On the well-posedness and decay rates of strong solutions to a multi-dimensional non-conservative viscous compressible two-fluid system

The present paper deals with the Cauchy problem of a multi-dimensional non-conservative viscous compressible two-fluid system. We first study the well-posedness of the model in spaces with critical regularity indices with respect to the scaling of the associated equations. In the functional setting as close as possible to the physical energy spaces, we prove the unique global solvability of strong solutions close to a stable equilibrium state. Furthermore, under a mild additional decay assumption involving only the low frequencies of the data, we establish the time decay rates for the constructed global solutions. The proof relies on an application of Fourier analysis to a complicated parabolic-hyperbolic system, and on a refined time-weighted inequality.


Introduction and main results
It is well known that models of two-phase or multiphase flows are widely applied to study the hydrodynamics in industry, for example, in manufacturing, engineering, and biomedicine, where the fluids under investigation contain more than one component. In fact, it has been estimated that over half of everything produced in a modern industrial society depends, to some degree, on a multiphase flow process for its optimum design and safe operation. In nature, there is a variety of different multiphase flow phenomena, such as sediment transport, geysers, volcanic eruptions, clouds, and rain [2,4]. In addition, models of multiphase flows also naturally appear in many contexts within biology, ranging from tumor biology and anticancer therapies to developmental biology and plant physiology [22]. transfer are relatively well understood; however, the thermofluid dynamics of two-phase flows is an order of magnitude more complicated than that of the single-phase flow due to the existence of a moving and deformable interface and its interactions with two phases [21,25,26].
From the viewpoint of partial differential equations, system (1.1) is a highly nonlinear system coupling between hyperbolic equations and parabolic equations. As a matter of fact, there is no diffusion on the mass conservation system satisfying hyperbolic equations, whereas velocity evolves according to the parabolic equations due to the viscosity phenomena. We should point out that the system (1.1) includes important single phase flow models such as the compressible Navier-Stokes equations when one of the two phases volume fraction tends to zero (i.e.,α + = 0 or α − = 0 ). As an extremely important system to describe compressible fluids (e.g., gas dynamics), the compressible Navier-Stokes equations have attracted a lot of attention among many analysts and many important results have been developed. Here we briefly review some of the most relevant papers about global well-posedness and large time behaviors of the solutions to the system.
Lions [18] proved the global existence of weak solutions for large initial data. However, the question of uniqueness of weak solutions remains open, even in the two dimensional case. Matsumura and Nishida [19,20] first studied the global existence of classical solutions to the compressible Navier-Stokes equations for data (ρ 0 , u 0 ) with high regularity order and close to a stable equilibrium in the 3D whole space and obtained the time decay rates based on the L 2 -framework. Later, Ponce [27] established the optimal L p (2 ≤ p ≤ ∞) decay rates. Applying Fourier analysis to the linearized homogeneous system and capturing the dissipation of the hyperbolic component in the solution, Kawashima [23,24] and Shizuta and Kawashima [28] developed a general approach to obtain the time-decay of solutions. It is worth mentioning that Li and Zhang [17] obtained the optimal L p time-decay rates for the compressible Navier-Stokes equations in three dimensions when initial data belong to some space H s ∩Ḃ −s 1,∞ (see Definition 2.2 for details)and s ∈ [0, 1]. Guo and Wang [15] obtained the optimal decay rates for the compressible Navier-Stokes equations when the initial data are close to a stable equilibrium state in negative Sobolev spaces by using a pure energy method. In [9], Danchin first proved the existence and uniqueness of the global strong solution for initial data close to a stable equilibrium state in critical Besov spaces. Later, Danchin [11] further established the time decay rates of the global solutions constructed in [9].
