WELL-POSEDNESS IN CRITICAL SPACES FOR A MULTI-DIMENSIONAL COMPRESSIBLE VISCOUS LIQUID-GAS TWO-PHASE FLOW MODEL

. This paper is dedicated to the study of the Cauchy problem for a compressible viscous liquid-gas two-phase ﬂow model in R N ( N ≥ 2). We concentrate on the critical Besov spaces based on the L p setting. We improve the range of Lebesgue exponent p , for which the system is locally well-posed, compared to [22]. Applying Lagrangian coordinates is the key to our state-ments, as it enables us to obtain the result by means of Banach ﬁxed point theorem.

For the more detailed explanations about the above model, one can refer to [5,17,21,23].
There are many results about the viscous liquid-gas two-fluid flow model. For the model (1) in 1D, where the liquid is incompressible and the gas is polytropic, the global well-posedness of weak solution to the free boundary value problem was studied in [11,13,25,26]. Specifically, when both of the two fluids are compressible, one could refer to [12]. In [23], Yao, Zhang and Zhu obtained the existence of global weak solutions to the 2D model when the initial energy is small and there is no initial vacuum, which can be seen as a generalization of the results in [12] from 1D to multi-dimension. Later on, in a bounded domain, Wen, Yao, Zhang and Zhu [21,24] proved the blow-up criterion in terms of the upper bound of the liquid mass for local strong solution to the 3D (or 2D) viscous liquid-gas two-phase flow model, in both cases when there is initial vacuum and no initial vacuum, respectively. In addition, the first author and collaborators [6] considered global well-posedness of the classical solutions for the 3D model when the initial data are only small in the energy-norm.
Let us mention that all of the above results were performed in the framework of Sobolev spaces. Inspired by [7] for the compressible Navier-Stokes equations, it is natural to study the system (1) in critical Besov spaces. We observe that system (1) is invariant by the transformation m = m(l 2 t, lx), n = n(l 2 t, lx), u = lu(l 2 t, lx) up to a change of the pressure law P = l 2 P . A critical space is a space in which the norm is invariant under the scaling ( e, f , g)(x) = (e(lx), f (lx), lg(lx)).
Recently, Hao and Li [14] studied the Cauchy problem (1) in Besov spaces based on the L 2 framework for all multi-dimensions. On one hand, they obtained the existence and uniqueness of the global strong solution for system (1) provided that the initial data are close to a constant equilibrium state. On the other hand, they also proved local well-posedness under the condition that initial data are slightly more regular. Very recently, motivated by the papers [2,3], Xu and Yuan [22] obtained the local well-posedness for large data in critical Besov spaces based on the general L p (1 < p ≤ N ) framework, which is a generalization of p = 2 in [14]. Moreover, compared with the local well-posedness result in [14], the one in [22] dose not need slightly more regular for initial data.
The purpose of this paper is to further improve the range of the Lebesgue exponent p in [22], for which system (1) is locally well-posed under the same conditions for initial data as in [22]. More precisely, we relax the restriction on the Lebesgue exponent p in Besov spaces from (1, N ] to (1, 2N ), which means that initial velocities in critical Besov spaces with negative indices generate a unique local solution.
Our main ideas come from the recent work dedicated to the compressible barotropic flow [8] and from the paper [9] concerning incompressible inhomogeneous fluids. Under the Lagrangian coordinates, the authors in [8,9] studied the corresponding local well-posedness problem in the critical Besov spaces setting. In addition, using this method, Chikami and Danchin [4] considered the full compressible Navier-Stokes system in critical Besov spaces. They improved the range of Lebesgue exponent p for which the system is locally well-posed, compared to [10]. Let us emphasize that this approach has already been successfully applied in the case of smooth data (see e.g. [15,16,19,20]).
As pointed out in [4,8,9], the motivation behind introducing Lagrangian coordinates is to effectively eliminate the hyperbolic part of the system, provided that the system of liquid mass and gas mass becomes explicitly solvable once the flow of the velocity field has been determined. Meanwhile, the velocity equation in Lagrangian coordinates remains of parabolic type (at least for small enough time), and the Banach fixed point theorem turns out to be applicable for obtaining the existence and uniqueness of the solution in the same class of spaces as in the Eulerian framework.
We organize the rest of the paper as follows. In the next section, we introduce the compressible viscous liquid-gas two-phase flow model in Lagrangian coordinates and present our main results. In Section 3, we list some results about the Besov spaces and Lagrangian coordinates that may be found in [1,8,9]. Section 4 is devoted to the proofs of Theorem 2.1 and Theorem 2.2 by Banach fixed point theorem.
Throughout the paper, C stands for a generic constant and we sometimes write A B as an equivalent to A ≤ CB. For X a Banach space, p ∈ [1, +∞] and T > 0, the notation L p (0, T ; X) or L p T (X) designates the set of measurable functions f : [0, T ] → X with t → f (t) X in L p (0, T ), endowed with the norm f L p T (X) := f X L p (0,T ) . We agree that C([0, T ]; X) denotes the set of continuous functions from [0, T ] to X.

