ , R− ) = 1 − + + − = − + − . ρ (R , R ) ρ (R , R )
ρ− (R+ , R− ) =
ρ+ (R+ , R− )
Based on the above, system (1.3) can be rewritten as follows: ∂t R± + div(R± u± ) = 0, − ′ + + + + + + 2 − + + +α f − ∂ (R u ) + div(R u ⊗ u ) + α C ρ ∇R + ρ + ρ ∇R t s2− = div{α+ [µ+ (∇u+ + ∇t u+ ) + λ+ divu+ Id]}, + ′ ∂t (R− u− ) + div(R− u− ⊗ u− ) + α− C 2 ρ− ∇R+ + ρ+ − ρ− αs2f ∇R− + − − − t − − − = div{α [µ (∇u + ∇ u ) + λ divu Id]}.
(1.10)
We consider the Cauchy problem of (1.10):
− − (R+ , u+ , R− , u− )(x, t) |t=0 = (R0+ , u+ 0 , R0 , u0 )(x),
x ∈ R3 ,
(1.11)
and + − > 0, |x| → ∞. u+ (x, t) → 0, u− (x, t) → 0, R+ (x, t) → R∞ > 0, R− (x, t) → R∞
(1.12)
+ − Without loss of generality, we assume R∞ = R∞ = 1. For the above problem (1.10)–(1.11), we obtain the following result.
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Theorem 1.1 ([18], Theorem 1.1) Under the condition −
η − s2− (1, 1) s2− (1, 1) < f ′ (1) < < 0, − α (1, 1) α− (1, 1)
(1.13)
where η is a positive, small fixed constant, there exists a constant ǫ such that if − − k (R0+ − 1, u+ 0 , R0 − 1, u0 ) kH 2 (R3 ) ≤ ǫ,
(1.14)
then the Cauchy problem (1.10)–(1.11) admits a unique solution (R+ , u+ , R− , u− ) globally in time in the sense that: R+ − 1, R− − 1 ∈ C 0 ([0, ∞); H 2 (R3 )) ∩ C 1 ([0, ∞); H 1 (R3 )),
u+ , u− ∈ C 0 ([0, ∞); H 2 (R3 )) ∩ C 1 ([0, ∞); L2 (R3 )).
− − 1 3 Moreover, if in addition the initial data (R0+ − 1, u+ 0 , R0 − 1, u0 ) is bounded in L (R ), the solution enjoys the following decay-in-time estimates: 3
k (R+ − 1, u+ , R− − 1, u− )(t) kL2 (R3 )) ≤ C(1 + t)− 4 , t ≥ 0,
(1.15)
k ∇(R , u , R , u )(t) kH 1 (R3 ) ≤ C(1 + t)
(1.16)
+
+
−
−
− 54
, t ≥ 0.
Remark 1.2 The proof of the theorem involves some new ideas. Usually, the proof consists of spectral analysis of the Green function for the corresponding linearized system and energy estimates of the solutions to the nonlinear system, refer for instance to [7, 8]. Here, we just need spectral analysis of the low-frequency part of the Green function and the energy estimates. Actually, we employ the energy method in the frequency space to get the decay rates of the low-frequency part so that we succeed to avoid some complicate analysis of the Green function, which is a 8 × 8 matrix. On the other hand, we notice that the high-frequency part can be handled directly by the energy estimates. Thus, the combination of the decay rates of the low-frequency part and the energy estimates show the decay rates of the solutions directly, even for initial data within the H 2 -framework. If the capillary effect is included, we consider the following equations: α+ + α− = 1, ∂ (α± ρ± ) + div(α± ρ± u± ) = 0, t
∂t (α± ρ± u± ) + div(α± ρ± u± ⊗ u± ) + α± ∇P ± (ρ± ) = div(ν ± α± ρ± (∇x u± + ∇tx u± )) + σ ± α± ρ± ∇△(α± ρ± ).
And we consider the two equal pressure functions: P = P (ρ+ ) = A+ (ρ+ )γ¯ − A− (ρ− )γ¯ , γ¯ ± > 1, i.e., f = 0. And (1.5)–(1.9) becomes (1.18)–(1.21): dα+ = dρ+ = dρ− =
α− s2+ α+ s2− + dR − dR− , α− ρ+ s2+ + α+ ρ− s2− α− ρ+ s2+ + α+ ρ− s2− R− (ρ+ )2 s2+
s2− (ρ− dR+ + ρ+ dR− ), + R+ (ρ− )2 s2−
s2+ (ρ− dR+ + ρ+ dR− ), R− (ρ+ )2 s2+ + R+ (ρ− )2 s2−
(1.17)
+
= P (ρ− ) =
(1.18) (1.19) (1.20)
and dP = C 2 (ρ− dR+ + ρ+ dR− ).
(1.21)
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Then the system (1.17) is equivalent to the following form ± ± ± ∂t R + div(R u ) = 0,
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∂t (R± u± ) + div(R± u± ⊗ u± ) + α± C 2 (ρ− ∇R+ + ρ+ ∇R− ) ±
±
±
= div[ν R (∇x u +
∇tx u± )]
±
±
(1.22)
±
+ σ R ∇△R .
We consider the problem (1.22) with initial value (1.11)–(1.12). For simplicity, set ν + = ν − = ν, σ + = σ − = σ. Introduce n± = R± − 1, then the initial value problem (1.22), (1.11) and (1.12) can be rewritten as ∂t n+ + divu+ = S1 , ∂ u+ + β ∇n+ + β ∇n− − ν∇u+ − σ∇△n+ = S , t 1 2 2 (1.23) − − ∂t n + divu = S3 , ∂t u− + β3 ∇n+ + β4 ∇n− − ν∇u− − σ∇△n− = S4 ,
with initial data
+ − − + + − − (n+ , u+ , n− , u− ) |t=0 = (n+ 0 , u0 , n0 , u0 )(x) = (R0 − 1, u0 , R0 − 1, u0 )(x),
x ∈ R3 , (1.24)
and the far-field behavior
2
−
(n+ , u+ , n− , u− )(x, t) → 0,
(1,1) Here β1 = C (1,1)ρ , β2 = β3 = C 2 (1, 1), β4 = ρ+ (1,1) β2 β3 = β32 ), and
S1 = S1 (n+ , u+ ) = −div(n+ u+ ),
S3 = S3 (n− , u− ) = −div(n− u− ),
2
|x| → ∞. +
C (1,1)ρ (1,1) ρ− (1,1)
(1.25) > 0 (which implies β1 β4 =
S2 = S2 (S21 , S22 , S23 )(n+ , u+ , n− , u− ), S4 = S4 (S41 , S42 , S43 )(n+ , u+ , n− , u− ),
and S2j (n+ , u+ , n− , u− )
S4j (n+ , u+ , n− , u− )
(C 2 ρ− )(n+ + 1, n− + 1) (C 2 ρ− )(1, 1) − ∂j n+ − (u+ · ∇)u+ =− j ρ+ (n+ + 1, n− + 1) ρ+ (1, 1) ν ν + + ∂k n+ ∂k u+ ∂k n+ ∂j u+ j + + k n +1 n +1 − [C 2 (n+ + 1, n− + 1) − C 2 (1, 1)]∂j n− , j = 1, 2, 3,
(C 2 ρ+ )(n+ + 1, n− + 1) (C 2 ρ+ )(1, 1) =− − ∂j n− − (u− · ∇)u− j ρ− (n+ + 1, n− + 1) ρ− (1, 1) ν ν + − ∂k n− ∂k u− ∂k n− ∂j u− j + − k n +1 n +1 2 + − 2 + − [C (n + 1, n + 1) − C (1, 1)]∂j n , j = 1, 2, 3.
For the above problem (1.23)–(1.25), we obtain the following result. Theorem 1.3 ([5], Theorem 1.1) For any integer s ≥ 3, there exists a constant δ > 0, such that if − + + k (n+ 0 , n0 ) kH s+1 (R3 )∩L1 (R3 ) + k (u0 , u0 ) kH s (R3 )∩L1 (R3 ) ≤ δ, then the initial value problem (1.23)–(1.25) admits a unique solution (n+ , u+ , n− , u− ) globally in time, which satisfies n+ , n− ∈ C 0 ([0, ∞); H s+1 (R3 )) ∩ C 1 ([0, ∞); H s (R3 )),
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u+ , u− ∈ C 0 ([0, ∞); H s (R3 )) ∩ C 1 ([0, ∞); H s−1 (R3 )), and for some c, C > 0 independent of t such that ||(n+ , n− )(t)||2H s+1 (R3 ) + ||(u+ , u− )(t)||2H s (R3 ) + ||(β3 ∇n+ + β4 ∇n− )(t)||2H s−1 (R3 ) Z t +c ||(n+ , n− )(τ )||2H s+2 (R3 ) + ||(u+ , u− )(τ )||2H s+1 (R3 ) 0 + ||(β3 ∇n+ + β4 ∇n− )(τ )||2H s (R3 ) dτ − 2 + − 2 ≤ C ||(n+ , n )|| + ||(u , u )|| 0 0 0 0 H s (R3 )∩L1 (R3 ) . H s+1 (R3 )∩L1 (R3 ) Moreover, the solution satisfies the decay-in-time estimates: 1
k
||∇k (n+ , n− )(t)||L2 (R3 ) ≤ C(1 + t)− 4 − 2 ,
0 ≤ k ≤ s − 1,
||∇ (u , u )(t)||L2 (R3 ) ≤ C(1 + t)
0 ≤ k ≤ s − 2,
k
+
− 34 − k 2
−
,
1
s
||∇s (n+ , n− )(t)||H 1 (R3 ) + ||∇s−1 (u+ , u− )(t)||H 1 (R3 ) ≤ C(1 + t) 4 − 2 ,
t ≥ 0.
Remark 1.4 The fraction densities converge to the equilibrium states at the L2 -rate 1 (1 + t)− 4 , and the k(∈ [1, s − 1]) order spatial derivatives of the fraction densities converge 1 3 k to zero at the L2 -rate (1 + t)− 4 − 2 , which is slower than the L2 -rate (1 + t)− 4 and L2 -rate 3 k (1 + t)− 4 − 2 for the compressible Navier-Stokes system (k = 0, 1) or Navier-Stokes-Korteweg system (k = 0, 1, 2), see [7, 43, 49]. This is caused by the non-conservation and complexity of the model (1.17).
2 2.1
Drift-flux Model Research Background-2
The drift-flux model is one of the commonly used models nowadays for the prediction of various two-phase flows. It was first developed by Zuber and Findlay [55]. It is used in chemical engineering to predict flow in bubble column reactors, in petroleum applications to model various wellbore operations related to drilling, production of oil and gas, and for the study of geothermal energy related problems and injection of CO2 . A one-dimensional transient drift-flux model can be written in the following form: ∂t (αl ρl ) + ∂x (αl ρl ul ) = 0, (2.1) ∂t (αg ρg ) + ∂x (αg ρg ug ) = 0, ∂t (αl ρl ul + αg ρg ug ) + ∂x (αl ρl u2l + αg ρg u2g + P (m, n)) = ∂x (ǫ∂x umix ) − q¯,
where m = αl ρl , n = αg ρg denote the masses of liquid and gas; The unknowns are αl , αg ∈ [0, 1] volume fractions of liquid and gas, satisfying αl + αg = 1; ρl , ρg the liquid and gas densities; P = P (m, n) common pressure for liquid and gas; ul , ug velocities of liquid and gas. In order to get a closed system, an algebraic equation called the slip relation which relates the two fluid velocities is added: ug = c0 umix + c1 , it implies: f (αg , ul , ug , ρl , ρg ) = ug − c0 umix − c1 = 0,
where c0 and c1 are flow dependent coefficients, c0 is referred to as the distribution parameter and c1 to as the drift velocity, umix = αg ug + αl ul ; ǫ = ǫ(m, n)(≥ 0) viscosity coefficient; q¯
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external force, such as the gravity and frictional force. For previous studies of the 1D model (2.1) with the slip relation ug − c0 umix − c1 = 0 with c0 > 1, c1 = 0, for the free boundary problem, we refer to [10, 22]. In [10], the local existence of weak solution was obtained whereas [22] gave a local existence of weak solution for a general slip (c0 > 1 and c1 > 0) and a global existence result for the special case c0 > 1, c1 = 0. The more general case c1 > 0 is important because it allows the model to describe, e.g. counter-current flow, where ul and ug possibly have different sign. Recently, Evje and Wen [20] obtained the global existence and unique of strong solution for the 1D initial boundary value problem. This work presents a first global existence result for the drift-flux model with a general slip law. Next, we give the main well-posedness results about the model (2.1). 2.2
Well-posedness Results
At first, we ignore the external force, i.e., q¯ = 0. Assume that c0 ≥ 1, c1 > 0, x ∈ (a, b(t)) (where b(t) separates the gas-liquid mixture and the gas region, satisfies: db dt = u(b(t), t), b(0) = b0 ), t > 0, and the pressure is given by γ n P (m, n) = . (2.2) ρl − m Introduce α∗g , α∗l as follows
α∗g =
1 , α∗l = 1 − α∗g . c0
Now, we introduce the notation: ρ = n + m − k ∗ , c = ρl − k ∗ = ρl α∗g , ρ(1, t) = n∗ .