Since a single phase flow model may be considered as a special case of a two-phase flow model in the limit when one of the two phases volume fractions tends to zero, the mathematical structure of the two-phase system is much more complex than that in the case of single phase flow model. So, extending the currently available results for single phase flow models to two-phase models is not an easy task. Nowadays, more and more researchers pay more attention to the mathematical problems of the generic two-phase model. In [2], Bresch et al. first established the existence of global weak solutions to the 3D generic two-fluid flow model with capillary pressure effects in terms of a third order derivative of α ± ρ ± . Based on detailed analysis of the Green function to the linearized system and on elaborate energy estimates to the nonlinear system, the authors in [8] obtained global existence of smooth solutions and the time decay rates to the 3D model where the initial data are close to an equilibrium state in H s (R 3 x )(s ≥ 3) with high Sobolev regularity and belong to L 1 (R 3 x ). More recently, Lai-Wen-Yao [16] studied the vanishing capillarity limit of the smooth solutions to the 3D model with unequal pressure functions. When a generic two-fluid flow model does not include capillary pressure effects, the model reduces to the system (1.1). Bresch-Huang-Li [3] extended the result in [2] and proved the existence of global weak solutions to (1.1) in one space dimension. In 2016, Evje-Wang-Wen [14] proved the global existence of strong solutions to the model (1.1) with constant viscosity coefficients and unequal pressure functions by the standard energy method under the condition that the initial data are close to the constant equilibrium state in H 2 (R 3 x ) and obtained the optimal decay rates for the constructed global strong solutions in L 2norm if the initial data belong to L 1 additionally. However, to the best of our knowledge, very few results have been established on the global well-posedness and the decay rates of strong solutions to a multi-dimensional non-conservative viscous compressible two-fluid systems in critical regularity framework. The purpose of this work is to investigate the mathematical properties of system (1.1) in critical regularity framework. More specifically, we address the question of whether available mathematical results such as the global well-posedness and time decay rate in critical Besov spaces to a single fluid governed by the compressible barotropic Navier-Stokes equations may be extended to multi-dimensional non-conservative viscous compressible two-fluid system.
First, we will derive another expression of the pressure gradient in terms of the gradients of α + ρ + and α − ρ − by using the pressure equilibrium assumption. The method comes from [2]. For the convenience of the reader, we also show some derivations in this part. The relation between the pressures of (1.1) implies the following differential identities where s ± denote the sound speed of each phase respectively. Let Resorting to (1.1) 1 , we have Combining with (1.4) and (1.5), we conclude that Substituting the above equality into (1.5), we obtain which give, for the pressure differential dP ± , Recalling α + + α − = 1, we get the following identity: (1.6) Then it follows from the pressure relation (1.1) 4 that (1.7) Differentiating ϕ with respect to ρ + , we have By the definition of R + , it is natural to look for ρ + which belongs to (R + , +∞). Since ϕ ′ > 0 in (R + , +∞) for any given R ± > 0, and ϕ : (R + , +∞) −→ (−∞, +∞), this determines that is the unique solution of the equation (1.7). Due to (1.5), (1.6) and (1.1) 1 , ρ − and α ± are defined as follows: Based on the above analysis, the system (1.1) is equivalent to the following form (1.8) In this paper, we are concerned with the Cauchy problem of the system (1.8) in R + × R N subject to the initial data and where R ± ∞ denote the background doping profile, and in the present paper R ± ∞ are taken as 1 for simplicity.