2.
Main results. Before introducing the Lagrangian system corresponding to (1), let us list some notational conventions.
For a C 1 function F : We denote by adj (A) the adjugate matrix of A, i.e., the transpose of the cofactor matrix of A. Given some matrix A, we define the "twisted" deformation tensor and divergence operator (acting on vector fields z) by the formulae D A (z) := 1 2 (Dz · A + A T · ∇z) and div A z := A T : ∇z = Dz : A.
Let X be the flow associated to the vector-field u, i.e., the solution to X(t, y) = y + t 0 u(τ, X(τ, y))dτ.
Denote m(t, y) := m(t, X(t, y)), n(t, y) := n(t, X(t, y)) and u(t, y) := u(t, X(t, y)) with (m, n, u) a solution of (1). Setting J := detDX and A := (D y X) −1 , using the chain rule and Proposition 6, one has that (m, n, u) satisfies Let us point out that since u(t, y) := u(t, X(t, y)), then it easily follows from the definition of flow X in (4) that Before presenting the main results of this paper, we first recall the definition of homogeneous Besov spaces. They could be defined through the use of a dyadic partition of unity in Fourier variables called homogeneous Littlewood-Paley decomposition. To this end, choose a radial function ϕ ∈ S(R N ) supported in The homogeneous frequency localization operator∆ j andṠ j are defined bẏ Let us denote the space Y (R N ) by the quotient space of S (R N )/P with the polynomials space P. The formal equality u = k∈Z∆ k u holds true for u ∈ Y (R N ) and is called the homogeneous Littlewood-Paley decomposition. We then define, for s ∈ R, 1 ≤ p, r ≤ +∞, the homogeneous Besov spacė Let m and n be two constants satisfying m > 0 and n ≥ 0. We shall obtain the existence and uniqueness of a local-in-time solution (m, n, u) for system (5), with .
Let us now state our main result.
Theorem 2.1. Let 1 < p < 2N and N ≥ 2. Let m > 0 and n ≥ 0 be two constants. Assume that the initial data satisfy Then system (5) admits a unique local solution (m, n, u) with m bounded away from In Eulerian coordinates, Theorem 2.1 means: Under the same assumptions as in Theorem 2.1 with 1 < p < 2N and N ≥ 2, system (1) has a unique local solution (m, n, u) with u ∈ E p (T ), 2 and m bounded away from 0.
Remark 1. In [22], the authors obtained the local existence and uniqueness of the solution to system (1) in critical Besov spaces based on the L p (1 < p ≤ N ) framework. Here, Theorem 2.2 improves the range of p from (1, N ] to (1, 2N ) under the same conditions as in [22]. In addition, since u 0 ∈ (Ḃ N p −1 p,1 ) N and 1 < p < 2N by means of Theorem 2.2, we infer that initial velocities in critical Besov spaces with negative indices produce a unique local solution.
3. Preliminaries. In this section, we present some technical results that have been used repeatedly in the next content.
3.1. linear parabolic systems with rough coefficients. The proof of the main results in this paper is based on the estimates that have been established recently in [8] for the following Lamé system with nonsmooth coefficients: (here both u and f are valued in R N ) when the following uniform ellipticity condition is satisfied: Proposition 1. (see [8]) Let a, b, λ and µ be bounded functions satisfying (9).
Assume that a∇µ, b∇λ, µ∇a and λ∇b are in L ∞ (0, T ;Ḃ N p −1 p,1 ) for some 1 < p < +∞, and that there exist some constants a, b, λ and µ satisfying 2āμ +bλ > 0 andāμ > 0, Then for any data u 0 ∈Ḃ Furthermore, there exist two constants η and C such that if q is so large as to satisfy dτ .