m−k∗ ρ ,
where k ∗ = ρl α∗l . And a∗ =
As [23], by using the Lagrangian coordinates transformation, and introducing the function ρ Q(c, ρ) = a∗ −cρ , satisfies 1 cρ Q(c, ρ) = Qc ct + Qρ ρt = Qρ ρt = + ρt a∗ − cρ (a∗ − cρ)2 a∗ a∗ ρ2 = ∗ ρ = − ux = −a∗ Q(c, ρ)2 ux . t (a − cρ)2 (a∗ − cρ)2
Consider the following initial boundary value problem ∂t c = 0, ∂t Q + a∗ Q2 ∂x u = 0,
2
∂t u + ∂x [P (c, Q) − u g(cQ) − uh(cQ) + j(cQ)] = ∂x [E(cQ)∂x u],
(2.3) x ∈ (0, 1), t > 0,
with cQ := E(cQ), P (c, Q) = [(1 − c)Q]γ , E(cρ) = cρ = a∗ 1 + cQ cρ cQ g(cρ) = k ∗ = a∗ α∗l := g(cQ), cρ + k ∗ α∗l + cQ c1 cρ cQ ∗ c1 h(cρ) = 2ρl = 2a := h(cQ), c0 k ∗ + cρ c0 α∗l + cQ 2 2 c1 1 c1 1 + cQ j(cρ) = ρ2l = ρ := j(cQ). l ∗ c0 k + cρ c0 α∗l + cQ
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Initial conditions are c(x, 0) = cˆ0 (x), Q(x, 0) = Q0 (x) = and boundary conditions are
ρ0 , u(x, 0) = u0 (x), x ∈ [0, 1], a∗ − cˆ0 ρ0 n∗ := Q∗ , c(1, t) = 0. a∗
u(0, t) = 0, Q(1, t) =
(2.4)
(2.5)
Assumptions on the parameters α, γ 0<α<
1 , γ > 1. 2
Assumptions on the initial data c˜ φα ≤ cˆ0 ≤ c˜2 φα , 0 < c˜1 ≤ c˜2 , sup cˆ0 < 1, A1 ≤ Q0 ≤ B1 , 1 Z 1 Q0 1 β ˜ G0 (x)dx ≤ M, ln Q∗ (1 + cˆ Q ) ≤ Cφ1 , β1 ∈ (0, α] ∩ 0, 2 − α , 0 0 0 1−α1 Q0 2 cˆ0 ln ∈ L2 , α1 ∈ (2α, 1) such that φ Q∗ (1 + cˆ0 Q0 ) x 1−α1 −α+2β1 p 2 φ cˆ0,x ∈ L2 , E(ˆ c0 Q0 )u0,x ∈ L2 ,
(2.6)
(2.7)
˜ where φ(x) = 1 − x, and G0 is the initial energy with positive constants c˜1 , c˜2 , A1 , A2 and C, as follows: 2 Z Q 1 2 p∗ − P (c, Q∗ ) ρl c1 cˆ0 Q0 P (c, s) − P (c, Q∗ ) G0 (x) = u0 + ds + − ∗ ∗ cˆ0 ln , 2 a ∗ s2 a∗ Q k a c0 a∗l + cˆ0 Q0 Q∗ where p∗ = [Q∗ ]γ . Moreover, the lower bound A of Q and the following relations 2 c1 µ2 c˜1 A ≤ , α∗ c 8 0 l √ A1 M A < , e4/˜c1 < 4 µ
bound M on the initial energy must satisfy the µ2 c˜1 A , 32 √ 4 M 1 , e4/˜c1 < + 1. 3 µ 6A M≤
(2.8)
The external pressure p∗ must obey
(p∗ )1/γ ≥ max{B1 (1 − sup cˆ0 ), 4A}.
(2.9)
Similarly, the upper bound B of Q and the bound M on the initial energy must satisfy the following relations: 2 ∗ γ B c1 c1 a γ ∗ ∗ ∗ ≥ C(p , c0 , c1 ) = p + a + ρl , (1 − sup cˆ0 ) 2 c0 c0 k∗ B 2 − 4δ (2.10) B1 ≤ , B ≤ , 2 3δ 1 2 12 2 c1 1 (2M ) 2 + 2 +1 1+ BM ≤ ln(1 + δ), µ˜ c1 µ c˜1 A c0 α∗l
for some δ > 0 and subject to the condition sup cˆ0 < 1. In addition, A and B are chosen such that: 3 2A ≤ Q0 ≤ B. (2.11) 4
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Finally, M must obey the smallness condition as follows: 1 2M 2 + 1 ≤ . µ2 c˜1 A 64
(2.12)
Theorem 2.1 ([23], Theorem 2.1 (existence)) Under the assumptions of (2.6)–(2.12), there exists a constant M0 > 0 such that (2.3)–(2.5) admits a global weak solution (c, Q, u) on [0, 1] × [0, T ] for any time for all M ≤ M0 in the sense that (A) The following regularity holhs
3
c, Q ∈ L∞ ([0, T ]; L∞ ) ∩ C 4 ([0, T ]; L2 ),
1
E(cQ)ux ∈ L∞ ([0, T ]; L2),
u ∈ L∞ ([0, T ]; L∞) ∩ C 2 ([0, T ]; L2 ).
Moreover, the following estimates hold Z A ≤ Q ≤ B, ||ux ||L4 (0,T ;L2 ) +
Z
1
1 0
Eu2x dx +
1−α1
φ(x)
0
Z tZ 0
1
0
u2s dxds ≤ C2 ,
Q c ln Q∗ (1 + cQ)
x
2 dx ≤ C2 ,
(2.13) (2.14)
for (x, t) ∈ [0, 1] × [0, T ], where C2 depends on A, B, M, c˜1 , c˜2 , α, β, γ, T and the initial data. (B) The following equations hold ∂t c = 0, ∂t Q + a∗ Q2 ∂x u = 0, a.e. (x, t) ∈ (0, 1) × (0, T ], (c, Q)(x, 0) = (ˆ c0 (x), Q0 (x)), a.e. (x, t) ∈ [0, 1], Z T Z 1 uϕt + P (c, Q) − u2 g(cQ) − uh(cQ) + j(cQ) 0 0 " # 2 Z 1 ρ l c1 ∗ − + p − µE(cQ)u )ϕ dxdt + u0 ϕ(x, 0)dx = 0, x x α∗l c0 0
(2.15)
for any test function ϕ ∈ C0∞ ((0, 1] × [0, T )). (C) Interface behavior
|u(x, t)| ≤ C2 x
r−1 r
,
(2.16)
for some r ∈ (1, 2) such that r(α + 1) < 2, and |Q(x, t) − Q∗ | ≤ C2 |x − 1|β1 ,
(2.17)
for β1 ∈ (0, α] ∩ (0, 12 − α]. Here (x, t) ∈ [0, 1] × [0, T ]. Theorem 2.2 ([23], Theorem 2.2 (uniqueness)) Under the conditions of Theorem 2.1 and by requiring that β1 = α, where 0 < α ≤ 14 , the weak solution is unique. Next, assume that c0 ≥ 1, c1 = 0, x ∈ (a, b(t)), t > 0, i.e., h(cQ) = j(cQ) = 0. Set E(c, cQ) = [cQ]β+1 , β > 0, and n∗ = 0 (ρ(1, t) = n∗ = 0). Consider the following initial boundary value problem: ∂t c = 0, (2.18) ∂t Q + a∗ Q2 ∂x u = 0, 2 ∂t u + ∂x [P (c, Q) − u g(cQ)] = ∂x [E(cQ)∂x u], x ∈ (0, 1), t > 0,
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with initial conditions c(x, 0) = cˆ0 (x), Q(x, 0) = Q0 (x) =
ρ0 (x) , u(x, 0) = u0 (x), x ∈ [0, 1], a∗ − cˆ0 ρ0 (x)
(2.19)
and with boundary condition u(0, t) = 0, (c, Q)(1, t) = 0.
(2.20)
˜ B ˜ and δ be positive constants, whereas α, β > 0 and γ > 1 Assumptions Let A1 , B1 , A, such that α α B1 φ 4 ≤ cˆ0 ≤ A1 φ 4 , B1 ≤ A1 , 1 1 3α Bδ ˜ ˜ γ−1 ˜ 3α ˜ γ−1 4 , φ 4 ≤ Q0 ≤ Aφ ≤ A, Bδ ! Z 1 (2.21) u20 (1 − cˆ0 )γ Qγ−1 0 + dx ≤ δ, 2 a∗ (γ − 1) 0 β+1 α (ˆ c0 Q0 ) 2 u0x ∈ L2 , φ1−αβ |(ˆ cβ+1 Qβ0 )x |2 ∈ L1 , φ1− 4 cˆ20,x ∈ L1 , 0 where φ(x) = 1 − x, and α, β, γ as well as the time decay exponents r1 , r2 , r3 , r4 > 0, satisfy the following 1 α(β + 1) < 1, α + 1 − 4αβ ≥ 0, α(4β + 1) ≤ 2, β ≤ , γ 2r β r (β + 1) 3 3 γ > max 5β + 2, 1 + r − 2r β , 1 + r − r (β + 1) , 3 2 3 2 2β γ − 1 1 γ ≥ , r3 < , r1 < , 2 − α(β + 1) γ γ−β (2.22) r2 + r1 (γ − 1 − β) > 1, r2 (β + 1) + r3 ≤ 1, 3r2 β ≤ r3 , r (β + 1) < r , r < r (2γ − β − 1), 2 3 4 1 r4 ≤ min{1 + r3 , r1 (γ − β) + r3 }, r2 (5β + 1) ≤ r4 , r2 > r1 .
Theorem 2.3 ([24], Theorem 3.1 (global weak solution)) Under the assumptions of (2.21)–(2.22), there exists a positive constant C0 such that if δ ≤ C0 , then (2.18)–(2.20) admits a unique solution (c, Q, u) on [0, 1] × [0, ∞) in the sense that: (A) the following regularity holds
c, Q ∈ L∞ ([0, T ]; L∞ ) ∩ C 1 ([0, T ]; L2),
1
E(cQ)ux ∈ L∞ ([0, T ]; L2 ),
u ∈ L∞ ([0, T ]; L∞) ∩ C 2 ([0, T ]; L2 ).
(B) the following equations hold ∂t c = 0, ∂t Q + a∗ Q2 ∂x u = 0, a.e. (x, t) ∈ (0, 1) × (0, T ], (c, Q)(x, 0) = (ˆ c0 (x), Q0 (x)), a.e. (x, t) ∈ [0, 1], Z TZ 1 Z uϕt + P (c, Q) − u2 g(cQ) − E(cQ)ux ϕx dxdt + 0
0
for any T > 0 and any test function ϕ ∈ C0∞ ((0, 1] × [0, T )).
0
1
u0 ϕ(x, 0)dx = 0,
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In the next section, we will consider the model (2.1) with c0 = 1 and c1 = 0, i.e., ug = ul = u, we call this model as the viscous liquid-gas two-phase flow model. We will use the appropriate αg ρg variable transformation, to rewrite the model in terms of ( αl ρl +α , αl ρl + αg ρg , u), which is g ρg αg ρg similar to the single-phase compressible Navier-Stokes equations, where αl ρl +α satisfies a g ρg transport equation.
3 3.1
Viscous Liquid-gas Two-phase Flow Model Research Background-3
Viscous liquid-gas two-phase flow model can be viewed to be simplified from the model, which are widely used within the petroleum industry to describe production and transport of oil and gas through long pipelines or well. This model is composed of two separate mass conservation equations corresponding to each of the two phases and one mixture momentum conservation equation in following form mt + div(mu) = 0, (3.1) nt + div(nu) = 0, T ((m + n)u)t + div((m + n)u ⊗ u) + ∇P (m, n) = div(µ(∇u + ∇u )) + ∇(λdivu) − q¯,
where m = αl ρl , n = αg ρg denote liquid mass and gas mass respectively; µ and λ are viscosity coefficients, satisfying: µ > 0, λ + d2 µ ≥ 0; The unknown variables αl , αg ∈ [0, 1] denote liquid and gas volume fractions, satisfying the fundamental relation: αl + αg = 1; Furthermore, ρl and ρg denote liquid and gas densities, respectively; u denotes velocity of liquid and gas; P = P (m, n) is common pressure for both phases, q¯ is the external force. The investigation of model (3.1) has been a topic during the last decade. There are many results about the numerical properties of this model or related model. However, there are few results providing insight into existence, uniqueness, regularity, asymptotic behavior and decay rate estimates concerning the two-phase liquid-gas models of the form (3.1). Let us review some previous works about the viscous liquid-gas two-phase flow model. For the model (3.1) in 1D, γ n when the liquid is incompressible and the gas is polytropic, i.e., P (m, n) = Cργl ρl −m , Evje and Karlsen [16] studied the existence and uniqueness of the global weak solution to the free mβ 1 boundary value problem with µ = µ(m) = k1 (ρl −m) β+1 , β ∈ (0, 3 ), when the fluids connected to vacuum state discontinuously. Evje, Fl˚ atten and Friis [13] also studied the model with nβ 1 µ = µ(m, n) = k1 (ρl −m) (β ∈ (0, )) in a free boundary setting when the fluids connected β+1 3 to vacuum state continuously, and obtained the global existence of the weak solution. If the acceleration terms in the mixture momentum equation was neglected, Evje and Karlsen [14] obtained the global existence of weak solution on the half line. If the external force and frictional force were included, see the related results in [27, 28]. Specifically, when both of the two fluids p are compressible, i.e., P (m, n) = C 0 (−b(m, n) + b(m, n)2 + c(n)), one can consult [15] for the global existence of strong solution to the 1D case; For multidimensional case, Yao, Zhang and Zhu [51] obtained the existence of the global weak solution to the 2D model when the initial energy is small. Furthermore, they proved a blow-up criterion in terms of the upper bound of the liquid mass for the strong solution to the 2D model in a smooth bounded domain, cf. [52]. For the Cauchy problem of a multi-dimensional viscous liquid-gas two-phase flow model, Hao
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and Li [30] obtained the global existence and uniqueness of strong solution for the initial data close to an equilibrium and the local in time existence and uniqueness of the solution with general initial data in the framework of Besov spaces. Concerning the well-posedness and large time behavior of solutions to the model (3.1) and related models, we refer the reader to the [9, 11, 12, 17, 29, 35, 47] and references therein. Next, we give the main well-posedness results about the model (3.1). 3.2
Well-posedness Results
At first, we consider the model (3.1) in 1D, in order to avoid some unsolved difficulties, we consider a simplified model obtained as follows: (1) Due to the fact that the liquid phase density is much higher than the gas phase density, typically to the order ρρgl = O(103 ), we can neglect the gas phase effects in the mixture momentum conservation equation. (2) Neglect the external forces, i.e., q¯ = 0. We assume that further the liquid is incompressible and the gas is polytropic, i.e., ρl = const, P = Cργg , γ > 1, C > 0. Then the model (3.1) can be simplified into the following model in the form ∂τ n + ∂ξ (nu) = 0, (3.2) ∂τ m + ∂ξ (mu) = 0, 2 ∂τ (mu) + ∂ξ (mu + P (m, n)) = ∂ξ (ǫ∂ξ u),
and
n . (3.3) ρl − m We will consider (3.2) in a free boundary value problem setting where the masses n and m initially occupy only a finite interval [a, b] ∈ R. That is P (m, n) = Cργl
n(ξ, 0) = n0 (ξ),
m(ξ, 0) = m0 (ξ),
u(ξ, 0) = u0 (ξ),
ξ ∈ [a, b],
and n(ξ, 0) = m(ξ, 0) = 0, ξ ∈ R \ [a, b]. The boundary conditions are given as (−P (m, n) + ǫu )(a(τ ), τ ) = 0, ξ (−P (m, n) + ǫuξ )(b(τ ), τ ) = 0,
(3.4)
(3.5)
) ) = u(a(τ ), τ ), a(0) = a; db(τ = where a(τ ) and b(τ ) are vacuum boundary, satisfying da(τ dτ dτ β m u(b(τ ), τ ), b(0) = b. As in [16], the viscosity coefficient ǫ = ǫ(m) = k1 (ρl −m)β+1 , where β ∈ (0, 1], k1 is a positive constant. In the Lagrangian coordinates, the free boundary value problem (3.2)–(3.5) becomes the following fixed boundary value problem ∂t n + mn∂x u = 0, (3.6) ∂t m + m2 ∂x u = 0, ∂t (mu) + ∂x (P (m, n)) = ∂x (ǫ(m)m∂x u), x ∈ (0, 1), t > 0,
with initial data
n(x, 0) = n0 (x), m(x, 0) = m0 (x), u(x, 0) = u0 (x), x ∈ [0, 1],
(3.7)
and the boundary conditions P (m, n) = ǫ(m)mux , x = 0, 1,
t ≥ 0.