At this stage, we are going to use scaling considerations for (1.1) to guess which spaces may be critical. One can check that if (α provided that the pressure laws P have been changed into λ 2 P . This suggests us to choose initial data (α + ρ + ) 0 , u + 0 , (α − ρ − ) 0 , u − 0 in critical spaces whose norm is invariant for all λ > 0 by the . Due to the mixed hyperbolic-parabolic property of the partial differential system (1.1), motivated by Danchin's excellent work in [9], the different dissipative mechanisms of low frequencies and high frequencies inspire us to deal with (α 2,1 (see Definitions 2.2 and 2.6 for details). However, we can not obtain the desired bounds directly in critical regularity framework if the convection terms are treated as perturbations. More precisely, there exists a difficulty coming from the convection terms u ± · ∇R ± in the transport equations without any diffusion in high frequencies, as one derivative loss about the function R ± will appear no matter how smooth is u ± if they are viewed as perturbation terms. To overcome the difficulty, employing the Littlewood-Paley theory and some commutator estimates, we shall, as in [9] for the standard barotropic Navier-Stokes equations, study a complicated hyperbolicparabolic linear system including convection terms and then deduce the smooth effect for (R + − 1, u + , R − − 1, u − ) in the low frequencies regime and the L 1 decay on the density R ± in the high frequencies regime. In particular, based on the damping effect of R ± , we further exploit the smooth effect for (u + , u − ) in the high frequencies regime with ∇R ± being viewed as perturbation terms and finally establish a uniform priori estimate for the complicated system (see the following Lemma 3.1 for details). Here, it should be pointed out that, different from the standard barotropic compressible Navier-Stokes equations, we need to make more careful analyses to cancel some mixed terms from the two-phase flows. Next, one may wonder how global strong solutions constructed above look like for large time. Under a suitable additional condition involving only the low frequencies of the data and in the L 2 -critical regularity framework, we exhibit the time decay rates for the constructed global strong solutions. In this part, our main ideas are based on the low-high frequency decomposition and a refined time-weighted energy functional. In low frequencies, making good use of Fourier localization analysis to a linearized parabolic-hyperbolic system in order to obtain smoothing effect of the Green function in the low-frequency part and avoid some complicate analysis of the Green function (see Lemma 6.1), which is a 8 × 8 matrix. In high frequencies, we can deal with the estimates of the nonlinear terms in the system employing the Fourier localization method and commutator estimates. Finally, in order to close the energy estimates, we exploit some decay estimates with gain of regularity for the high frequencies of ∇u ± . Now we state our main results as follows: Then there exists a constant η > 0 such that if then the Cauchy problem (1.8)-(1.9) admits a unique global solution satisfying that for all t ≥ 0, . (1.13) (1.14) then the global solution (R + − 1, u + , R − − 1, u − ) given by Theorem 1.1 satisfies for all t ≥ 0, .
(1.16) Remark 1.3. In Theorem 1.2, we obtain the time decay rates for multi-dimensional non-conservative viscous compressible two-fluid system (1.1) in critical regularity framework. Additionally, the regularity index s can take both negative and nonnegative values, rather than only nonnegative integers, which improves the classical decay results in high Sobolev regularity, such as [14] when In fact, for the solution (R + −1, u + , R − −1, u − ) constructed in Theorem 1.1, applying to homogeneous Littlewood-Paley decomposition for R + − 1, we have Based on Bernstein's inequalities and the low-high frequencies decomposition, we may write If follows from Inequality (1.15) and definitions of D(t) and α that and that, because we have α ≥ N 4 + s 2 for all s ≤ min{2, N/2}, This yields the following desired result for R + − 1 In particular, taking s = 0 leads back to the standard optimal L 1 -L 2 decay rate of (R + −1, u + , R − − 1, u − ) as in [14] when N = 3. 2,∞ (R 3 ), our results in Theorem 1.2 extend the known conclusions in [14]. In particular, our condition involves only the low frequencies of the data and is based on the L 2 (R 3 x )-norm framework. In particular, the decay rates of strong solutions is in the so-called critical Besov spaces in any dimension N ≥ 2 and the dimension of space is more extensive and is not limited to N = 3. Remark 1.5. In this paper, we can not deal with the case of the model (1.1) with unequal pressure functions as in [14] since we take advantages of the symmetrizers methods in our process of the proof.
Notations. We assume C be a positive generic constant throughout this paper that may vary at different places and denote A ≤ CB by A B. We shall also use the following notations Noting the small overlap between low and high frequencies, we have z ℓ The homogeneous frequency localization operators∆ q andṠ q are defined bẏ With our choice of ϕ, one can easily verify thaṫ We denote the space Z ′ (R N ) by the dual space of Z(R N ) = {f ∈ S(R N ); D αf (0) = 0; ∀α ∈ N N multi-index}. It also can be identified by the quotient space of S ′ (R N )/P with the polynomials space P. The formal equality holds true for f ∈ Z ′ (R N ) and is called the homogeneous Littlewood-Paley decomposition.
The following Bernstein's inequalities will be frequently used.
