BLOW-UP CRITERION FOR AN INCOMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM WITH DIFFERENT DENSITIES

. This paper is concerned with a coupled Navier-Stokes/Allen-Cahn system describing a diﬀuse interface model for two-phase ﬂow of viscous incom- pressible ﬂuids with diﬀerent densities in a bounded domain Ω ⊂ R N ( N = 2 , 3). We establish a criterion for possible break down of such solutions at ﬁnite time in terms of the temporal integral of both the maximum norm of the deformation tensor of velocity gradient and the square of maximum norm of gradient of phase ﬁeld variable in 2D. In 3D, the temporal integral of the square of maximum norm of velocity is also needed. Here, we suppose the initial density function ρ 0 has a positive lower bound.

1. Introduction. In this paper, we investigate a diffusive interface model, which describes the motion of a mixture of two viscous incompressible fluids. Especially, the fluids have not "matched densities" but "different densities". In this model the sharp interfaces are replaced by narrow transition layers. The latter feature has the advantage to deal with interfaces that merge, reconnect and hit conditions. A phase field variable χ is introduced and a mixing energy is defined in terms of χ and its spatial gradient. The model consists of Navier-Stokes equations governing the fluid velocity coupled with a convective Allen-Cahn equation for the change of the concentration caused by diffusion. The effects of phase trasitions can also be described by different modified convective Cahn-Hilliard or other types of dynamics, see [3,9,15].
In this paper, we are interested in the following coupled Navier-Stokes/Allen-Cahn system for viscous incompressible fluids with different densities ρ t + div(ρu) = 0, (ρu) t + div(ρu ⊗ u) + ∇p = div(2η(χ)Du) − δdiv(∇χ ⊗ ∇χ), divu = 0, (ρχ) t + div(ρuχ) = −µ, ρµ = −δ∆χ + ρ ∂f ∂χ for (x, t) ∈ Ω × (0, +∞), where Ω is a bounded domain in R N (N = 2, 3) with smooth boundary ∂Ω, ρ ≥ 0 is the total density, u denotes the mean velocity of the fluid mixture, Du = 1 2 (∇u + ∇u T ), p is the pressure, χ represents the concentration difference of the two fluids, µ is the chemical potential, η(χ) > 0 is the viscosity of the mixture, the free energy density satisfies double-well structure , positive constant δ denotes the width of the interface. The usual Kronecker product is denoted by ⊗, i.e. (a ⊗ b) ij = a i b j for a, b ∈ R N . The equations (1) 1−3 are nonhomogeneous incompressible Navier-Stokes equations, which have an extra term ∇χ ⊗ ∇χ describing capillary effect related to the free energy The equations (1) 4−5 are Allen-Cahn equations. Obviously, the system (1) is a highly nonlinear system coupling hyperbolic equations with parabolic equations. The diffuse interface models for two-phase flow of incompressible viscous fluids with "matched densities" have been extensively studied. We refer the readers to [1,4,12,19,20,21] for details. It is evident that, the densities in two fluids are often quite different. Within our knowledge, there are only a few theoretical results available to compressible models. For compressible Navier-Stokes/Allen-Cahn system, Feireisl et al. [10] proved the existence of weak solutions in 3D. In [8], we obtained the global well-posedness in 1D with constant mobility. We prove the existence of the initial boundary value problem in various regularity classes, as well as uniqueness for strong solutions. For compressible Navier-Stokes/Cahn-Hilliard system, Abels and Feireisl [2] derived the existence of weak solutions.
In this paper, we investigate the Navier-Stokes/Allen-Cahn system for two fluids with non-matched densities, but the velocity u satisfies the divergence-free condition divu = 0, i.e. the fluids are incompressible and with different densities. Following our works in [14], where the existence of unique local strong solution has been obtained, we deal with the main mechanism for possible breakdown of such a local strong solution. We supplement the system (1) with the following initial conditions the usual no-slip boundary condition on the velocity and Neumann boundary condition on the phase field variable where n is the unit outward normal vector of ∂Ω.
Notations. For p ≥ 1, denote L p = L p (Ω) as the L p space with the norm · L p . For k ≥ 1 and p ≥ 1, denote W k,p = W k,p (Ω) for a Sobolev space, whose norm is denoted by · W k,p , and specially H k = W k,2 (Ω).
The existence and uniqueness of local strong solutions have been proved in [14].
Then there exist a time T * > 0, a constant c = c(c 0 , T * ) and a unique strong solution (ρ, u, p, χ, µ) of the problem (1)- (3) in Ω × (0, T * ]. The existence of global solutions is closely related to the estimate for ∇ρ L ∞ (Q T ) , which is the main difficulty we can not handle. Besides the global existence, another interesting question is the main mechanism of possible break down of local strong solutions. For such question, the pioneering work is obtained by Beale, Kato and Majda [6], they proved that the maximum norm of the vorticity ∇ × u controls the breakdown of smooth solutions of the 3D Euler equations. Later, Ponce [16] derived the same blow-up criterion when the vorticity was substituted by the deformation tensor Du, that is, a solution remains smooth if T 0 Du L ∞ dt remains bounded. The works on blow-up criterion for incompressible Navier-Stokes equation, we refer the readers to Serrin [17] and Struwe [18] for example. Recently, for nonhomogeneous incompressible Navier-Stokes equations, i.e. the velocity is divergence-free, but the density is not assumed to be a constant, Kim [13] established a weak Serrin class blow-up criterion.
Motivated by these works, we will establish in this paper the blow-up criterion of breakdown of local strong solutions in finite time. Our main result is as follows. Theorem 1.3. Let (ρ, u, p, χ, µ) be a strong solution of the initial boundary value problem (1)-(3). If 0 < T * < +∞ is the maximum time of existence, then Remark. We should point out that, because of the appearance of the density ρ in Allen-Cahn equation (1) 4,5 , we can not handle the vacuum state. Moreover, our proof strongly depends on the divergence-free condition, which ensure the solutions being away from vacuum. So the conclusions in this paper can not be extended to corresponding compressible system directly.
Since the constant δ play no role in the analysis, we assume henceforth that δ = 1. Throughout this paper, we assume that η(s) ∈ C 1 (R) and there exist positive constants η, η andη, such that Moreover, we denote by A B if there exists a positive constant C such that A ≤ CB.
holds for some constant M 0 > 0, then there exists a positive constant C depending only on ρ 0 , u 0 , χ 0 , T * and M 0 such that In terms of the a priori estimates (9), we can prove that T * is not the maximum time, which is the desired contradiction. Firstly, we deal with the bounds of ρ and ∇ρ, which is very important in the proof of the following estimates.
The following estimate is crucial in the proof of the forthcoming lemmas.
We assume that the hypotheses in Lemma 2.1 hold. Then for any 2 ≤ r ≤ 6, there holds Proof. From (1) 4,5 we have Differentiating the above equation with respect to x yields Multiplying (22) by r|∇χ| r−2 ∇χ (2 ≤ r ≤ 6) and integrating the result over Ω, we get We calculate the first term on the right hand side Putting (24) into (23), integrating by parts and using (1) 1 , we have d dt Ω ρ|∇χ| r dx + r(r − 1) In what follows, we estimate I i (i = 1, 2, 3, 4, 5, 6) one by one in dimension three for example.