(3.8)
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n Remark 3.1 Introduce the variables Q(m) = ρlm −m and c = m , we can rewrite the initial boundary value problem (3.6)–(3.8) in the form ∂t c = 0, ∂t Q(m) + ρl Q(m)2 ∂x u = 0, ∂t u + ∂x P (c, m) = ∂x (E(Q(m))∂x u) , x ∈ (0, 1), t > 0,
with initial data
c(x, 0) = cˆ0 (x), Q(x, 0) = Q0 (x), u(x, 0) = u0 (x),
x ∈ [0, 1],
and the boundary conditions P (c, Q(m)) = E(Q(m))ux , at x = 0, 1, t ≥ 0, where P (c, Q(m)) = cγ Q(m)γ ,
γ > 1,
E(Q(m)) = mε(m) = Q(m)β+1 , which is similar to the model of single-phase Navier-Stokes equations. Therefore, we can apply technique in studying Navier-Stokes equations to deal with the related problem for the viscous liquid-gas two-phase flow model. Yao and Zhu [53] improved the previous result of Evje and Karlsen [16] from β ∈ (0, 13 ) to β ∈ (0, 1], and got the global existence of weak solution, regularity of the solutions and the asymptotic behavior result. The result of global existence of weak solution is as follows. Theorem 3.2 ([53], Theorem 2.2 (Existence and uniqueness)) Under the assumptions of (A1 ) inf n0 (x) > 0, sup n0 (x) < ∞, inf m0 (x) > 0, sup m0 (x) < ρl , n0 , m0 ∈ x∈[0,1]
x∈[0,1]
x∈[0,1]
x∈[0,1]
W 1,∞ ([0, 1]), n0 (A2 ) u0 ∈ L∞ ([0, 1]), (E(m0 )u0x )x ∈ L2 ([0, 1]), [( ρl −m )γ ]x ∈ L2 ([0, 1]), 0 (A3 ) γ > 1, 0 < β ≤ 1, the initial boundary problem (3.6)–(3.8) possesses a unique global weak solution (n(x, t), m(x, t), u(x, t)) satisfying for any T > 0 C0 0 < C(T ) ≤ m(x, t) ≤ ρl (< ρl ), (x, t) ∈ [0, 1] × [0, T ], (3.9) 1 + C0 and 0 < C(T ) ≤ n(x, t) ≤ C, (x, t) ∈ [0, 1] × [0, T ]. (3.10) On the other hand, for the viscous liquid-gas model with constant viscosity coefficient when both the initial liquid and gas masses connect to vacuum continuously, Yao and Zhu [54] used a new technique to get the upper and lower bounds of gas and liquid masses n and m, then got the global existence of weak solution by the line method. For this case, the boundary conditions (3.8) is replaced by n = m = 0, x = 0, 1, t ≥ 0. (3.11) For simplicity, set ǫ = 1, Cργl = 1, then (3.3) becomes γ n . P (m, n) = ρl − m Next, we give the definition of weak solution.
(3.12)
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Definition 3.3 ([54], Definition 1) A pair of functions (n(x, t), m(x, t), u(x, t)) is called a global weak solution to the initial boundary value problem (3.6), (3.7) and (3.11), if for any T >0 1
n, m ∈ C([0, T ]; H 1 ([0, 1])) ∩ C 2 ([0, T ]; L2([0, 1])), u ∈ C([0, T ]; H 1 ([0, 1])) ∩ C 1 ([0, T ]; L2([0, 1])), 1
mux ∈ C([0, T ]; H 1 ([0, 1])) ∩ C 2 ([0, T ]; L2 ([0, 1])), and lim m(x, t) = lim n(x, t) = lim m(x, t) = lim n(x, t) = 0.
x→0+
x→0+
x→1−
x→1−
Furthermore, the following equations hold ∂t n + mn∂x u = 0, ∂t m + m2 ∂x u = 0, (n, m)(x, 0) = (n0 (x), m0 (x)), for a.e. x ∈ (0, 1) and any t ≥ 0,
and
Z
0
∞
Z
0
1
(uψt + (P (m, n) − mux )ψx )dxdt +
Z
1
u0 (x)ψ(x, 0)dx = 0,
0
for any test functions ψ(x, t) ∈ C0∞ (Ω), with Ω = {(x, t) : 0 ≤ x ≤ 1, t ≥ 0}.
Theorem 3.4 ([54], Theorem 1) Under the assumptions of (A1 ) m0 (x) > 0, x ∈ (0, 1); m0 (0) = m0 (1) = 0; m0 (x) ∈ C([0, 1]); sup m0 (x) < ρl ; x∈[0,1]
n0 (x) > 0, x ∈ (0, 1); n0 (0) = n0 (1) = 0; n0 (x) ∈ C([0, 1]); cˆ0 (x) ∈ C 1 ([0, 1]); cˆ0 (0) = cˆ0 (1) = 0, n0 (x), where cˆ0 (x) = m 0 R1 1/2 (A2 ) u0 ∈ L2 ([0, 1]); (m0 u0x )x ∈ L2 ([0, 1]); 0 < 0 m01(x) dx < ∞; m0 u0x ∈ L2 ([0, 1]); m0x ∈ L2 ([0, 1]), R1 R m (x) 1 γ−2 (A3 ) exp 2 0 u20 (x) + 2 0 0 cˆγ0 (x) (ρsl −s)γ ds dx 2 m0 (x) ≤ ρl (1 − δ). Here δ ∈ (0, 1) is a positive constant and satisfies: 1 − δ > maxx∈[0,1] αl,0 (x), and where αl,0 (x) = sup m0 (x) < ρl implies sup αl,0 (x) < 1, x∈[0,1]
m0 (x) ρl ,
and
x∈[0,1]
(A4 ) γ > 1, the initial boundary value problem (3.6), (3.7) and (3.11) possesses a unique global weak solution (m(x, t), n(x, t), u(x, t)) defined by Definition 3.3. Next, we consider (3.1) in 3D. We neglect the gas phase effects in the mixture momentum conservation equation and external force, i.e., study the following initial boundary value problem mt + div(mu) = 0, (3.13) nt + div(nu) = 0, (mu)t + div(mu ⊗ u) + ∇P (m, n) = div(µ(∇u + ∇uT )) + ∇(λdivu),
with initial conditions
(m, n, u) |t=0 = (m0 , n0 , u0 ),
¯ x ∈ Ω,
(3.14)
and with the boundary conditions u(x, t) = 0,
(x, t) ∈ ∂Ω × [0, ∞),
(3.15)
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where Ω ⊆ R3 is a smooth bounded domain; The viscous coefficients µ, λ are constants; Pressure P satisfies p P (m, n) = C 0 (−b(m, n) + b(m, n)2 + c(n)),
with C 0 = 12 α2l , k0 = ρl,0 −
Pl,0 α2l
> 0, α0 =
αg 2 , αl
b(m, n) = k0 − m − c(n) = 4k0
αg αl
αg αl
and 2
2
n = k0 − m − α0 n,
n = 4k0 α0 n.
For the problem (3.13)–(3.15), Wen, Yao and Zhu [50] proved the local existence of strong solution and established the blow-up criterion, when there was initial vacuum. If the liquid mass R was upper bounded, we could obtain a high integrability of the velocity, sup Ω m|u|r dx ≤ C, 0≤t≤T
for some r ∈ (3, 4]. Moreover, in order to overcome the singularity brought by the pressure P (m, n) when there is vacuum, we needed the assumption 0 ≤ s0 m0 ≤ n0 ≤ s0 m0 , where s0 and s0 were positive constants. Theorem 3.5 ([50], Theorem 1.1 (local existence)) Let Ω be a bounded smooth domain in R3 and q ∈ (3, 6]. Assume that the initial data m0 , n0 , u0 satisfy m0 , n0 ∈ W 1,q (Ω), u0 ∈ ¯ where s , s¯0 are positive constants. The following H01 (Ω)∩H 2 (Ω), 0 ≤ s0 m0 ≤ n0 ≤ s¯0 m0 in Ω, 0 compatible condition is also valid: −µ△u0 − (µ + λ)∇divu0 + ∇P (m0 , n0 ) =
√ m0 g, for some g ∈ L2 (Ω).
(3.16)
Then, there exist a T0 > 0 and a unique strong solution (m, n, u) to the problem (3.13)–(3.15), such that 0 ≤ s0 m ≤ n ≤ s¯0 m,
(m, n) ∈ C([0, T0 ]; W 1,q (Ω)),
(mt , nt ) ∈ L∞ (0, T0 ; Lq (Ω)),
P ∈ L∞ (0, T0 ; W 1,q (Ω)), u ∈ L∞ (0, T0 ; H01 (Ω) ∩ L2 (0, T0 ; W 2,q (Ω)), √ mut ∈ L∞ (0, T0 ; L2 (Ω)), ut ∈ L2 (0, T0 ; H01 (Ω)).
Theorem 3.6 ([50], Theorem 1.2 (blow-up citerion)) Under the assumptions of Theorem 3.5, if T ∗ < ∞ is the maximal existence time for the strong solution (m, n, u)(x, t) to the problem problem (3.13)–(3.15) stated in Theorem 3.5, then lim sup ||m||L∞ (0,T ;L∞ (Ω)) = ∞,
(3.17)
T →T ∗
provided that λ <
25 3 µ.
Remark 3.7 The analysis in Theorem 3.6 can be applied to study a blow-up criterion of the strong solution to compressible Navier-Stokes equations for 25 3 µ > λ, i.e., lim sup kρ(t)kL∞ (0,T ;L∞ (Ω)) = ∞, T →T ∗
which improves the corresponding result about Navier-Stokes equations in [48] where 7µ > λ. If we don’t neglect the gas phase effects in the mixture momentum conservation equation,
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and consider the following initial boundary value problem mt + (mu)x = 0, nt + (nu)x = 0, ((m + n)u)t + [(m + n)u2 ]x + P (m, n)x = µuxx , (x, t) ∈ (0, 1) × (0, ∞),
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(3.18)
the viscosity coefficient µ is constant, the pressure P satisfies p P (m, n) = C 0 (−b(m, n) + b(m, n)2 + c(n)), with initial conditions
m(x, 0) = m0 (x), n(x, 0) = n0 (x), (m + n)u(x, 0) = M0 (x),
x ∈ [0, 1],
(3.19)
and with boundary conditions u(0, t) = u(1, t) = 0, t ≥ 0.
(3.20)
We give a precise definition of global weak solutions. Definition 3.8 ([26], Definition 1.1) We call (m, n, u) : (0, 1) × (0, +∞) → R+ × R+ × R a global weak solution of (3.18)–(3.20) if for any 0 < T < +∞: (1) m, n ∈ L∞ ((0, 1) × (0, T )), (m + n)u2 ∈ L∞ (0, T ; L1(0, 1)), m, n ≥ 0 a.e., in (0, 1) × (0, T ), u ∈ L2 (0, T ; H01 (0, 1)); (2) (m, n, u) satisfy the system (3.18) in the sense of distribution; (3) (m, n, (m + n)u)(x, 0) = (m0 (x), n0 (x), M0 (x)), a.e. x ∈ (0, 1). Theorem 3.9 ([26], Theorem 1.1 (global existence)) If
inf m0 ,
x∈(0,1)
inf n0 ≥ 0, m0 ,
x∈(0,1)
0 n0 ∈ L∞ (0, 1), and √mM0 +n ∈ L2 (0, 1), then there exists a global weak solution (m, n, u) : 0 (0, 1) × [0, +∞) → R+ × R+ × R to (3.18)–(3.20).
Remark 3.10 Note that we do not need the conditions s0 m0 ≤ n0 ≤ s0 m0 for some s0 , s0 ≥ 0 typically made use of in previous literature [15, 50–52]. This implies that transition to single phase is allowed, i.e., one of the two phases can completely occupy some regions.
4
Open Problems
• Compressible nonconservative two-fluid model Compared with the single-phase flow (i.e., compressible Navier-Stokes equations), the twofluid model has its own challenges by means of the nonconservative structure in the pressure terms. More specifically, when one looks for the global solutions with large initial data in the sense of distribution, the spatial derivatives in the pressure terms can not all be shifted to the test functions. In view of the fact, one has to obtain some estimates of the spatial derivatives of density or the related. This makes that two viscosity coefficients have to be equivalent to the corresponding density, i.e., µ± ∼ ρ± , λ± ∼ ρ± , see [3] for 1D case. To our best knowledge, it is still open for the case of multi-dimensions and for that µ± ∼ (ρ± )α , λ± ∼ (ρ± )β with more general α, β ∈ [0, ∞). • Drift-flux model The main difference between two-fluid model and drift-flux model is that two fluids are considered as a whole in the momentum equations for the latter case. However, in this case,
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some other equations have to be added to the system for completeness. The so called “slip law” is commonly used. The main challenge is that the global entropy estimate is difficult to find though the momentum equation is of the conservative form. This leads to an open problem whether the global weak solutions with large initial data exist or not for more general slip law. In fact, when c0 = 1 and c1 = 0 for large initial data, and c0 ≥ 1, c1 ≥ 0 for small initial data, it is known that the global weak solutions exist, see [26], and Theorems 2.1 and 3.2 respectively. Here c0 and c1 are constants. • Viscous liquid-gas two-phase flow model
The viscous liquid-gas two-phase flow model can be considered as a simplified case from the two-fluid model when two velocity fields and two pressure functions are equal respectively. Although there are some results achieved for the simplified case particularly, there are still some open problems. For example, whether the global smooth solution with large initial data in high dimensions exists or not, or equivalently whether the smooth solution blows up in finite time.