Obviously, L 1 T (Ḃ s p,1 ) = L 1 T (Ḃ s p,1 ). By a direct application of Minkowski's inequality, we have the following relations between these spaces To deal with functions with different regularities for high frequencies and low frequencies, motivated by [9,11], it is more effective to work in hybrid Besov spaces. We remark that using hybrid Besov spaces has been crucial for proving global well-posedness for compressible systems in critical spaces (see [7,9,11]).
Remark 2.7. Some properties about the hybrid Besov spaces are as follows • Interpolation: for s 1 , s 2 , t 1 , t 2 ∈ R and θ ∈ [0, 1], we have We have the following properties for the product in Besov spaces and hybrid Besov spaces.
[10] For all s 1 , s 2 > 0, there exists a positive universal constant such that For all s 1 , s 2 ≤ N 2 such that min{s 1 + t 1 , s 2 + t 2 } > 0, there exists a positive universal constant such that For the composition of the binary functions, we have the following estimates.
. Moreover, there exists a C depending only on s, p, N and F such that Then, there exists a C depending only on s, p, N and F such that Throughout this paper, the following estimates for the convection terms arising in the linearized systems will be used frequently.
[11] Let 1 ≤ p, p 1 ≤ ∞, and σ ∈ R. There exists a constant C > 0 depending only on σ such that for all q ∈ Z, we have Then for all T > 0 the following a priori estimate is fulfilled .
T (Ḃ s p,r ) and u is the solution of the following transport equation We finish this subsection by listing an elementary but useful inequality.
3 Reformulation of the System (1.8) and and A priori estimates for the linearized system with convection terms 3.1 Reformulation of the System (1.8) To make it more convenient to study, we need some reformulations of (1.8). More precisely, taking a change of variables by Then, the system (1.8) can be rewritten as with initial data and the source terms are where we define the nonlinear functions of (c + , c − ) by .
3.2 A priori estimates for the linearized system with convection terms Next, we investigate some a priori estimates for the following linearized system with We will establish a uniform estimate for a mixed hyperbolic-parabolic linear system (3.12) with convection terms. What is crucial in this work is to exploit the smoothing effects on the velocity u + , u + and the L 1 decay on c + , c − , which play a key role to control the pressure term in the proof of the Theorem 1.1.
Let T > 0 and (c + , u + , c − , u − ) be a solution of the system (3.12). Then the following estimates Proof. Applying the operator∆ q to the system (3.12), we deduce that Taking the L 2 -scalar product of the first equation of (3.14) with∆ q c + and (−∆∆ q c + ), the second equation with∆ q u + , the third equation with∆ q c − and (−∆∆ q c − ) and the fourth equation witḣ ∆ q u − respectively, we obtain the following six identities: where (· | ·) stands for the L 2 inner product.
In order to obtain a second energy estimate, we need to derive some identities involving Taking the L 2 -scalar product of the first equation of (3.14) with∆ q c − and the third equation with∆ q c + and then summing the results, which yields On the other hand, applying the operator ∇ to the first equation in (3.2) and taking the L 2 scalar product with∆ q u + , then calculating the scalar product of the second equation in (3.2) with ∇∆ q c + , and then summing up the results, we get and employing Young's inequality, we obtain Using further β 2 2 = β 2 3 = β 1 β 4 and Young's inequality, and choosing M 2 = β 1 β 2 , we get Then, there exist two positive constants c 1 and c 2 such that Thus, for some fix q 0 , Combining with (3.15)-(3.23), it yields, with the help of Proposition 2.11, that 1 2 with (γ q ) q∈Z in the unit sphere of ℓ 1 (Z).