3.2.
Estimates for product and composition. For the proofs of the following propositions, one can see [1,7,18], and Appendix of [8,9].
The following product law holds: uv Ḃσ Proposition 4. If a 1 and a 2 belong toḂ and there exists a function of two variables C depending only on s, N and G, and such that The chain rule states that D y H(y) = D x H(X(y)) · D y X(y) and ∇ y H(y) = ∇ y X(y) · ∇ x H(X(y)).
Proposition 5. (See [8,9]) Let X be a globally bi-Lipschitz diffeomorphism of R N and (s, p) with 1 ≤ p < ∞ and − N p < s ≤ N p . Then a → a • X is a self-map oveṙ B s p,1 in the following cases: The supremum is taken over those functions ψ inḂ t p,1 with norm 1.
Proposition 6. (See [8,9]) Let K be a C 1 -scalar function over R N and H be a C 1 -vector field. If X is a C 1 -diffeomorphism such that J := det(D y X) > 0, then where adj (D y X) is the adjugate matrix of D y X.
From Proposition 6, we can obtain the following relations: 3.4. Estimates of flow. In the following, we recall the flow estimates that have been proved in [8,9].
then for all t ∈ [0, T ], we have . Furthermore, if w is a vector field such that Dw ∈ L 1 (0, T ;Ḃ N p p,1 ) then .

HAIBO CUI, QUNYI BIE AND ZHENG-AN YAO
4. Proof of the main theorem. In this section, with the help of Proposition 1, we give the proofs of Theorem 2.1 and Theorem 2.2 through Banach fixed theorem. From now on, to simplify the notation, we drop the bars of the Lagrangian coordinates. Denoting system (5) thus writes with where we have used that m = J −1 v m 0 and n = J −1 v n 0 . In order to solve (5) locally, it suffices to show that the map with u the solution to has a fixed point in E p (T ) for small enough T . Let u L be the solution to the linear system corresponding to (18) with m 0 ≡ 1, i.e., L 1 u L = 0, , u L | t=0 = u 0 .

4.1.
Proof of Theorem 2.1. We claim that the Banach fixed point theorem applies to the map Φ defined in (21) in some closed ball B Ep(T ) (u L , R) with suitably small T and R. Denoting u := u − u L , we obtain that u satisfy Since (7) is fulfilled, we set Because the spaceḂ N p p,1 embeds in the set of bounded continuous functions, then there exists some q ∈ Z so that min inf In order to prove Theorem 2.1, due to Proposition 1, it suffices to study the right hand side of (23) 1 . In what follows, we suppose that for a small enough c, it holds that Proposition 1 and the definition of the multiplier space M(Ḃ N p −1 p,1 ) (see (13)) yield that for some constant C In order to deal with the first term of the r.h.s. of (25), it follows from (L 1 − L m0 )u L = (m −1 0 − 1)div 2µD(u L ) + λdiv u L Id and composition inequalities (11) and (12) that .
Terms I 1 and I 2 have been bounded in [8] as follows: .
Next, we focus on the estimate of I 3 (v). Recall that the pressure is given by