References [1] Bear J. Dynamics of Fluids in Porous Media, Environmental Science Series. New York: Elsevier, 1972; reprinted with corrections, New York: Dover, 1988 [2] Bresch D, Desjardins B, Ghidaglia J M, Grenier E. Global weak solutions to a generic two-fluid model. Arch Ration Mech Anal, 2010, 196: 599–629 [3] Bresch D, Huang X D, Li J. Global weak solutions to one-dimensional non-conservative viscous compressible two-phase system. Comm Math Phys, 2012, 309: 737–755 [4] Chen Z Z, Zhao H J. Existence and nonlinear stability of stationary solutions to the full compressible Navier-Stokes-Korteweg system. J Math Pures Appl, 2014, 101: 330–371 [5] Cui H B, Wang W J, Yao L, Zhu C J. Decay rates for a nonconservative compressible generic two-fluid model. SIAM J Math Anal, 2016, 48: 470–512 [6] Danchin R, Desjardins B. Existence of solutions for compressible fluid models of Korteweg type. Ann Inst H Poincar´ e Anal Non Lin´ eaire, 2001, 18: 97–133 [7] Duan R J, Ukai S, Yang T, Zhao H J. Optimal convergence rates for the compressible Navier-Stokes equations with potential forces. Math Models Methods Appl Sci, 2007, 17: 737–758 [8] Duan R J, Liu H X, Ukai S, Yang T. Optimal Lp -Lq convergence rate for the compressible Navier-Stokes equations with potential force. J Differential Equations, 2007, 238: 220–233 [9] Evje S. Global weak solutions for a compressible gas-liquid model with well-formation interaction. J Differential Equations, 2011, 251: 2352–2386 [10] Evje S. Weak solutions for a gas-liquid model relevant for describing gas-kick in oil wells. SIAM J Math Anal, 2011, 43: 1887–1922 [11] Evje S. A compressible two-phase model with pressure-dependent well-reservoir iteraction. SIAM J Math Anal, 2013, 45: 518–546 [12] Evje S. Genuine two-phase flow dynamics with a free interface separating gas-liquid mixture from gas. SIAM J Math Anal, 2013, 45: 2894–2923 [13] Evje S, Fl˚ atten T, Friis H A. Global weak solutions for a viscous liquid-gas model with transition to single-phase gas flow and vacuum. Nonlinear Anal, 2009, 70: 3864–3886 [14] Evje S, Karlsen K H. Analysis of a compressible gas-liquid model motivated by oil well control operations. Acta Math Sci, 2012, 32B(1): 295–314 [15] Evje S, Karlsen K H. Global existence of weak solutions for a viscous two-phase model. J Differential Equations, 2008, 245: 2660–2703 [16] Evje S, Karlsen K H. Global weak solutions for a viscous liquid-gas model with singular pressure law. Comm Pure Appl Anal, 2009, 8: 1867–1894 [17] Evje S, Liu Q Q, Zhu C J. Asymptotic stability of the compressible gas-liquid model with well-formation interaction and gravity. J Differential Equations, 2014, 257: 3226–3271
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[18] Evje S, Wang W J, Wen H Y. Global well-Posedness and decay rates of strong solutions to a non-conservative compressible two-fluid model. Arch Ration Mech Anal, 2016, 221: 1285–1316 [19] Evje S, Wen H Y. Analysis of a compressible two-fluid Stokes system with constant viscosity. J Math Fluid Mech, 2015, 17: 423–436 [20] Evje S, Wen H Y. Global solutions of a viscous gas-liquid model with unequal fluid velocities in a closed conduit. SIAM J Math Anal, 2015, 47: 381–406 [21] Evje S, Wen H Y. Stability of a compressible two-fluid hyperbolic-elliptic system arising in fluid mechanics. Nonlinear Anal Real World Appl, 2016, 31: 610–629 [22] Evje S, Wen H Y. Weak solutions of a gas-liquid drift-flux model with general slip law for wellbore operators. Discrete Contin Dyn, System-A, 2013, 33: 4497–4530 [23] Evje S, Wen H Y. Weak solutions of a two-phase Navier-Stokes model with a general slip law. J Funct Anal, 2015, 268: 93–139 [24] Evje S, Wen H Y. On the large time behavior of the compressible gas-liquid drift-flux model with slip. Math Models Methods Appl Sci, 2015, 25: 2175-2215 [25] Evje S, Wen H Y, Yao L. Global solutions to a one-dimensional non-conservative viscous compressible two-phase gas-liquid model. Discrete Contin Dyn, System-A, 2016, 36: 1927–1955 [26] Evje S, Wen H Y, Zhu C J. On global solutions to the viscous liquid-gas model with unconstrained transition to single-phase flow. Math Models Methods Appl Sci, 2017, 27: 323–346 [27] Friis H A, Evje S. Global weak solutions for a gas-liquid model with external forces and general pressure law. SIAM J Appl Math, 2011, 71: 409–442 [28] Friis H A, Evje S. Well-posedness of a compressible gas-liquid model with a friction term important for well control operations. SIAM J Appl Math, 2011, 71: 2014–2047 [29] Friis H A, Evje S. Asymptotic behavior of a compressible two-phase model with well-formation interaction. J Differential Equations, 2013, 254: 3957–3993 [30] Hao C C, Li H L. Well-posedness for a multidimensional viscous liquid-gas two-phase flow model. SIAM J Math Anal, 2012, 44: 1304–1332 [31] Hattori H, Li D. Solutions for two-dimensional system for materials of Korteweg type. SIAM J Math Anal, 1994, 25: 85–98 [32] Hattori H, Harumi H, Li D. Global solutions of a high-dimensional system for Korteweg materials. J Math Anal Appl, 1996, 198: 84–97 [33] Hoff D, Zumbrun K. Multi-dimentional diffusion waves for the Navier-Stokes equations of compressible flow. Indiana Univ Math J, 1995, 44: 603–676 [34] Hoff D, Zumbrun K. Pointwise decay estimates for multidimentional Navier-Stokes diffusion waves. Z Angew Math Phys, 1997, 48: 597–614 [35] Hong G Y, Zhu C J. Optimal decay rates on the solution to the compressible gas-liquid drift-flux model with slip. Math Models Methods Appl Sci, 2018, 28(2): 337–385 [36] Ishii M. Thremo-Fluid Dynamic Theory of Two-Phase Flow. Paris: Eyrolles, 1975 [37] Kagei Y, Kobayashi T. On large-time behavior of solutions to the compressible Navier-Stokes equations in the half space in R3 . Arch Ration Mech Anal, 2002, 165: 89–159 [38] Kagei Y, Kobayashi T. Asymptotic behavior of solutions to the compressible Navier-Stokes equations on the half space. Arch Ration Mech Anal, 2005, 177: 231–330 [39] Kobayashi T. Some estimates of solutions for the equations of motion of compressible viscous fluid in an exterior domain in R3 . J Differential Equations, 2002, 184: 587–619 [40] Kobayashi T, Shibata Y. Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in R3 . Comm Math Phys, 1999, 200: 621–659 [41] Lai J, Wen H Y, Yao L. Vanishing capillarity limit of the non-conservative compressible two-fluid model. Discrete Contin Dyn, System-B, 2017, 22: 1361–1392 [42] Li H L, Zhang T. Large time behavior of isentropic compressible Navier-Stokes system in R3 . Math Methods Appl Sci, 2011, 34: 670–682 [43] Liu T P, Wang W K. The pointwise estimates of diffusion waves for the Navier-Stokes equations in odd multi-dimensions. Comm Math Phys, 1998, 196: 145–173 [44] Matsumura A, Nishida T. The initial value problem for the equation of motion of viscous and heatconductive gases. J Math Kyoto Univ, 1980, 20: 67–104 [45] Matsumura A, Nishida T. The initial value problem for the equation of motion of compressible viscous and
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heat-conductive fluids. Proc Japan Acad, Ser A, Math Sci, 1979, 55: 337–342 [46] Prosperetti A, Tryggvason G. Computational Methods for Multiphase Flow. Cambridge University Press, 2007 [47] Solem, S, Evje S. Relaxation limit of a compressible gas-liquid model with well-reservoir interaction. Z Angew Math Phys, 2017, 68: 23–25 [48] Sun Y Z, Wang C, Zhang Z F. A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navier-Stokes equations. J Math Pures Appl, 2011, 95: 36–47 [49] Wang Y J, Tan Z. Optimal decay rates for the compressible fluid models of Korteweg type. J Math Anal Appl, 2011, 379: 256–271 [50] Wen H Y, Yao L, Zhu C J. A blow-up criterion of strong solution to a 3D viscous liquid-gas two-phase flow model with vacuum. J Math Pures Appl, 2012, 97: 204–229 [51] Yao L, Zhang T, Zhu C J. Existence and asymptotic behavior of global weak solutions to a 2D viscous liquid-gas two-phase flow model. SIAM J Math Anal, 2010, 42: 1874–1897 [52] Yao L, Zhang T, Zhu C J. A blow-up criterion for a 2D viscous liquid-gas two-phase flow model. J Differential Equations, 2011, 250: 3362–3378 [53] Yao L, Zhu C J. Free boundary value problem for a viscous two-phase model with mass-dependent viscosity. J Differential Equations, 2009, 247: 2705–2739 [54] Yao L, Zhu C J. Existence and uniqueness of global weak solution to a two-phase flow model with vacuum. Math Ann, 2011, 349: 903–928 [55] Zuber N, Findlay J A. Average volumetric concentration in two-phase flow systems. J Heat Transf, 1965, 87: 453–468
REVIEW ON MATHEMATICAL ANALYSIS OF SOME TWO-PHASE FLOW MODELS∗
§)
Huanyao WEN (
1
[)
Lei YAO (
2
Á )
Changjiang ZHU (
1†
1. School of Mathematics, South China University of Technology, Guangzhou 510641, China 2. School of Mathematics and Center for Nonlinear Studies, Northwest University, Xi’an 710127, China E-mail : [email protected]; [email protected]; [email protected]
In Memory of Professor Xiaqi DING Abstract The two-phase flow models are commonly used in industrial applications, such as nuclear, power, chemical-process, oil-and-gas, cryogenics, bio-medical, micro-technology and so on. This is a survey paper on the study of compressible nonconservative two-fluid model, drift-flux model and viscous liquid-gas two-phase flow model. We give the research developments of these three two-phase flow models, respectively. In the last part, we give some open problems about the above models. Key words
compressible nonconservative two-fluid model; drift-flux model; viscous liquidgas two-phase flow model; well-posedness
2010 MR Subject Classification
1 1.1
76T10; 76N10
Compressible Nonconservative Two-fluid Model Research background-1
One of the typical two-phase flow models, called compressible nonconservative two-fluid model, is ± ± ± ± ± ∂t (α ρ ) + div(α ρ u ) = 0, (1.1) ∂t (α± ρ± u± ) + div(α± ρ± u± ⊗ u± ) + α± ∇P ± (ρ± ) = div(α± τ ± ) + σ ± α± ρ± ∇△(α± ρ± ) − f ± u± + I(u+ − u− ) + α± ρ± g.
The variable α+ , α− ∈ [0, 1] are the volume fraction of liquid and gas, satisfy α+ + α− = 1; ± ρ± , u± , P ± (ρ± ) = A± (ρ± )γ¯ are, respectively, the densities, the velocities and the pressures of each phase, and the two pressure functions satisfy P + (ρ+ ) − P − (ρ− ) = f (α− ρ− ), where ∗ Received February 5, 2018. Wen was supported by the National Natural Science Foundation of China (11722104, 11671150), and by GDUPS (2016) and the Fundamental Research Funds for the Central Universities of China (D2172260). Yao was supported by the National Natural Science Foundation of China (11571280, 11331005) and FANEDD No. 201315. Zhu was supported by the National Natural Science Foundation of China (11331005, 11771150). † Corresponding author: Changjiang ZHU.
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A± > 0, γ¯ ± > 1 are constants, f (·) is a known smooth function, it can be a zero function (in this case, the two pressures are equal); τ ± := µ± (∇x u± + ∇tx u± ) + λ± divu± Id are the viscous stress tensors, µ± and λ± are shear and bulk viscosity coefficients, satisfying µ± > 0, 2µ± + dλ± ≥ 0; σ ± are the capillary coefficients; The last three terms on the right-hand side of momentum equations, denote wall frictional forces, interfacial forces and gravities, f ± , I, g are wall frictional coefficients, interfacial coefficient and gravity constant, respectively. This model is commonly used in industrial applications, such as nuclear, power, oil-and-gas, micro-technology and so on, here we can refer to [2]. The classical modeling technique consists of performing a volume average to derive a model without free surface, that is, a two-fluid model. Furthermore, internal viscous and capillary forces cannot be neglected for some applications such as, for instance, wave breaking. Here, we include a capillary pressure term, i.e., we do not assume that the two phase pressures P + and P − are equal. The assumption about non-equal pressure functions P + 6= P − is quite natural. This amounts to including capillary pressure forces and is commonly included in modeling of two-phase flow in porous media. For more information about this model and related models, see for instance [1, 36, 46] and references therein. There are many numerical results about this model and related models, see [46] and references therein. If we ignore frictional forces between wall and fluids, interfacial friction and gravity, then system (1.1) takes the simplified form ± ± ± ± ± ∂t (α ρ ) + div(α ρ u ) = 0, (1.2) ∂t (α± ρ± u± ) + div(α± ρ± u± ⊗ u± ) + α± ∇P ± (ρ± ) = div(α± τ ± ) + σ ± α± ρ± ∇△(α± ρ± ).