Thus, it follows Thus, by Gronwall's inequality, we have It follows that from Proposition 2.11 , which implies, with the help of (3.26), that

Global existence for initial data near equilibrium
In this section, we show that if the initial data satisfy for some sufficiently small η, then there exists a positive constant M such that This uniform estimate will enable us to extend the local solution (R + −1, u + , R − −1, u − ) obtained within an iterative scheme as in [9] to a global one. To this end, we use a contradiction argument. Define with M to be determined later. Suppose that T 0 < ∞. We apply the linear estimates in Lemma 3.1 to the solutions of the reformulated system (3.1) such that for all t ∈ [0, T 0 ], the following estimates hold In what follows, we derive some estimates for the nonlinear terms H 11 −H 41 . First, by Proposition 2.9, we have (H 11 , H 31 ) , thus, thanks to Proposition 2.8, we easily infer According to Proposition 2.10(ii) and interpolation inequality, we have Hence, we gather that Similarly, we also have Substituting (4.2)-(4.4) into (4.1), we obtain that Choose M = 8C 1 , for sufficiently small η such that which implies that This is a contradiction with the definition of T 0 . As a consequence, we conclude that T 0 = ∞.

Uniqueness
In this section, we prove the uniqueness of the solution for the system (1.8)-(1.9). First, let us recall the Osgood Lemma (see [?]), which allows us to deduce uniqueness of the solution in the critical case.
[?] Let f ≥ 0 be a measurable function, γ be a locally integrable function and µ be a positive, continuous and non decreasing function which verifies the following condition Let also a be a positive real number and let f satisfy the inequality Then, (i) if a is equal to zero, the function f vanishes; (ii) if a is not zero, then we have Next, we need the following result of logarithmic interpolation.
We assume that (c + 1 , u + 1 , c − 1 , u − 1 ), (c + 2 , u + 2 , c − 2 , u − 2 ) are two solutions of the system (3.1) with the same initial data satisfying (1.12) On the other hand, if η is sufficiently small, we have The continuity in time for c ± 2 thus yields the existence of a time T > 0 such that From the embedding theorem and (1.12), we have Then (δc + , δu + , δc − , δu − ) satisfies the following system In what follows, we set . To begin with, we shall prove uniqueness on the time interval [0, T ] by estimating (δc + , δu + , δc − , δu − ) in the following functional space: We apply Proposition 2.15 to get for any t ∈ [0, T ], dτ.
Applying Remark 2.14 to the second equation and the fourth equation of (5.2), we have δu + Employing Proposition 2.8 and Proposition 2.10, we have (5.8)

Time decay estimates
In this section, we will establish the time decay rates of the global strong solutions constructed in Theorem 1.1. We divide the proof into several steps.
Step 1: Low frequencies We first exhibit the smoothing properties of the system (3.1) in the low frequencies regime.
The key to these remarkable properties is given by the following lemma. ) and ν ± i (i = 1, 2) respectively, such that the following inequality holds for all t ≥ 0, Proof. By the same derivation process of (3.25), in the case Thus, Furthermore, taking advantage of the Duhamel formula, we can readily deduce (6.1).
Denoting by A(D) the semi-group associated to the system (3.1), we have for all q ∈ Z,  Based on (6.3) and (6.4), we get for all q ≤ q 0 , Hence, multiplying by t N 4 + s 2 2 qs and summing up on q ≤ q 0 , we get As for any σ > 0 there exists a constant C σ so that sup t≥0 q∈Z We get from (6.5) and (6.6) that for s > −N/2, Furthermore, it is obvious that for s > −N/2, Thus, from (6.1) and (6.7), we have We claim that for all s ∈ (−N/2, 2] and t ≥ 0, then To bound the term with H 1 , we use the following decomposition: Now, from Hölder's inequality, the embeddingḂ 0 2,1 ֒→ L 2 , the definitions of D(t), α and Lemma 2.16, one may write for all s ∈ (ε − N 2 , 2], The term c + div (u + ) ℓ may be treated along the same lines, and we have Regarding the term with c + div (u + ) h , we get for all t ≥ 2 that From the definitions of X(t) and D(t), we obtain and, using the fact that τ ≈ τ when τ ≥ 1, we get Thus, for t ≥ 2, we conclude that The case t ≤ 2 is obvious as t ≈ 1 and t − τ ≈ 1 for 0 ≤ τ ≤ t ≤ 2, and

X(t)D(t).
(6.13) From (6.10)-(6.13), we get The term H 3 may be treated along the same lines, and we obtain Next, to bound the first term of H i 2 , we write that where g + stands for some smooth function vanishing at 0.
