A new blowup criterion for strong solutions to a viscous liquid-gas two-phase flow model with vacuum in three dimensions

In this paper, we establish a new blowup criterion for the strong 
solutions in a smooth bounded domain $\Omega\subset\mathbb{R}^3$. In 
[13], Wen, Yao, and Zhu prove that if the strong 
solutions blow up at finite time $T^*$, the mass in 
$L^\infty(\Omega)$ norm must concentrate at $T^*$. Here we extend 
Wen, Yao, and Zhu's work in the sense of the concentration of mass 
in $BMO(\Omega)$ norm at $T^*$. The method can be applied to study 
the blow-up criterion in terms of the concentration of density in 
$BMO(\Omega)$ norm for 
 the strong solutions to compressible Navier-Stokes equations in smooth bounded domains. Therefore, as a 
 byproduct, we can also improves the corresponding result about Navier-Stokes equations in 
[11]. Moreover, the appearance of vacuum is allowed 
in the paper.

density, respectively. As stated in [4], ρ l and ρ g satisfy the equations of state, i.e., ρ l = ρ l,0 + P −P l,0 a 2 l , ρ g = P a 2 g , where a l , a g are sonic speeds, respectively, in the liquid and gas, and P l,0 and ρ l,0 are the reference pressure and density given as constants. u denotes velocity of the liquid and gas. P is the common pressure for both phases, which satisfies P (m, n) = C 0 −b(m, n) + b(m, n) 2 + c(n) , with C 0 = 1 2 a 2 l , k 0 = ρ l,0 − Let us briefly review some previous works about the viscous liquid-gas two-phase flow model (1). More precisely, when both of the two fluids are compressible (liquid is considered slightly compressible), Evje and Karlsen [4] derived the first work on the global existence of weak solutions of Cauchy problem of (1) in one dimension. In [4], both of m and n are positive initially. In higher dimensions, Yao, Zhang and Zhu [17] obtained the existence of the global weak solution to the two-dimensional case of (1) when the initial energy is small, and both of m and n are positive initially. Furthermore, they [18] established a blow-up criterion in terms of the upper bound of the liquid mass for the strong solutions to the model in a smooth bounded domain of R 2 . In [18], the authors only deal with the case: there is no initial vacuum, i.e., m 0 > 0, n 0 > 0. The global well-posedness of classical solutions with small initial energy in R 3 can be referred to Cui-Wen-Yin's work [2]. In the framework of Besov space, please refer to [7].
When vacuum is allowed, i.e., m 0 ≥ 0 and n 0 ≥ 0, things become more difficult. For example, we can not estimate 1 m or 1 n when we do the estimates. By using the iteration arguments, Wen, Yao, and Zhu obtained the local well-posedness of strong solutions of system (1) with vacuum in a smooth bounded domain of R 3 . Moreover, Wen, Yao, Zhu derived a blowup criterion for the strong solutions in terms of the concentration of m in L ∞ (Ω) at the finite maximal time T * for existence. For the global well-posedness of the strong solutions with small initial energy in three dimensions, please refer to [6]. When the liquid is incompressible and the gas is polytropic, i.e., P (m, n) = Cρ γ l n ρ l − m γ , please refer to [5,15,3,16] and references therein.
However, it is still unknown that the global smooth solutions with large initial data in three dimensions exist or not. This motivates us to study some sharp blowup criteria which can give some insight into the mechanics of possible blowing up.
Before stating our main result, we would like to mention the definition of strong solutions here.
Actually, given some suitable initial data, the local existence and uniqueness of the strong solutions of (1)-(3) has been obtained by Wen, Yao, Zhu in their work [13]. More precisely, Let Ω be a bounded smooth domain in R 3 and q ∈ (3,6]. Assume that the initial data m 0 , n 0 , u 0 satisfy m 0 , n 0 ∈ W 1,q (Ω), u 0 ∈ H 1 0 (Ω) ∩ H 2 (Ω), 0 ≤ s 0 m 0 ≤ n 0 ≤ s 0 m 0 in Ω, where s 0 and s 0 are positive constants. The following compatible condition is also valid: Then, there exist a T 0 > 0 and a unique strong solution (m, n, u) to the problem Furthermore, under the assumption Wen, et al in [13,14] established the blow-up criterion for the strong solutions: provided that (8) holds.
In this paper, we establish a new blow-up criterion for the strong solutions. Our main result is stated as follows: provided 29µ 3 > λ. Here BM O(Ω) denotes the John-Nirenberg's space of bounded mean oscillation whose norm is given by (18) and (19).