Many models are related to system (1.2) such as compressible Navier-Stokes equations (α+ ≡ 0 or α− ≡ 0 and σ ± = 0), Navier-Stokes-Korteweg equations (α+ ≡ 0 or α− ≡ 0), etc. There is a huge literature on the investigation of global existence and large time behavior of smooth solutions in relation to these models. Here we mention several of the most relevant papers. Matsumura and Nishida [44, 45] first considered the global existence of smooth solutions to the compressible Navier-Stokes equations in multidimensional whole space and obtained that the global solutions tended to its equilibrium state, they also obtained the decay rate; Hoff and Zumbrun [33, 34] considered the Green’s function of the compressible isentropic NavierStokes equations with the artificial viscosity and showed the convergence of global solutions to diffusion waves. Later, Liu and Wang [43] investigated the properties of Green’s function for the isentropic Navier-Stokes system in odd dimension. They showed an interesting pointwise convergence of global solution to the diffusion waves with the optimal time decay rate, where the important phenomena of the weaker Huygens’principle was also justified due to the dispersion effects of compressible viscous fluids in multidimensional odd space. Recently, Li and Zhang [42] obtained the optimal Lp time-decay rate for isentropic Navier-Stokes equations in three −s dimensions when initial data belonged to some space H l ∩ B˙ 1,∞ (l ≥ 4, s ∈ [0, 1]). The same decay property also appears in the half space and exterior domain, see [37–40]. Near the constant equilibrium states, Hattori and Li [31, 32] considered the local and global existence of the smooth solution of the compressible Navier-Stokes-Korteweg system for multidimensional model in Sobolev space; Danchin and Desjardins [6] proved existence and uniqueness results of suitably smooth solution in the Besov space frame; Next, Wang and Tan
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[49] obtained the optimal time-decay of the system, which is the same as that of the NavierStokes system. Recently, Based on the weighted L2 -method and some delicate L1 estimates on solutions to the linearized problem, Chen and Zhao [4] studied the existence and uniqueness of stationary solutions of the compressible nonisentropic fluid model of Korteweg type by the contraction mapping principle. Furthermore, they also obtained the stability of the stationary solution. In recent years, some efforts were made on the existence and large time behavior of global solution to the nonconservative compressible two-fluid model (1.2). For the equal pressures, Bresch, Desjardins, Ghidaglia and Grenier [2] considered the existence of global weak solution in the periodic domain when 1 < γ¯ ± < 6; Bresch, Huang and Li [3] showed the existence of global weak solution in one-dimensional case without capillary effect (i.e., σ ± =0), when γ¯ ± > 1. Recently, Lai, Wen and Yao [41] obtained the vanishing capillary coefficient limit to the nonconservative compressible two-fluid model (1.2), where two pressure functions are unequal. We refer also to the recent work [25] which was in a gas-liquid context where a polytropic gas law was used for the gas phase whereas the liquid was assumed to be incompressible, and the global existence of strong solutions and large time behavior were obtained for both the initial boundary value problem and initial value problem. For more results about the compressible nonconservative two-fluid model (1.2), see [19, 21]. Next, we give the main well-posedness results about the model (1.2). By using the detailed analysis of the Green’s function to the linearized system and elaborate energy estimates to the nonlinear system, the time-decay estimates of classical or strong solutions of nonconservative viscous compressible generic two-fluid model with or without capillary effect in R3 were obtained in [5, 18], the equal and unequal pressures were considered respectively. 1.2
Well-posedness Results We consider the following compressible nonconservative two-fluid model in R3 : α+ + α− = 1, ∂ (α± ρ± ) + div(α± ρ± u± ) = 0, t
∂t (α± ρ± u± ) + div(α± ρ± u± ⊗ u± ) + α± ∇P ± (ρ± ) = div(α± τ ± ), + + P (ρ ) − P − (ρ− ) = f (α− ρ− ), ±
(1.3)
±
) ± where dP ± = s2± dρ± , s2± := dP ¯ ± P ρ(ρ , and s± denote the sound speed of each ± dρ± (ρ ) = γ ± ± ± phase separately. Introduce the variable R = α ρ , then ±
dρ+ =
1 (dR+ − ρ+ dα+ ), α+
dρ− =
1 (dR− − ρ− dα+ ). α−
(1.4)
Furthermore, from (1.4) and dP + − dP − = df (α− ρ− ), we obtain α− s2 α+ α− dα = − + 2 + + − 2 dR+ − − + 2 α ρ s+ + α ρ s− α ρ s+ + α+ ρ− s2− +
s2− ′ + f dR− . α−
Substitute (1.5) into (1.4), we get the following expressions: − ′ ρ+ ρ− s2− − + + +α f − dρ+ = − + 2 2 ρ dR + ρ + ρ dR , R (ρ ) s+ + R+ (ρ− )2 s2− s2−
(1.5)
(1.6)
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ρ+ ρ− s2+ dρ = − + 2 2 R (ρ ) s+ + R+ (ρ− )2 s2− −
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+ ′ − + + −α f − , ρ dR + ρ − ρ dR s2+
(1.7)
which give dP
−
dP
+
and
where we denote C 2 := Define
+ ′ − + + −α f − , = C ρ dR + ρ − ρ dR s2+
(1.8)
− ′ − + + +α f − = C ρ dR + ρ + ρ , dR s2−
(1.9)
2
2
s2− s2+ . α− ρ+ s2+ +α+ ρ− s2−
ϕ(ρ+ ) = P + (ρ+ ) − P −
R − ρ+ + ρ − R+
− f (R− ) = 0.
It is easy to see that in (R+ , +∞), ϕ′ (ρ+ ) = s2+ + s2−
R− R+ > 0, (ρ+ − R+ )2
and ϕ: (R+ , +∞) 7→ (−∞, +∞). Applying the implicit function theorem to (1.3)4 , there exists ρ+ = ρ+ (R+ , R− ) such that + ϕ(ρ ) = 0. Furthermore, we can define R− ρ+ (R+ , R− ) , − R+ R+ α+ (R+ , R− ) = + + − , ρ (R , R ) R+ R− α− (R+ , R− ) = 1 − + + − = − + − . ρ (R , R ) ρ (R , R )
ρ− (R+ , R− ) =
ρ+ (R+ , R− )
Based on the above, system (1.3) can be rewritten as follows: ∂t R± + div(R± u± ) = 0, − ′ + + + + + + 2 − + + +α f − ∂ (R u ) + div(R u ⊗ u ) + α C ρ ∇R + ρ + ρ ∇R t s2− = div{α+ [µ+ (∇u+ + ∇t u+ ) + λ+ divu+ Id]}, + ′ ∂t (R− u− ) + div(R− u− ⊗ u− ) + α− C 2 ρ− ∇R+ + ρ+ − ρ− αs2f ∇R− + − − − t − − − = div{α [µ (∇u + ∇ u ) + λ divu Id]}.
(1.10)
We consider the Cauchy problem of (1.10):
− − (R+ , u+ , R− , u− )(x, t) |t=0 = (R0+ , u+ 0 , R0 , u0 )(x),
x ∈ R3 ,
(1.11)
and + − > 0, |x| → ∞. u+ (x, t) → 0, u− (x, t) → 0, R+ (x, t) → R∞ > 0, R− (x, t) → R∞
(1.12)
+ − Without loss of generality, we assume R∞ = R∞ = 1. For the above problem (1.10)–(1.11), we obtain the following result.
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Theorem 1.1 ([18], Theorem 1.1) Under the condition −
η − s2− (1, 1) s2− (1, 1) < f ′ (1) < < 0, − α (1, 1) α− (1, 1)
(1.13)
where η is a positive, small fixed constant, there exists a constant ǫ such that if − − k (R0+ − 1, u+ 0 , R0 − 1, u0 ) kH 2 (R3 ) ≤ ǫ,
(1.14)
then the Cauchy problem (1.10)–(1.11) admits a unique solution (R+ , u+ , R− , u− ) globally in time in the sense that: R+ − 1, R− − 1 ∈ C 0 ([0, ∞); H 2 (R3 )) ∩ C 1 ([0, ∞); H 1 (R3 )),
u+ , u− ∈ C 0 ([0, ∞); H 2 (R3 )) ∩ C 1 ([0, ∞); L2 (R3 )).
− − 1 3 Moreover, if in addition the initial data (R0+ − 1, u+ 0 , R0 − 1, u0 ) is bounded in L (R ), the solution enjoys the following decay-in-time estimates: 3
k (R+ − 1, u+ , R− − 1, u− )(t) kL2 (R3 )) ≤ C(1 + t)− 4 , t ≥ 0,
(1.15)
k ∇(R , u , R , u )(t) kH 1 (R3 ) ≤ C(1 + t)
(1.16)
+
+
−
−
− 54
, t ≥ 0.
Remark 1.2 The proof of the theorem involves some new ideas. Usually, the proof consists of spectral analysis of the Green function for the corresponding linearized system and energy estimates of the solutions to the nonlinear system, refer for instance to [7, 8]. Here, we just need spectral analysis of the low-frequency part of the Green function and the energy estimates. Actually, we employ the energy method in the frequency space to get the decay rates of the low-frequency part so that we succeed to avoid some complicate analysis of the Green function, which is a 8 × 8 matrix. On the other hand, we notice that the high-frequency part can be handled directly by the energy estimates. Thus, the combination of the decay rates of the low-frequency part and the energy estimates show the decay rates of the solutions directly, even for initial data within the H 2 -framework. If the capillary effect is included, we consider the following equations: α+ + α− = 1, ∂ (α± ρ± ) + div(α± ρ± u± ) = 0, t
∂t (α± ρ± u± ) + div(α± ρ± u± ⊗ u± ) + α± ∇P ± (ρ± ) = div(ν ± α± ρ± (∇x u± + ∇tx u± )) + σ ± α± ρ± ∇△(α± ρ± ).
And we consider the two equal pressure functions: P = P (ρ+ ) = A+ (ρ+ )γ¯ − A− (ρ− )γ¯ , γ¯ ± > 1, i.e., f = 0. And (1.5)–(1.9) becomes (1.18)–(1.21): dα+ = dρ+ = dρ− =
α− s2+ α+ s2− + dR − dR− , α− ρ+ s2+ + α+ ρ− s2− α− ρ+ s2+ + α+ ρ− s2− R− (ρ+ )2 s2+
s2− (ρ− dR+ + ρ+ dR− ), + R+ (ρ− )2 s2−
s2+ (ρ− dR+ + ρ+ dR− ), R− (ρ+ )2 s2+ + R+ (ρ− )2 s2−
(1.17)
+
= P (ρ− ) =
(1.18) (1.19) (1.20)
and dP = C 2 (ρ− dR+ + ρ+ dR− ).
(1.21)
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Then the system (1.17) is equivalent to the following form ± ± ± ∂t R + div(R u ) = 0,
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∂t (R± u± ) + div(R± u± ⊗ u± ) + α± C 2 (ρ− ∇R+ + ρ+ ∇R− ) ±
±
±
= div[ν R (∇x u +
∇tx u± )]
±
±
(1.22)
±
+ σ R ∇△R .
We consider the problem (1.22) with initial value (1.11)–(1.12). For simplicity, set ν + = ν − = ν, σ + = σ − = σ. Introduce n± = R± − 1, then the initial value problem (1.22), (1.11) and (1.12) can be rewritten as ∂t n+ + divu+ = S1 , ∂ u+ + β ∇n+ + β ∇n− − ν∇u+ − σ∇△n+ = S , t 1 2 2 (1.23) − − ∂t n + divu = S3 , ∂t u− + β3 ∇n+ + β4 ∇n− − ν∇u− − σ∇△n− = S4 ,
with initial data
+ − − + + − − (n+ , u+ , n− , u− ) |t=0 = (n+ 0 , u0 , n0 , u0 )(x) = (R0 − 1, u0 , R0 − 1, u0 )(x),
x ∈ R3 , (1.24)
and the far-field behavior
2
−
(n+ , u+ , n− , u− )(x, t) → 0,
(1,1) Here β1 = C (1,1)ρ , β2 = β3 = C 2 (1, 1), β4 = ρ+ (1,1) β2 β3 = β32 ), and
S1 = S1 (n+ , u+ ) = −div(n+ u+ ),
S3 = S3 (n− , u− ) = −div(n− u− ),
2
|x| → ∞. +
C (1,1)ρ (1,1) ρ− (1,1)
(1.25) > 0 (which implies β1 β4 =
S2 = S2 (S21 , S22 , S23 )(n+ , u+ , n− , u− ), S4 = S4 (S41 , S42 , S43 )(n+ , u+ , n− , u− ),
and S2j (n+ , u+ , n− , u− )
S4j (n+ , u+ , n− , u− )
(C 2 ρ− )(n+ + 1, n− + 1) (C 2 ρ− )(1, 1) − ∂j n+ − (u+ · ∇)u+ =− j ρ+ (n+ + 1, n− + 1) ρ+ (1, 1) ν ν + + ∂k n+ ∂k u+ ∂k n+ ∂j u+ j + + k n +1 n +1 − [C 2 (n+ + 1, n− + 1) − C 2 (1, 1)]∂j n− , j = 1, 2, 3,
(C 2 ρ+ )(n+ + 1, n− + 1) (C 2 ρ+ )(1, 1) =− − ∂j n− − (u− · ∇)u− j ρ− (n+ + 1, n− + 1) ρ− (1, 1) ν ν + − ∂k n− ∂k u− ∂k n− ∂j u− j + − k n +1 n +1 2 + − 2 + − [C (n + 1, n + 1) − C (1, 1)]∂j n , j = 1, 2, 3.
For the above problem (1.23)–(1.25), we obtain the following result. Theorem 1.3 ([5], Theorem 1.1) For any integer s ≥ 3, there exists a constant δ > 0, such that if − + + k (n+ 0 , n0 ) kH s+1 (R3 )∩L1 (R3 ) + k (u0 , u0 ) kH s (R3 )∩L1 (R3 ) ≤ δ, then the initial value problem (1.23)–(1.25) admits a unique solution (n+ , u+ , n− , u− ) globally in time, which satisfies n+ , n− ∈ C 0 ([0, ∞); H s+1 (R3 )) ∩ C 1 ([0, ∞); H s (R3 )),
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u+ , u− ∈ C 0 ([0, ∞); H s (R3 )) ∩ C 1 ([0, ∞); H s−1 (R3 )), and for some c, C > 0 independent of t such that ||(n+ , n− )(t)||2H s+1 (R3 ) + ||(u+ , u− )(t)||2H s (R3 ) + ||(β3 ∇n+ + β4 ∇n− )(t)||2H s−1 (R3 ) Z t +c ||(n+ , n− )(τ )||2H s+2 (R3 ) + ||(u+ , u− )(τ )||2H s+1 (R3 ) 0 + ||(β3 ∇n+ + β4 ∇n− )(τ )||2H s (R3 ) dτ − 2 + − 2 ≤ C ||(n+ , n )|| + ||(u , u )|| 0 0 0 0 H s (R3 )∩L1 (R3 ) . H s+1 (R3 )∩L1 (R3 ) Moreover, the solution satisfies the decay-in-time estimates: 1
k
||∇k (n+ , n− )(t)||L2 (R3 ) ≤ C(1 + t)− 4 − 2 ,
0 ≤ k ≤ s − 1,
||∇ (u , u )(t)||L2 (R3 ) ≤ C(1 + t)
0 ≤ k ≤ s − 2,
k
+
− 34 − k 2
−
,
1
s
||∇s (n+ , n− )(t)||H 1 (R3 ) + ||∇s−1 (u+ , u− )(t)||H 1 (R3 ) ≤ C(1 + t) 4 − 2 ,
t ≥ 0.