YINGHUA LI, SHIJIN DING AND MINGXIA HUANG
Moreover, by using Cauchy inequality, we have Putting all these estimates into (25) and choosing ε > 0 small enough, we obtain d dt Ω ρ|∇χ| r dx + Applying Gronwall's inequality and using (7), (8), (12), (13), we derive (20). The case of dimension two is similar. Therefore, the proof of this lemma is complete. Then we continue to do some estimates for χ and u.
Multiplying the above equation by u t , integrating the result over Ω, and recalling (1) 1,3 , we get where we have used (12), (26) and Korn's inequality (19) in the last step. From which we have In the following, we deal with the term ∇χ t 2 L 2 . Differentiating (21) with respect to t gives Multiplying (40) by χ tt , integrating the result over Ω and noticing where we have used (10), (11), (12) and (26). It follows that Multiplying (41) by η/C, adding the resulting inequality to (39) and by using (6) yield From the equations (1) 2,4,5 and the assumptions on the initial data ρ 0 , u 0 , χ 0 , applying Gronwall's inequality, by (7) and (26), we obtain(38). Thus we complete the proof of this lemma. In terms of the results for the stationary Stokes equation, we derive the higher order estimates for u and p.
At last, we deal with the higher order estimates for ρ, χ and µ.
Moreover, differentiating (1) 1 with respect to x, we can derive that ∇ρ t L r ∇u L r ∇ρ L ∞ + u L ∞ ∇ 2 ρ L r ≤ C.