Remark 1.
Under the conditions of Proposition 1, we can prove Lemma 3.1 which implies that (9) is equivalent to Observe that for a given function f , it is easy to verify for some known positive constantC independent of t and m. Therefore, it is obvious that our main result (10) is a relaxation of Wen et al's result (9).
Remark 2. Unlike [14], the domain here is bounded. The estimates of the effective viscous flux and the vorticity in [14] are not valid any more, since we do not know anything about the boundary conditions of them. Here we use some ideas of Sun et al [11] to handle some challenges due to the bounded domain.
Remark 3. Our methods can also be applied to compressible Navier-Stokes equations in bounded domains. Therefore, as a byproduct, we extend the blowup criterion in terms of the concentration of the density in L ∞ (Ω) in Sun et al's work [11] for the strong solutions of compressible Navier-Stokes equations in bounded domains.

2.
Preliminaries. In this section, like in [13], we give some useful lemmas which will be used afterwards, where N = 3.
Lemma 2.1. Let Ω ⊂ R N be an arbitray bounded domain with piecewise smooth boundaries. Then the following inequality is valid for every function u ∈ W 1,p 0 (Ω) or u ∈ W 1,p (Ω), (11) is also valid for p = ∞. The positive constant C 1 in inequality (11) depends on N , p, r , α and the domain Ω but independent of the function u.

Lemma 2.2.
Let Ω ⊂ R N be an arbitrary bounded domain with piecewise smooth boundaries. Then the following inequality is valid for every function u ∈ W 1,p (Ω): where N , p, r , p and α are the same as those in Lemma 2.1. The positive constant C 2 in inequality (12) depends on N , p, r , α and the domain Ω but independent of the function u.
The above two lemmas can be found in [9,12] and the references therein. Next, we give some L p (p ∈ (1, ∞)) regularity estimates for the solution of the following boundary problem: Here . From (4), we know that (13) is a strong elliptic system. If F ∈ W −1,2 (Ω), then there exists an unique weak solution U ∈ H 1 0 (Ω). In the subsequent context, we will use L −1 F to denote the unique solution U of the system (13) with F belonging to some suitable space such as W −1,p (Ω). The following two lemmas can be found in [11] and references therein: ∞), and U be a solution of (13). Then there exists a constant C depending only on µ, λ, p, N and Ω such that . Then there exists a constant C depending on p, N and the Lipschitz property of the domain Ω such that Here BM O(Ω) denotes the John-Nirenberg's space of bounded mean oscillation whose norm is defined by with the semi-norm where Ω r (x) = B r (x) ∩ Ω, B r (x) is the ball with center x and radius r and d is the diameter of Ω. For a measurable subset E of R N , |E| denotes its Lebesgue measure and The last lemma in the section can be found in [1]: It is easy to verify from (21) that lim sup for any q 1 < ∞ and some positive constant M 1 .
In this section, we denote by C a generic positive constant which may depend on µ, λ, Ω, m 0 , n 0 , u 0 , M , T * , and the parameters in the expression of P in (5). For simplicity, we omit the domain of the spatial integrability and spatial norm when it does not cause any confusion.
Similar to Lemma 5.1 and Lemma 5.2 in [13], we get the following lemmas. In lemma 3.2, in order to relax the restriction 25µ 3 > λ in [13], we use the technique in [14]. For later use, as in [11], we denote w = u − h, where h is the unique solution to From Lemma 2.3, we get for any p ∈ (1, ∞) (1) 3 , (3) and (23) imply where for any q 1 < ∞ and p 1 < 6.
Proof. Similar to Lemma 4.3 in [13], we can easily obtain P m L ∞ (Q T ) + P n L ∞ (Q T ) ≤ C where we have used Lemma 3.1. Next we aim to show P L ∞ (0,T ;L q 1 ) ≤ C. A direct calculus yields that This together with (22) completes the proof of (26) 1 .
In the following we prove (26) 2 .
[ where we have used the boundness of P m and P n . This together with (21) and (26) 1 completes the proof of (26) 2 .
Proof. (1) 3 can be rewritten as Differentiating (37) with respect to t, and using (1) 1 , we conclude Multiplying (38) byu, integrating the resulting equation over Ω, and using integration by parts, we obtain
where we have used integration by parts and Hölder inequality.
In the following, we give the estimates of the derivatives of m and n. (∇m, ∇n)(t) L q ≤ C.
Proof. Differentiating the equation (1) 1 with respect to x i , then multiplying both sides of the resulting equation by q|∂ i m| q−2 ∂ i m, we get Integrating (45) over Ω, we obtain d dt Similarly, d dt ∇n L q ≤ C ∇u L ∞ ∇n L q + C ∇ 2 un L q .
Due to Lemma 3.5 and Sobolev inequality, we immediately give the following result. By Proposition 2 and Corollary 2, T * is not the maximum time of existence of strong solution, which is the desire contradiction. Thus, the proof of Theorem 1.2 is completed.