Vanishing capillarity limit of the non-conservative compressible two-fluid model

In this paper, we consider the non-conservative compressible two-fluid model with constant viscosity coefficients and unequal pressure function in $\mathbb{R}^3$, we study the vanishing capillarity limit of the smooth solution to the initial value problem. We first establish the uniform estimates of global smooth solution with respect to the capillary coefficients $σ^+$ and $σ^-$, then by the Lion-Aubin lemma, we can obtain the unique smooth solution of the 3D non-conservative compressible two-fluid model with the capillary coefficients converges globally in time to the smooth solution of the 3D generic two-fluid model as $σ^+$ and $σ^-$ tend to zero. Also, we give the convergence rate estimates with respect to the capillary coefficients $σ^+$ and $σ^-$ for any given positive time.

Then the system (1.1) can be rewritten as follows: (1.11) We consider the Cauchy problem of (1.11):
The following are the main results. Firstly, the global existence of smooth solution to the problem (1.11)-(1.12) is stated in the following theorem.
where η is a positive, small fixed constant, and there exists a constant δ >0, where δ is small enough, then the problem (1.11)-(1.12) admits a solution globally in time and satisfies where C is a positive constant independent of σ + and σ − .
Compare with [11], here we need to show the uniform estimates of the global smooth solution for the problem (1.11)-(1.12) with respect to the capillary coefficients σ + and σ − .
Here (R + ,ū + ,R − ,ū − )(x,t) is the global in time solution of the problem (1.14)- (1.15) in [0,T], satisfies , which is a global solution for t > 0, and satisfies the following estimate for any t ∈ [0, T ], where C is a positive constant independent of σ + and σ − .
Finally, we give the convergence rate estimates from the solution of the problem (1.11)-(1.12) to the solution of the problem (1.14)-(1.15) for any given positive time.
The rest of this paper is described as follows. In the section 2, we will give some useful lemmas which will be used later. In sections 3, we will reformulate the original problem. In the section 4, we will establish the uniform estimates of the global-in-time solution. Finally, section 5 is devoted to the proof of the main results (Theorems 1.1-1.3).
Notation. Throughout this paper, C > 0 are generic positive constants independent of capillary coefficient σ ± , which may vary in different estimates. ∇ = (∂ x1 , ∂ x2 , ∂ x3 ) and for a multi-index α = (α 1 , α 2 , α 3 ), ∂ α x = (∂ α1 x1 , ∂ α2 x2 , ∂ α3 x3 ) and |α| = α 1 + α 2 + α 3 . L p (R 3 )(1 ≤ p ≤ ∞) denotes the space of measurable functions whose p-powers are integrable on R 3 , with the norm · L p = ( we denote the norm of L 2 (R 3 ) by · for simplicity, and L ∞ (R 3 ) is the space of bounded measurable functions on R 3 , with the norm · L ∞ = ess sup x | · |. H k (R 3 ) is the sobolev space, k ∈ Z + , which stands for the space of L 2 (R 3 )-functions whose derivatives (in the sense of distribution) up to kth order are also L 2 (R 3 )-functions .∇ k f denotes a set composed of all k order partial derivatives with respect to the variable x of the function f . The notation " ·, · " stands for the inner-product in L 2 (R 3 ).

Preliminaries.
In this section, we will give some useful lemmas which can be used in the next section. Then (2) f L 6 ≤ C ∇f ; ( and where p 1 , p 2 , p 3 , p 4 ∈ [1, ∞] and 3. Reformation of the original problem. In the section, we will reformulate the problem (1.11)-(1.12) in terms of the perturbed variables. Without loss of generality, we can take the constant equilibrium statesR ± =1 and set n ± = R ± − 1, then the system (1.11)-(1.12) can rewritten as follows: with initial date as |x| → ∞, where , β 3 = ς 2 (1, 1), and the source terms are where we define the nonlinear functions of (n + , n − ) by .
Note that this will be used in Lemma 4.1 which is crucial for the proof of Proposition 4.1 below.

4.
The a priori estimates. The global existence of the solution (R + , u + , R − , u − ) to the steady state (1, 0, 1, 0) can be easily translated into the global existence of the perturbed solution (n + , u + , n − , u − ). The global smooth solution (n + , u + , n − , u − ) to the problem (3.1)-(3.2) is constructed by the combination of the local existence and the a a priori estimate. The proof of the existence of local solution is standard, refer for instance to [18] and we mainly concern about the uniform a priori estimates of (n + , u + , n − , u − ).
In the following, we assume that there exists a sufficiently small δ > 0, such that the following a priori assumptions holds By (4.1) and the Sobolev inequality, we have For some positive constant C, we have for k ≥ 1. These will be often used in the rest of this paper.
If δ is small enough, then there exists a positive constant C independent of σ + and σ − , such that for t ∈ [0, T ], it holds that Next, we give the proof of Proposition 4.1. For the sake of clarity, we divide the proof into the subsequent three lemmas.
3) for δ > 0 small enough, and some positive constant C, which is independent of σ + and σ − .
up and then integrating the resulting equality over R 3 by parts, we have First, by using (4.4) 1 and (4.4) 2 , we have Similarly, we have Then, by using integration by part, Hölder inequality, Lemma 2.2, the a priori assumption (4.1) and Young inequality, we have In the same way, we get Now, we estimate I 3 and I 8 by using (4.4) 1 and (4.4) 2 , we have and Then, one deduces that By using the Young inequality and Hölder inequality, it holds that and In the end, for I 5 and I 10 , by using integration by parts, (4.4) 1 and (4.4) 2 , we get and Putting the estimates of I 1 -I 10 into (4.5), taking δ sufficiently small, and noticing that we get (4.3). Thus, the proof of Lemma 4.1 is completed.
Next, we will give the higher-order derivative estimates.
Lemma 4.2. For k = 1, 2, 3, it holds that and for δ > 0 small enough, and some positive constants C, which are independent of σ + and σ − .

12)
and d dt for δ > 0 small enough, and some positive constants C, which are independent of σ + and σ − .
Proof of Proposition 4.1. Since where η is a positive, small fixed constant, then β 2 = ς 2 (1, 1)+ where 1 is defined by Since , there exists a constant D 1 independent of δ, σ + , σ − and β 2 , such that Then (4.17) + 1≤k≤3 d 2 (4.7) gives d dt 1 (t) + + 1 2 where then by using the smallness of δ, we have , thus there exists a positive constant D 2 , such that Next, we take η sufficiently small, such that the sum of the second, third, f if th, and the seventh terms on LHS of (4.21) for some positive constant D 3 . Denote From the above, there exists a constantC independent of δ, σ + and σ − , such that In the end, there exists a positive constantC 1 independent of δ, σ + and σ − , such that the following inequalities holds (4.22) provided that δ is small enough, and β 2 is sufficiently small. Then by the Gronwall inequality, we complete the proof of Proposition 4.1.

5.
Proof of the main results. In this section, we will give the proofs of Theorems 1.1-1.3.
Proof of Theorem 1.1. Theorem 1.1 is a direct conclusion of the combination of the local existence and the global a priori estimate in Proposition 4.1, we omit the proof here, see [23] for the details.