Remark 1.4 The fraction densities converge to the equilibrium states at the L2 -rate 1 (1 + t)− 4 , and the k(∈ [1, s − 1]) order spatial derivatives of the fraction densities converge 1 3 k to zero at the L2 -rate (1 + t)− 4 − 2 , which is slower than the L2 -rate (1 + t)− 4 and L2 -rate 3 k (1 + t)− 4 − 2 for the compressible Navier-Stokes system (k = 0, 1) or Navier-Stokes-Korteweg system (k = 0, 1, 2), see [7, 43, 49]. This is caused by the non-conservation and complexity of the model (1.17).
2 2.1
Drift-flux Model Research Background-2
The drift-flux model is one of the commonly used models nowadays for the prediction of various two-phase flows. It was first developed by Zuber and Findlay [55]. It is used in chemical engineering to predict flow in bubble column reactors, in petroleum applications to model various wellbore operations related to drilling, production of oil and gas, and for the study of geothermal energy related problems and injection of CO2 . A one-dimensional transient drift-flux model can be written in the following form: ∂t (αl ρl ) + ∂x (αl ρl ul ) = 0, (2.1) ∂t (αg ρg ) + ∂x (αg ρg ug ) = 0, ∂t (αl ρl ul + αg ρg ug ) + ∂x (αl ρl u2l + αg ρg u2g + P (m, n)) = ∂x (ǫ∂x umix ) − q¯,
where m = αl ρl , n = αg ρg denote the masses of liquid and gas; The unknowns are αl , αg ∈ [0, 1] volume fractions of liquid and gas, satisfying αl + αg = 1; ρl , ρg the liquid and gas densities; P = P (m, n) common pressure for liquid and gas; ul , ug velocities of liquid and gas. In order to get a closed system, an algebraic equation called the slip relation which relates the two fluid velocities is added: ug = c0 umix + c1 , it implies: f (αg , ul , ug , ρl , ρg ) = ug − c0 umix − c1 = 0,
where c0 and c1 are flow dependent coefficients, c0 is referred to as the distribution parameter and c1 to as the drift velocity, umix = αg ug + αl ul ; ǫ = ǫ(m, n)(≥ 0) viscosity coefficient; q¯
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external force, such as the gravity and frictional force. For previous studies of the 1D model (2.1) with the slip relation ug − c0 umix − c1 = 0 with c0 > 1, c1 = 0, for the free boundary problem, we refer to [10, 22]. In [10], the local existence of weak solution was obtained whereas [22] gave a local existence of weak solution for a general slip (c0 > 1 and c1 > 0) and a global existence result for the special case c0 > 1, c1 = 0. The more general case c1 > 0 is important because it allows the model to describe, e.g. counter-current flow, where ul and ug possibly have different sign. Recently, Evje and Wen [20] obtained the global existence and unique of strong solution for the 1D initial boundary value problem. This work presents a first global existence result for the drift-flux model with a general slip law. Next, we give the main well-posedness results about the model (2.1). 2.2
Well-posedness Results
At first, we ignore the external force, i.e., q¯ = 0. Assume that c0 ≥ 1, c1 > 0, x ∈ (a, b(t)) (where b(t) separates the gas-liquid mixture and the gas region, satisfies: db dt = u(b(t), t), b(0) = b0 ), t > 0, and the pressure is given by γ n P (m, n) = . (2.2) ρl − m Introduce α∗g , α∗l as follows
α∗g =
1 , α∗l = 1 − α∗g . c0
Now, we introduce the notation: ρ = n + m − k ∗ , c = ρl − k ∗ = ρl α∗g , ρ(1, t) = n∗ .
m−k∗ ρ ,
where k ∗ = ρl α∗l . And a∗ =
As [23], by using the Lagrangian coordinates transformation, and introducing the function ρ Q(c, ρ) = a∗ −cρ , satisfies 1 cρ Q(c, ρ) = Qc ct + Qρ ρt = Qρ ρt = + ρt a∗ − cρ (a∗ − cρ)2 a∗ a∗ ρ2 = ∗ ρ = − ux = −a∗ Q(c, ρ)2 ux . t (a − cρ)2 (a∗ − cρ)2
Consider the following initial boundary value problem ∂t c = 0, ∂t Q + a∗ Q2 ∂x u = 0,
2
∂t u + ∂x [P (c, Q) − u g(cQ) − uh(cQ) + j(cQ)] = ∂x [E(cQ)∂x u],
(2.3) x ∈ (0, 1), t > 0,
with cQ := E(cQ), P (c, Q) = [(1 − c)Q]γ , E(cρ) = cρ = a∗ 1 + cQ cρ cQ g(cρ) = k ∗ = a∗ α∗l := g(cQ), cρ + k ∗ α∗l + cQ c1 cρ cQ ∗ c1 h(cρ) = 2ρl = 2a := h(cQ), c0 k ∗ + cρ c0 α∗l + cQ 2 2 c1 1 c1 1 + cQ j(cρ) = ρ2l = ρ := j(cQ). l ∗ c0 k + cρ c0 α∗l + cQ
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Initial conditions are c(x, 0) = cˆ0 (x), Q(x, 0) = Q0 (x) = and boundary conditions are
ρ0 , u(x, 0) = u0 (x), x ∈ [0, 1], a∗ − cˆ0 ρ0 n∗ := Q∗ , c(1, t) = 0. a∗
u(0, t) = 0, Q(1, t) =
(2.4)
(2.5)
Assumptions on the parameters α, γ 0<α<
1 , γ > 1. 2
Assumptions on the initial data c˜ φα ≤ cˆ0 ≤ c˜2 φα , 0 < c˜1 ≤ c˜2 , sup cˆ0 < 1, A1 ≤ Q0 ≤ B1 , 1 Z 1 Q0 1 β ˜ G0 (x)dx ≤ M, ln Q∗ (1 + cˆ Q ) ≤ Cφ1 , β1 ∈ (0, α] ∩ 0, 2 − α , 0 0 0 1−α1 Q0 2 cˆ0 ln ∈ L2 , α1 ∈ (2α, 1) such that φ Q∗ (1 + cˆ0 Q0 ) x 1−α1 −α+2β1 p 2 φ cˆ0,x ∈ L2 , E(ˆ c0 Q0 )u0,x ∈ L2 ,
(2.6)
(2.7)
˜ where φ(x) = 1 − x, and G0 is the initial energy with positive constants c˜1 , c˜2 , A1 , A2 and C, as follows: 2 Z Q 1 2 p∗ − P (c, Q∗ ) ρl c1 cˆ0 Q0 P (c, s) − P (c, Q∗ ) G0 (x) = u0 + ds + − ∗ ∗ cˆ0 ln , 2 a ∗ s2 a∗ Q k a c0 a∗l + cˆ0 Q0 Q∗ where p∗ = [Q∗ ]γ . Moreover, the lower bound A of Q and the following relations 2 c1 µ2 c˜1 A ≤ , α∗ c 8 0 l √ A1 M A < , e4/˜c1 < 4 µ
bound M on the initial energy must satisfy the µ2 c˜1 A , 32 √ 4 M 1 , e4/˜c1 < + 1. 3 µ 6A M≤
(2.8)
The external pressure p∗ must obey
(p∗ )1/γ ≥ max{B1 (1 − sup cˆ0 ), 4A}.
(2.9)
Similarly, the upper bound B of Q and the bound M on the initial energy must satisfy the following relations: 2 ∗ γ B c1 c1 a γ ∗ ∗ ∗ ≥ C(p , c0 , c1 ) = p + a + ρl , (1 − sup cˆ0 ) 2 c0 c0 k∗ B 2 − 4δ (2.10) B1 ≤ , B ≤ , 2 3δ 1 2 12 2 c1 1 (2M ) 2 + 2 +1 1+ BM ≤ ln(1 + δ), µ˜ c1 µ c˜1 A c0 α∗l
for some δ > 0 and subject to the condition sup cˆ0 < 1. In addition, A and B are chosen such that: 3 2A ≤ Q0 ≤ B. (2.11) 4
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Finally, M must obey the smallness condition as follows: 1 2M 2 + 1 ≤ . µ2 c˜1 A 64
(2.12)
Theorem 2.1 ([23], Theorem 2.1 (existence)) Under the assumptions of (2.6)–(2.12), there exists a constant M0 > 0 such that (2.3)–(2.5) admits a global weak solution (c, Q, u) on [0, 1] × [0, T ] for any time for all M ≤ M0 in the sense that (A) The following regularity holhs
3
c, Q ∈ L∞ ([0, T ]; L∞ ) ∩ C 4 ([0, T ]; L2 ),
1
E(cQ)ux ∈ L∞ ([0, T ]; L2),
u ∈ L∞ ([0, T ]; L∞) ∩ C 2 ([0, T ]; L2 ).
Moreover, the following estimates hold Z A ≤ Q ≤ B, ||ux ||L4 (0,T ;L2 ) +
Z
1
1 0
Eu2x dx +
1−α1
φ(x)
0
Z tZ 0
1
0
u2s dxds ≤ C2 ,
Q c ln Q∗ (1 + cQ)
x
2 dx ≤ C2 ,
(2.13) (2.14)
for (x, t) ∈ [0, 1] × [0, T ], where C2 depends on A, B, M, c˜1 , c˜2 , α, β, γ, T and the initial data. (B) The following equations hold ∂t c = 0, ∂t Q + a∗ Q2 ∂x u = 0, a.e. (x, t) ∈ (0, 1) × (0, T ], (c, Q)(x, 0) = (ˆ c0 (x), Q0 (x)), a.e. (x, t) ∈ [0, 1], Z T Z 1 uϕt + P (c, Q) − u2 g(cQ) − uh(cQ) + j(cQ) 0 0 " # 2 Z 1 ρ l c1 ∗ − + p − µE(cQ)u )ϕ dxdt + u0 ϕ(x, 0)dx = 0, x x α∗l c0 0
(2.15)
for any test function ϕ ∈ C0∞ ((0, 1] × [0, T )). (C) Interface behavior
|u(x, t)| ≤ C2 x
r−1 r
,
(2.16)
for some r ∈ (1, 2) such that r(α + 1) < 2, and |Q(x, t) − Q∗ | ≤ C2 |x − 1|β1 ,
(2.17)
for β1 ∈ (0, α] ∩ (0, 12 − α]. Here (x, t) ∈ [0, 1] × [0, T ]. Theorem 2.2 ([23], Theorem 2.2 (uniqueness)) Under the conditions of Theorem 2.1 and by requiring that β1 = α, where 0 < α ≤ 14 , the weak solution is unique. Next, assume that c0 ≥ 1, c1 = 0, x ∈ (a, b(t)), t > 0, i.e., h(cQ) = j(cQ) = 0. Set E(c, cQ) = [cQ]β+1 , β > 0, and n∗ = 0 (ρ(1, t) = n∗ = 0). Consider the following initial boundary value problem: ∂t c = 0, (2.18) ∂t Q + a∗ Q2 ∂x u = 0, 2 ∂t u + ∂x [P (c, Q) − u g(cQ)] = ∂x [E(cQ)∂x u], x ∈ (0, 1), t > 0,
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with initial conditions c(x, 0) = cˆ0 (x), Q(x, 0) = Q0 (x) =
ρ0 (x) , u(x, 0) = u0 (x), x ∈ [0, 1], a∗ − cˆ0 ρ0 (x)
(2.19)
and with boundary condition u(0, t) = 0, (c, Q)(1, t) = 0.
(2.20)
˜ B ˜ and δ be positive constants, whereas α, β > 0 and γ > 1 Assumptions Let A1 , B1 , A, such that α α B1 φ 4 ≤ cˆ0 ≤ A1 φ 4 , B1 ≤ A1 , 1 1 3α Bδ ˜ ˜ γ−1 ˜ 3α ˜ γ−1 4 , φ 4 ≤ Q0 ≤ Aφ ≤ A, Bδ ! Z 1 (2.21) u20 (1 − cˆ0 )γ Qγ−1 0 + dx ≤ δ, 2 a∗ (γ − 1) 0 β+1 α (ˆ c0 Q0 ) 2 u0x ∈ L2 , φ1−αβ |(ˆ cβ+1 Qβ0 )x |2 ∈ L1 , φ1− 4 cˆ20,x ∈ L1 , 0 where φ(x) = 1 − x, and α, β, γ as well as the time decay exponents r1 , r2 , r3 , r4 > 0, satisfy the following 1 α(β + 1) < 1, α + 1 − 4αβ ≥ 0, α(4β + 1) ≤ 2, β ≤ , γ 2r β r (β + 1) 3 3 γ > max 5β + 2, 1 + r − 2r β , 1 + r − r (β + 1) , 3 2 3 2 2β γ − 1 1 γ ≥ , r3 < , r1 < , 2 − α(β + 1) γ γ−β (2.22) r2 + r1 (γ − 1 − β) > 1, r2 (β + 1) + r3 ≤ 1, 3r2 β ≤ r3 , r (β + 1) < r , r < r (2γ − β − 1), 2 3 4 1 r4 ≤ min{1 + r3 , r1 (γ − β) + r3 }, r2 (5β + 1) ≤ r4 , r2 > r1 .
Theorem 2.3 ([24], Theorem 3.1 (global weak solution)) Under the assumptions of (2.21)–(2.22), there exists a positive constant C0 such that if δ ≤ C0 , then (2.18)–(2.20) admits a unique solution (c, Q, u) on [0, 1] × [0, ∞) in the sense that: (A) the following regularity holds
c, Q ∈ L∞ ([0, T ]; L∞ ) ∩ C 1 ([0, T ]; L2),
1
E(cQ)ux ∈ L∞ ([0, T ]; L2 ),
u ∈ L∞ ([0, T ]; L∞) ∩ C 2 ([0, T ]; L2 ).
(B) the following equations hold ∂t c = 0, ∂t Q + a∗ Q2 ∂x u = 0, a.e. (x, t) ∈ (0, 1) × (0, T ], (c, Q)(x, 0) = (ˆ c0 (x), Q0 (x)), a.e. (x, t) ∈ [0, 1], Z TZ 1 Z uϕt + P (c, Q) − u2 g(cQ) − E(cQ)ux ϕx dxdt + 0
0
for any T > 0 and any test function ϕ ∈ C0∞ ((0, 1] × [0, T )).