Then, I 3 (v) is written as
where Attention is now focused on bounding K 1 and K 2 . Owing to Proposition 7,(24) and composition inequalities (11) and (12), it follows that Thus one arrives at .
Putting all the above information together, we conclude that .
Since v belongs to the ball B Ep(T ) (u L , R), decomposing v into v + u L and keeping in mind that v satisfies (24), we have Therefore, if we first choose R so that for a small enough constant η, and then take T so that then we may conclude that Φ maps B Ep(T ) (u L , R) into itself.
Second step. Contraction estimates. We consider two vector-fields v 1 and v 2 in B Ep(T ) (u L , R) and set u 1 := Φ(v 1 ) and u 2 := Φ(v 2 ). Let δu := u 2 − u 1 and δv := v 2 − v 1 . To simplify the notation, we set To prove that Φ is contractive, it is mainly a matter of applying Proposition 1 to . Then one gets, provided that C m0,q T ≤ log 2, .
The first two terms of the r.h.s. of (30) have been estimated in [8], then we list as follows: for j = 1, 2. As for the last term of (30), a straightforward calculation based on Proposition 8 ensures that .

Indeed, recall that
To bound H 1 , the way to deal with P (J −1 2 m 0 , J −1 2 n 0 ) is in accordance with (26). Then it follows from Proposition 8 that In order to deal with H 2 , let us first notice that . Applying Proposition 8, composition inequalities (11) and (12) again yields Hence .
Finally, we get .
Given that v 1 and v 2 are in B Ep(T ) (u L , R) with suitably small T and R, we end up with One can thus conclude that Φ admits a unique fixed point in B Ep(T ) (u L , R). Consider two triplets (m 1 0 , n 1 0 , u 1 0 ) and (m 2 0 , n 2 0 , u 2 0 ) of data satisfying the assumptions of Theorem 2.1. Let (m 1 , n 1 , u 1 ) and (m 2 , n 2 , u 2 ) be two solutions in E p (T ) corresponding to those data. Setting δu := u 2 − u 1 , we thus get Here, I 1 and I 2 correspond to the quantities that have been defined previously in (20), and . Applying Proposition 1 to (31), we can estimate each term on the r.h.s. of (31), exactly as in the estimates of the second step. Here, for the first three terms on the r.h.s. of (31), we only list the estimates as follows (see also [8]): .
To deal with term G 3 , we first recall that Then, combining composition, flow and product estimates yields .
Collecting all the above estimates, we conclude that for t ≤ T , .
As in [8], the constant C m 1 0 ,m 2 0 depends only on m 1 0 through its norm, for the integer q used in Proposition 1 corresponds to m 1 0 only. So if δm 0 is small enough then the above inequality implies that Since Du 1 → 0 as t → 0, a bootstrap argument yields that, for small enough t, δu 0 and (δm 0 , δn 0 ), Concerning the liquid mass and the gas mass, we have here, in the last line, we have used (32). Therefore, we may conclude to both uniqueness and continuity of the flow map on a small time interval. Note that in the case where both initial data coincide, then we may iterate the proof and obtain the uniqueness on the whole time interval then the map X defined in (6) is a C 1 (and in fact locallyḂ N p +1 p,1 ) diffeomorphism over R N and the triplet (m, n, u) := (m • X −1 , n • X −1 , u • X −1 ) satisfies (1) and has the same regularity as (m, n, u).
Conversely, given that some solution (m, n, u) to system (5)  As above, the relations (14), (15) and (16) hold whenever 1 < p < 2N . Then (m, n, u) is a solution to system (1) and Proposition 5 ensures that (m, n, u) has the desired regularity. p,1 ) and u in the space E p (T ). If T is small enough then (34) is satisfied, so Proposition 9 provides that (m, n, u) := (m • X −1 , n • X −1 , u • X −1 ) is a solution of (1) in the desired functional space.
To prove uniqueness, we consider two solutions (m 1 , n 1 , u 1 ) and (m 2 , n 2 , u 2 ) corresponding to the same data (m 0 , n 0 , u 0 ), and execute the Lagrangian change of variable (pertaining to the flow of u 1 and u 2 respectively). The obtained triplets