Similar to (6.14), we have To bound the term with (u + · ∇)u + i , we employ the following decomposition: For the term (u + · ∇)(u + i ) ℓ , we have Regarding the term with (u + · ∇)(u + i ) h , we have for all t ≥ 2 that From the definitions of X(t) and D(t), we get and, using the fact that τ ≈ τ when τ ≥ 1, Therefore, for t ≥ 2, we deduce that The case t ≤ 2 is obvious as t ≈ 1 and t − τ ≈ 1 for 0 ≤ τ ≤ t ≤ 2, and (6.17) To deal with the term µ + h + (c + , c − )∂ j c + ∂ j u + i , we take the following decomposition: We bound the two terms L 1 and L 2 as follows respectively, and Thus, We deal with the two terms M 1 and M 2 in the following, if t ≥ 2, Using the definitions of X(t) and D(t), we obtain and, employing the fact that τ ≈ τ when τ ≥ 1, we have For the term M 2 , we have Remembering the definitions of X(t) and D(t), we obtain Therefore, for t ≥ 2, we obatin The case t ≤ 2 is obvious as t ≈ 1 and t − τ ≈ 1 for 0 ≤ τ ≤ t ≤ 2, and (6.20) From (6.18)-(6.20), we finally conclude that Similarly, we also obtain the corresponding estimates of other terms Here, we omit the details.
From the low-high frequency decomposition for µ + l + (c + , c − )∂ 2 j u + i , we have where l + stands for some smooth function vanishing at 0. Thus, To handle the term µ + l + (c + , c − )∂ 2 j (u + i ) h , we consider the cases t ≥ 2 and t ≤ 2 respectively. When t ≥ 2, then we have From the definitions of X(t) and D(t), we obtain and, using the fact that τ ≈ τ when τ ≥ 1, we have Thus, for t ≥ 2, we arrive at The case t ≤ 2 is obvious as t ≈ 1 and t − τ ≈ 1 for 0 ≤ τ ≤ t ≤ 2, (6.24) From (6.22)-(6.24), we get Similarly, The term H 4 may be treated along the same lines, and we have Thus, we complete the proof of (6.9). Combining with (6.7) and (6.9), we conclude that for all Step 2: High frequencies Now, the starting point is Inequality (3.24) which implies that for q ≥ q 0 and for some After time integration, we have and thus, by multiplying both sides by 2 q( N 2 −1) , taking the supremum on [0, T ], and then summing up over q ≥ q 0 , Bounding the sum, for 0 ≤ t ≤ 2, and taking advantage of Proposition 2.12, we end up with For the term ∇H 31 , along the same lines, we have Combining interpolation inequality and Hölder's inequality, we deduce that where g + stands for some smooth function vanishing at 0.
The estimates of other terms such as j and λ + k + (c + , c − )∂ i c − ∂ j u + j are similar to (6.32). Here, we omit the details.
Similarly, (2), Therefore, for the case t ≤ 2, Let us finally consider the [1, t] part of the integral for 2 ≤ t ≤ T. We shall use repeatedly the following inequalities which is straightforward as regards to the high frequencies of u ± and stem from for the low frequencies of u ± .
Regarding the contribution of S 1 q , by Lemma 2.16 we first notice that . Now, the product laws in tilde spaces ensures that .
From (6.36) and (6.40), we also see that To bound the term with S 2 q , we use the following fact that Hence, thanks to Lemma 2.16 and Proposition 2.12, we have .
The first term on the right-side of the above inequality may be bounded from (6.36), and the high frequencies of the last one on the right-side are obviously bounded by D(T ). To bound the term , if N ≤ 6 we have the following inequality If N ≥ 7, Then plugging (6.34), (6.35) and (6.49) into (6.28) yields + X 2 (T ) + X 3 (T ) + D 2 (T ) + D 4 (T ). (6.50) Step 3: Decay estimates with gain of regularity for the high frequencies of ∇u + , ∇u − .
We deduce from Remark 2.14 that , whence, using the bounds given by Theorem 1.1, we have .
Next, the product and composition estimates adapted to tilde spaces give Similarly, we have Furthermore, from (6.36) and the definition of X(t), we have