0
1
u0 ϕ(x, 0)dx = 0,
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In the next section, we will consider the model (2.1) with c0 = 1 and c1 = 0, i.e., ug = ul = u, we call this model as the viscous liquid-gas two-phase flow model. We will use the appropriate αg ρg variable transformation, to rewrite the model in terms of ( αl ρl +α , αl ρl + αg ρg , u), which is g ρg αg ρg similar to the single-phase compressible Navier-Stokes equations, where αl ρl +α satisfies a g ρg transport equation.
3 3.1
Viscous Liquid-gas Two-phase Flow Model Research Background-3
Viscous liquid-gas two-phase flow model can be viewed to be simplified from the model, which are widely used within the petroleum industry to describe production and transport of oil and gas through long pipelines or well. This model is composed of two separate mass conservation equations corresponding to each of the two phases and one mixture momentum conservation equation in following form mt + div(mu) = 0, (3.1) nt + div(nu) = 0, T ((m + n)u)t + div((m + n)u ⊗ u) + ∇P (m, n) = div(µ(∇u + ∇u )) + ∇(λdivu) − q¯,
where m = αl ρl , n = αg ρg denote liquid mass and gas mass respectively; µ and λ are viscosity coefficients, satisfying: µ > 0, λ + d2 µ ≥ 0; The unknown variables αl , αg ∈ [0, 1] denote liquid and gas volume fractions, satisfying the fundamental relation: αl + αg = 1; Furthermore, ρl and ρg denote liquid and gas densities, respectively; u denotes velocity of liquid and gas; P = P (m, n) is common pressure for both phases, q¯ is the external force. The investigation of model (3.1) has been a topic during the last decade. There are many results about the numerical properties of this model or related model. However, there are few results providing insight into existence, uniqueness, regularity, asymptotic behavior and decay rate estimates concerning the two-phase liquid-gas models of the form (3.1). Let us review some previous works about the viscous liquid-gas two-phase flow model. For the model (3.1) in 1D, γ n when the liquid is incompressible and the gas is polytropic, i.e., P (m, n) = Cργl ρl −m , Evje and Karlsen [16] studied the existence and uniqueness of the global weak solution to the free mβ 1 boundary value problem with µ = µ(m) = k1 (ρl −m) β+1 , β ∈ (0, 3 ), when the fluids connected to vacuum state discontinuously. Evje, Fl˚ atten and Friis [13] also studied the model with nβ 1 µ = µ(m, n) = k1 (ρl −m) (β ∈ (0, )) in a free boundary setting when the fluids connected β+1 3 to vacuum state continuously, and obtained the global existence of the weak solution. If the acceleration terms in the mixture momentum equation was neglected, Evje and Karlsen [14] obtained the global existence of weak solution on the half line. If the external force and frictional force were included, see the related results in [27, 28]. Specifically, when both of the two fluids p are compressible, i.e., P (m, n) = C 0 (−b(m, n) + b(m, n)2 + c(n)), one can consult [15] for the global existence of strong solution to the 1D case; For multidimensional case, Yao, Zhang and Zhu [51] obtained the existence of the global weak solution to the 2D model when the initial energy is small. Furthermore, they proved a blow-up criterion in terms of the upper bound of the liquid mass for the strong solution to the 2D model in a smooth bounded domain, cf. [52]. For the Cauchy problem of a multi-dimensional viscous liquid-gas two-phase flow model, Hao
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and Li [30] obtained the global existence and uniqueness of strong solution for the initial data close to an equilibrium and the local in time existence and uniqueness of the solution with general initial data in the framework of Besov spaces. Concerning the well-posedness and large time behavior of solutions to the model (3.1) and related models, we refer the reader to the [9, 11, 12, 17, 29, 35, 47] and references therein. Next, we give the main well-posedness results about the model (3.1). 3.2
Well-posedness Results
At first, we consider the model (3.1) in 1D, in order to avoid some unsolved difficulties, we consider a simplified model obtained as follows: (1) Due to the fact that the liquid phase density is much higher than the gas phase density, typically to the order ρρgl = O(103 ), we can neglect the gas phase effects in the mixture momentum conservation equation. (2) Neglect the external forces, i.e., q¯ = 0. We assume that further the liquid is incompressible and the gas is polytropic, i.e., ρl = const, P = Cργg , γ > 1, C > 0. Then the model (3.1) can be simplified into the following model in the form ∂τ n + ∂ξ (nu) = 0, (3.2) ∂τ m + ∂ξ (mu) = 0, 2 ∂τ (mu) + ∂ξ (mu + P (m, n)) = ∂ξ (ǫ∂ξ u),
and
n . (3.3) ρl − m We will consider (3.2) in a free boundary value problem setting where the masses n and m initially occupy only a finite interval [a, b] ∈ R. That is P (m, n) = Cργl
n(ξ, 0) = n0 (ξ),
m(ξ, 0) = m0 (ξ),
u(ξ, 0) = u0 (ξ),
ξ ∈ [a, b],
and n(ξ, 0) = m(ξ, 0) = 0, ξ ∈ R \ [a, b]. The boundary conditions are given as (−P (m, n) + ǫu )(a(τ ), τ ) = 0, ξ (−P (m, n) + ǫuξ )(b(τ ), τ ) = 0,
(3.4)
(3.5)
) ) = u(a(τ ), τ ), a(0) = a; db(τ = where a(τ ) and b(τ ) are vacuum boundary, satisfying da(τ dτ dτ β m u(b(τ ), τ ), b(0) = b. As in [16], the viscosity coefficient ǫ = ǫ(m) = k1 (ρl −m)β+1 , where β ∈ (0, 1], k1 is a positive constant. In the Lagrangian coordinates, the free boundary value problem (3.2)–(3.5) becomes the following fixed boundary value problem ∂t n + mn∂x u = 0, (3.6) ∂t m + m2 ∂x u = 0, ∂t (mu) + ∂x (P (m, n)) = ∂x (ǫ(m)m∂x u), x ∈ (0, 1), t > 0,
with initial data
n(x, 0) = n0 (x), m(x, 0) = m0 (x), u(x, 0) = u0 (x), x ∈ [0, 1],
(3.7)
and the boundary conditions P (m, n) = ǫ(m)mux , x = 0, 1,
t ≥ 0.
(3.8)
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n Remark 3.1 Introduce the variables Q(m) = ρlm −m and c = m , we can rewrite the initial boundary value problem (3.6)–(3.8) in the form ∂t c = 0, ∂t Q(m) + ρl Q(m)2 ∂x u = 0, ∂t u + ∂x P (c, m) = ∂x (E(Q(m))∂x u) , x ∈ (0, 1), t > 0,
with initial data
c(x, 0) = cˆ0 (x), Q(x, 0) = Q0 (x), u(x, 0) = u0 (x),
x ∈ [0, 1],
and the boundary conditions P (c, Q(m)) = E(Q(m))ux , at x = 0, 1, t ≥ 0, where P (c, Q(m)) = cγ Q(m)γ ,
γ > 1,
E(Q(m)) = mε(m) = Q(m)β+1 , which is similar to the model of single-phase Navier-Stokes equations. Therefore, we can apply technique in studying Navier-Stokes equations to deal with the related problem for the viscous liquid-gas two-phase flow model. Yao and Zhu [53] improved the previous result of Evje and Karlsen [16] from β ∈ (0, 13 ) to β ∈ (0, 1], and got the global existence of weak solution, regularity of the solutions and the asymptotic behavior result. The result of global existence of weak solution is as follows. Theorem 3.2 ([53], Theorem 2.2 (Existence and uniqueness)) Under the assumptions of (A1 ) inf n0 (x) > 0, sup n0 (x) < ∞, inf m0 (x) > 0, sup m0 (x) < ρl , n0 , m0 ∈ x∈[0,1]
x∈[0,1]
x∈[0,1]
x∈[0,1]
W 1,∞ ([0, 1]), n0 (A2 ) u0 ∈ L∞ ([0, 1]), (E(m0 )u0x )x ∈ L2 ([0, 1]), [( ρl −m )γ ]x ∈ L2 ([0, 1]), 0 (A3 ) γ > 1, 0 < β ≤ 1, the initial boundary problem (3.6)–(3.8) possesses a unique global weak solution (n(x, t), m(x, t), u(x, t)) satisfying for any T > 0 C0 0 < C(T ) ≤ m(x, t) ≤ ρl (< ρl ), (x, t) ∈ [0, 1] × [0, T ], (3.9) 1 + C0 and 0 < C(T ) ≤ n(x, t) ≤ C, (x, t) ∈ [0, 1] × [0, T ]. (3.10) On the other hand, for the viscous liquid-gas model with constant viscosity coefficient when both the initial liquid and gas masses connect to vacuum continuously, Yao and Zhu [54] used a new technique to get the upper and lower bounds of gas and liquid masses n and m, then got the global existence of weak solution by the line method. For this case, the boundary conditions (3.8) is replaced by n = m = 0, x = 0, 1, t ≥ 0. (3.11) For simplicity, set ǫ = 1, Cργl = 1, then (3.3) becomes γ n . P (m, n) = ρl − m Next, we give the definition of weak solution.
(3.12)
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Definition 3.3 ([54], Definition 1) A pair of functions (n(x, t), m(x, t), u(x, t)) is called a global weak solution to the initial boundary value problem (3.6), (3.7) and (3.11), if for any T >0 1
n, m ∈ C([0, T ]; H 1 ([0, 1])) ∩ C 2 ([0, T ]; L2([0, 1])), u ∈ C([0, T ]; H 1 ([0, 1])) ∩ C 1 ([0, T ]; L2([0, 1])), 1
mux ∈ C([0, T ]; H 1 ([0, 1])) ∩ C 2 ([0, T ]; L2 ([0, 1])), and lim m(x, t) = lim n(x, t) = lim m(x, t) = lim n(x, t) = 0.
x→0+
x→0+
x→1−
x→1−
Furthermore, the following equations hold ∂t n + mn∂x u = 0, ∂t m + m2 ∂x u = 0, (n, m)(x, 0) = (n0 (x), m0 (x)), for a.e. x ∈ (0, 1) and any t ≥ 0,
and
Z
0
∞
Z
0
1
(uψt + (P (m, n) − mux )ψx )dxdt +
Z
1
u0 (x)ψ(x, 0)dx = 0,
0
for any test functions ψ(x, t) ∈ C0∞ (Ω), with Ω = {(x, t) : 0 ≤ x ≤ 1, t ≥ 0}.
Theorem 3.4 ([54], Theorem 1) Under the assumptions of (A1 ) m0 (x) > 0, x ∈ (0, 1); m0 (0) = m0 (1) = 0; m0 (x) ∈ C([0, 1]); sup m0 (x) < ρl ; x∈[0,1]
n0 (x) > 0, x ∈ (0, 1); n0 (0) = n0 (1) = 0; n0 (x) ∈ C([0, 1]); cˆ0 (x) ∈ C 1 ([0, 1]); cˆ0 (0) = cˆ0 (1) = 0, n0 (x), where cˆ0 (x) = m 0 R1 1/2 (A2 ) u0 ∈ L2 ([0, 1]); (m0 u0x )x ∈ L2 ([0, 1]); 0 < 0 m01(x) dx < ∞; m0 u0x ∈ L2 ([0, 1]); m0x ∈ L2 ([0, 1]), R1 R m (x) 1 γ−2 (A3 ) exp 2 0 u20 (x) + 2 0 0 cˆγ0 (x) (ρsl −s)γ ds dx 2 m0 (x) ≤ ρl (1 − δ). Here δ ∈ (0, 1) is a positive constant and satisfies: 1 − δ > maxx∈[0,1] αl,0 (x), and where αl,0 (x) = sup m0 (x) < ρl implies sup αl,0 (x) < 1, x∈[0,1]
m0 (x) ρl ,
and
x∈[0,1]
(A4 ) γ > 1, the initial boundary value problem (3.6), (3.7) and (3.11) possesses a unique global weak solution (m(x, t), n(x, t), u(x, t)) defined by Definition 3.3. Next, we consider (3.1) in 3D. We neglect the gas phase effects in the mixture momentum conservation equation and external force, i.e., study the following initial boundary value problem mt + div(mu) = 0, (3.13) nt + div(nu) = 0, (mu)t + div(mu ⊗ u) + ∇P (m, n) = div(µ(∇u + ∇uT )) + ∇(λdivu),
with initial conditions
(m, n, u) |t=0 = (m0 , n0 , u0 ),
¯ x ∈ Ω,
(3.14)
and with the boundary conditions u(x, t) = 0,
(x, t) ∈ ∂Ω × [0, ∞),
(3.15)
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where Ω ⊆ R3 is a smooth bounded domain; The viscous coefficients µ, λ are constants; Pressure P satisfies p P (m, n) = C 0 (−b(m, n) + b(m, n)2 + c(n)),
with C 0 = 12 α2l , k0 = ρl,0 −
Pl,0 α2l
> 0, α0 =
αg 2 , αl
b(m, n) = k0 − m − c(n) = 4k0
αg αl
αg αl
and 2
2
n = k0 − m − α0 n,
n = 4k0 α0 n.
For the problem (3.13)–(3.15), Wen, Yao and Zhu [50] proved the local existence of strong solution and established the blow-up criterion, when there was initial vacuum. If the liquid mass R was upper bounded, we could obtain a high integrability of the velocity, sup Ω m|u|r dx ≤ C, 0≤t≤T
for some r ∈ (3, 4]. Moreover, in order to overcome the singularity brought by the pressure P (m, n) when there is vacuum, we needed the assumption 0 ≤ s0 m0 ≤ n0 ≤ s0 m0 , where s0 and s0 were positive constants. Theorem 3.5 ([50], Theorem 1.1 (local existence)) Let Ω be a bounded smooth domain in R3 and q ∈ (3, 6]. Assume that the initial data m0 , n0 , u0 satisfy m0 , n0 ∈ W 1,q (Ω), u0 ∈ ¯ where s , s¯0 are positive constants. The following H01 (Ω)∩H 2 (Ω), 0 ≤ s0 m0 ≤ n0 ≤ s¯0 m0 in Ω, 0 compatible condition is also valid: −µ△u0 − (µ + λ)∇divu0 + ∇P (m0 , n0 ) =
√ m0 g, for some g ∈ L2 (Ω).
(3.16)
Then, there exist a T0 > 0 and a unique strong solution (m, n, u) to the problem (3.13)–(3.15), such that 0 ≤ s0 m ≤ n ≤ s¯0 m,
(m, n) ∈ C([0, T0 ]; W 1,q (Ω)),
(mt , nt ) ∈ L∞ (0, T0 ; Lq (Ω)),
P ∈ L∞ (0, T0 ; W 1,q (Ω)), u ∈ L∞ (0, T0 ; H01 (Ω) ∩ L2 (0, T0 ; W 2,q (Ω)), √ mut ∈ L∞ (0, T0 ; L2 (Ω)), ut ∈ L2 (0, T0 ; H01 (Ω)).
Theorem 3.6 ([50], Theorem 1.2 (blow-up citerion)) Under the assumptions of Theorem 3.5, if T ∗ < ∞ is the maximal existence time for the strong solution (m, n, u)(x, t) to the problem problem (3.13)–(3.15) stated in Theorem 3.5, then lim sup ||m||L∞ (0,T ;L∞ (Ω)) = ∞,
(3.17)
T →T ∗
provided that λ <
25 3 µ.
Remark 3.7 The analysis in Theorem 3.6 can be applied to study a blow-up criterion of the strong solution to compressible Navier-Stokes equations for 25 3 µ > λ, i.e., lim sup kρ(t)kL∞ (0,T ;L∞ (Ω)) = ∞, T →T ∗
which improves the corresponding result about Navier-Stokes equations in [48] where 7µ > λ. If we don’t neglect the gas phase effects in the mixture momentum conservation equation,
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and consider the following initial boundary value problem mt + (mu)x = 0, nt + (nu)x = 0, ((m + n)u)t + [(m + n)u2 ]x + P (m, n)x = µuxx , (x, t) ∈ (0, 1) × (0, ∞),
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(3.18)
the viscosity coefficient µ is constant, the pressure P satisfies p P (m, n) = C 0 (−b(m, n) + b(m, n)2 + c(n)), with initial conditions
m(x, 0) = m0 (x), n(x, 0) = n0 (x), (m + n)u(x, 0) = M0 (x),
x ∈ [0, 1],
(3.19)
and with boundary conditions u(0, t) = u(1, t) = 0, t ≥ 0.
(3.20)
We give a precise definition of global weak solutions. Definition 3.8 ([26], Definition 1.1) We call (m, n, u) : (0, 1) × (0, +∞) → R+ × R+ × R a global weak solution of (3.18)–(3.20) if for any 0 < T < +∞: (1) m, n ∈ L∞ ((0, 1) × (0, T )), (m + n)u2 ∈ L∞ (0, T ; L1(0, 1)), m, n ≥ 0 a.e., in (0, 1) × (0, T ), u ∈ L2 (0, T ; H01 (0, 1)); (2) (m, n, u) satisfy the system (3.18) in the sense of distribution; (3) (m, n, (m + n)u)(x, 0) = (m0 (x), n0 (x), M0 (x)), a.e. x ∈ (0, 1). Theorem 3.9 ([26], Theorem 1.1 (global existence)) If
inf m0 ,
x∈(0,1)
inf n0 ≥ 0, m0 ,
x∈(0,1)
0 n0 ∈ L∞ (0, 1), and √mM0 +n ∈ L2 (0, 1), then there exists a global weak solution (m, n, u) : 0 (0, 1) × [0, +∞) → R+ × R+ × R to (3.18)–(3.20).
Remark 3.10 Note that we do not need the conditions s0 m0 ≤ n0 ≤ s0 m0 for some s0 , s0 ≥ 0 typically made use of in previous literature [15, 50–52]. This implies that transition to single phase is allowed, i.e., one of the two phases can completely occupy some regions.
4
Open Problems
• Compressible nonconservative two-fluid model Compared with the single-phase flow (i.e., compressible Navier-Stokes equations), the twofluid model has its own challenges by means of the nonconservative structure in the pressure terms. More specifically, when one looks for the global solutions with large initial data in the sense of distribution, the spatial derivatives in the pressure terms can not all be shifted to the test functions. In view of the fact, one has to obtain some estimates of the spatial derivatives of density or the related. This makes that two viscosity coefficients have to be equivalent to the corresponding density, i.e., µ± ∼ ρ± , λ± ∼ ρ± , see [3] for 1D case. To our best knowledge, it is still open for the case of multi-dimensions and for that µ± ∼ (ρ± )α , λ± ∼ (ρ± )β with more general α, β ∈ [0, ∞). • Drift-flux model The main difference between two-fluid model and drift-flux model is that two fluids are considered as a whole in the momentum equations for the latter case. However, in this case,
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some other equations have to be added to the system for completeness. The so called “slip law” is commonly used. The main challenge is that the global entropy estimate is difficult to find though the momentum equation is of the conservative form. This leads to an open problem whether the global weak solutions with large initial data exist or not for more general slip law. In fact, when c0 = 1 and c1 = 0 for large initial data, and c0 ≥ 1, c1 ≥ 0 for small initial data, it is known that the global weak solutions exist, see [26], and Theorems 2.1 and 3.2 respectively. Here c0 and c1 are constants. • Viscous liquid-gas two-phase flow model
The viscous liquid-gas two-phase flow model can be considered as a simplified case from the two-fluid model when two velocity fields and two pressure functions are equal respectively. Although there are some results achieved for the simplified case particularly, there are still some open problems. For example, whether the global smooth solution with large initial data in high dimensions exists or not, or equivalently whether the smooth solution blows up in finite time.
References [1] Bear J. Dynamics of Fluids in Porous Media, Environmental Science Series. New York: Elsevier, 1972; reprinted with corrections, New York: Dover, 1988 [2] Bresch D, Desjardins B, Ghidaglia J M, Grenier E. Global weak solutions to a generic two-fluid model. Arch Ration Mech Anal, 2010, 196: 599–629 [3] Bresch D, Huang X D, Li J. Global weak solutions to one-dimensional non-conservative viscous compressible two-phase system. Comm Math Phys, 2012, 309: 737–755 [4] Chen Z Z, Zhao H J. Existence and nonlinear stability of stationary solutions to the full compressible Navier-Stokes-Korteweg system. J Math Pures Appl, 2014, 101: 330–371 [5] Cui H B, Wang W J, Yao L, Zhu C J. Decay rates for a nonconservative compressible generic two-fluid model. SIAM J Math Anal, 2016, 48: 470–512 [6] Danchin R, Desjardins B. Existence of solutions for compressible fluid models of Korteweg type. Ann Inst H Poincar´ e Anal Non Lin´ eaire, 2001, 18: 97–133 [7] Duan R J, Ukai S, Yang T, Zhao H J. Optimal convergence rates for the compressible Navier-Stokes equations with potential forces. Math Models Methods Appl Sci, 2007, 17: 737–758 [8] Duan R J, Liu H X, Ukai S, Yang T. Optimal Lp -Lq convergence rate for the compressible Navier-Stokes equations with potential force. J Differential Equations, 2007, 238: 220–233 [9] Evje S. Global weak solutions for a compressible gas-liquid model with well-formation interaction. J Differential Equations, 2011, 251: 2352–2386 [10] Evje S. Weak solutions for a gas-liquid model relevant for describing gas-kick in oil wells. SIAM J Math Anal, 2011, 43: 1887–1922 [11] Evje S. A compressible two-phase model with pressure-dependent well-reservoir iteraction. SIAM J Math Anal, 2013, 45: 518–546 [12] Evje S. Genuine two-phase flow dynamics with a free interface separating gas-liquid mixture from gas. SIAM J Math Anal, 2013, 45: 2894–2923 [13] Evje S, Fl˚ atten T, Friis H A. Global weak solutions for a viscous liquid-gas model with transition to single-phase gas flow and vacuum. Nonlinear Anal, 2009, 70: 3864–3886 [14] Evje S, Karlsen K H. Analysis of a compressible gas-liquid model motivated by oil well control operations. Acta Math Sci, 2012, 32B(1): 295–314 [15] Evje S, Karlsen K H. Global existence of weak solutions for a viscous two-phase model. J Differential Equations, 2008, 245: 2660–2703 [16] Evje S, Karlsen K H. Global weak solutions for a viscous liquid-gas model with singular pressure law. Comm Pure Appl Anal, 2009, 8: 1867–1894 [17] Evje S, Liu Q Q, Zhu C J. Asymptotic stability of the compressible gas-liquid model with well-formation interaction and gravity. J Differential Equations, 2014, 257: 3226–3271
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H.Y. Wen et al: REVIEW ON MATHEMATICAL ANALYSIS OF SOME TWO-PHASE
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[18] Evje S, Wang W J, Wen H Y. Global well-Posedness and decay rates of strong solutions to a non-conservative compressible two-fluid model. Arch Ration Mech Anal, 2016, 221: 1285–1316 [19] Evje S, Wen H Y. Analysis of a compressible two-fluid Stokes system with constant viscosity. J Math Fluid Mech, 2015, 17: 423–436 [20] Evje S, Wen H Y. Global solutions of a viscous gas-liquid model with unequal fluid velocities in a closed conduit. SIAM J Math Anal, 2015, 47: 381–406 [21] Evje S, Wen H Y. Stability of a compressible two-fluid hyperbolic-elliptic system arising in fluid mechanics. Nonlinear Anal Real World Appl, 2016, 31: 610–629 [22] Evje S, Wen H Y. Weak solutions of a gas-liquid drift-flux model with general slip law for wellbore operators. Discrete Contin Dyn, System-A, 2013, 33: 4497–4530 [23] Evje S, Wen H Y. Weak solutions of a two-phase Navier-Stokes model with a general slip law. J Funct Anal, 2015, 268: 93–139 [24] Evje S, Wen H Y. On the large time behavior of the compressible gas-liquid drift-flux model with slip. Math Models Methods Appl Sci, 2015, 25: 2175-2215 [25] Evje S, Wen H Y, Yao L. Global solutions to a one-dimensional non-conservative viscous compressible two-phase gas-liquid model. Discrete Contin Dyn, System-A, 2016, 36: 1927–1955 [26] Evje S, Wen H Y, Zhu C J. On global solutions to the viscous liquid-gas model with unconstrained transition to single-phase flow. Math Models Methods Appl Sci, 2017, 27: 323–346 [27] Friis H A, Evje S. Global weak solutions for a gas-liquid model with external forces and general pressure law. SIAM J Appl Math, 2011, 71: 409–442 [28] Friis H A, Evje S. Well-posedness of a compressible gas-liquid model with a friction term important for well control operations. SIAM J Appl Math, 2011, 71: 2014–2047 [29] Friis H A, Evje S. Asymptotic behavior of a compressible two-phase model with well-formation interaction. J Differential Equations, 2013, 254: 3957–3993 [30] Hao C C, Li H L. Well-posedness for a multidimensional viscous liquid-gas two-phase flow model. SIAM J Math Anal, 2012, 44: 1304–1332 [31] Hattori H, Li D. Solutions for two-dimensional system for materials of Korteweg type. SIAM J Math Anal, 1994, 25: 85–98 [32] Hattori H, Harumi H, Li D. Global solutions of a high-dimensional system for Korteweg materials. J Math Anal Appl, 1996, 198: 84–97 [33] Hoff D, Zumbrun K. Multi-dimentional diffusion waves for the Navier-Stokes equations of compressible flow. Indiana Univ Math J, 1995, 44: 603–676 [34] Hoff D, Zumbrun K. Pointwise decay estimates for multidimentional Navier-Stokes diffusion waves. Z Angew Math Phys, 1997, 48: 597–614 [35] Hong G Y, Zhu C J. Optimal decay rates on the solution to the compressible gas-liquid drift-flux model with slip. Math Models Methods Appl Sci, 2018, 28(2): 337–385 [36] Ishii M. Thremo-Fluid Dynamic Theory of Two-Phase Flow. Paris: Eyrolles, 1975 [37] Kagei Y, Kobayashi T. On large-time behavior of solutions to the compressible Navier-Stokes equations in the half space in R3 . Arch Ration Mech Anal, 2002, 165: 89–159 [38] Kagei Y, Kobayashi T. Asymptotic behavior of solutions to the compressible Navier-Stokes equations on the half space. Arch Ration Mech Anal, 2005, 177: 231–330 [39] Kobayashi T. Some estimates of solutions for the equations of motion of compressible viscous fluid in an exterior domain in R3 . J Differential Equations, 2002, 184: 587–619 [40] Kobayashi T, Shibata Y. Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in R3 . Comm Math Phys, 1999, 200: 621–659 [41] Lai J, Wen H Y, Yao L. Vanishing capillarity limit of the non-conservative compressible two-fluid model. Discrete Contin Dyn, System-B, 2017, 22: 1361–1392 [42] Li H L, Zhang T. Large time behavior of isentropic compressible Navier-Stokes system in R3 . Math Methods Appl Sci, 2011, 34: 670–682 [43] Liu T P, Wang W K. The pointwise estimates of diffusion waves for the Navier-Stokes equations in odd multi-dimensions. Comm Math Phys, 1998, 196: 145–173 [44] Matsumura A, Nishida T. The initial value problem for the equation of motion of viscous and heatconductive gases. J Math Kyoto Univ, 1980, 20: 67–104 [45] Matsumura A, Nishida T. The initial value problem for the equation of motion of compressible viscous and
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Vol.38 Ser.B
heat-conductive fluids. Proc Japan Acad, Ser A, Math Sci, 1979, 55: 337–342 [46] Prosperetti A, Tryggvason G. Computational Methods for Multiphase Flow. Cambridge University Press, 2007 [47] Solem, S, Evje S. Relaxation limit of a compressible gas-liquid model with well-reservoir interaction. Z Angew Math Phys, 2017, 68: 23–25 [48] Sun Y Z, Wang C, Zhang Z F. A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navier-Stokes equations. J Math Pures Appl, 2011, 95: 36–47 [49] Wang Y J, Tan Z. Optimal decay rates for the compressible fluid models of Korteweg type. J Math Anal Appl, 2011, 379: 256–271 [50] Wen H Y, Yao L, Zhu C J. A blow-up criterion of strong solution to a 3D viscous liquid-gas two-phase flow model with vacuum. J Math Pures Appl, 2012, 97: 204–229 [51] Yao L, Zhang T, Zhu C J. Existence and asymptotic behavior of global weak solutions to a 2D viscous liquid-gas two-phase flow model. SIAM J Math Anal, 2010, 42: 1874–1897 [52] Yao L, Zhang T, Zhu C J. A blow-up criterion for a 2D viscous liquid-gas two-phase flow model. J Differential Equations, 2011, 250: 3362–3378 [53] Yao L, Zhu C J. Free boundary value problem for a viscous two-phase model with mass-dependent viscosity. J Differential Equations, 2009, 247: 2705–2739 [54] Yao L, Zhu C J. Existence and uniqueness of global weak solution to a two-phase flow model with vacuum. Math Ann, 2011, 349: 903–928 [55] Zuber N, Findlay J A. Average volumetric concentration in two-phase flow systems. J Heat Transf, 1965, 87: 453–468