On global classical solutions to one-dimensional compressible Navier–Stokes/Allen–Cahn system with density-dependent viscosity and vacuum

In this paper, by using the energy estimates, the structure of the equations, and the properties of one dimension, we establish the global existence and uniqueness of strong and classical solutions to the initial boundary value problem of compressible Navier–Stokes/Allen–Cahn system in one-dimensional bounded domain with the viscosity depending on density. Here, we emphasize that the time does not need to be bounded and the initial vacuum is still permitted. Furthermore, we also show the large time behavior of the velocity.


Introduction
The Navier-Stokes/Allen-Cahn system, which is a combination of the compressible Navier-Stokes equations with an Allen-Cahn phase field description, is considered in this paper. Mathematically, in one dimension, this model reads as follows [5] (cf. [1]): for (t, x) ∈ (0, +∞) × [0, 1]. Here, ρ, u, and χ represent the density of the fluid, the mean velocity of the fluid mixture, and the concentration of one selected constituent, respectively; μ is the chemical potential, √ δ represents the thickness of the interfacial region. The viscous coefficient ν(ρ) > 0 satisfies 0 <ν ≤ ν(ρ).
Before stating our main results, we review some previous works on this topic. For 1dimensional compressible Navier-Stokes/Allen-Cahn system, Ding et al. [5] established the existence and uniqueness of local and global classical solutions for initial data ρ 0 without vacuum states. Besides, Ding et al. [6] proved the existence and uniqueness of global strong solutions to (1)-(4) with free boundary conditions and with the lower bound of the initial density. Yin et al. [18] investigated the large time behavior of the solutions to the inflow problem in the half space, and they obtained that the nonlinear wave is asymptotically stable if the initial data has a small perturbation. Recently, Luo et al. [15] (see also [14]) proved that the system tends to the rarefaction wave time-asymptotically, where the strength of the rarefaction wave is not required to be small. Chen et al. [2] established the global strong and classical solutions with initial vacuum in bounded domains. After that, Chen et al. [4] established the blowup criterion of the strong solutions with the viscosity depending on the density and the concentration of one selected constituent. Very recently, Yan et al. [17] considered the global existence of strong solutions with the phase variable dependent viscosity and the temperature dependent heat-conductivity without vacuum.
For the multi-dimensional compressible Navier-Stokes/Allen-Cahn system, Kotschote [11] established the local existence of a unique strong solution without initial vacuum. Later on, Feireisl et al. [8] proved the existence of weak solutions in 3D, where the density ρ is a measurable function, and they [9] obtained the global weak solutions in the bounded domain of R 3 without any restriction on the initial data for γ > 6, which was extended to γ > 2 by Chen et al. [3]. Hosek et al. [10] considered the weak-strong uniqueness result in a bounded domain of R 3 under the incompressibility assumption, which is relying on the relative entropy method. Very recently, Feireisl et al. [7] proved that the model is thermodynamically consistent, particularly, a variant of the relative energy inequality holds. At the same time, they obtained the weak-strong uniqueness principle and showed the low Mach number limit to the standard incompressible model. For more related results, we refer the readers to Zheng et al. [19], Liu et al. [12], and Ma et al. [16].
Although considerable progress has been made to the compressible Navier-Stokes/ Allen-Cahn system, one of the natural questions is whether one could obtain the global classical solutions without any small assumption on the initial data or perturbations, where the time t could tend to +∞? Motivated by [13], we give a partial answer to this question.
Our first main result in the paper is the following.
Then there exists a global strong solution (ρ, u, χ) to the initial boundary value problem (1)- (7) such that, for all T ∈ (0, +∞), Especially, the density can remain uniformly bounded for all time, that is, and The following result means that the strong solution obtained by Theorem 1 is a classical solution provided that the initial data (ρ 0 , u 0 , χ 0 ) satisfies some additional conditions. Theorem 2 Assume ν(ρ) ∈ C 2 [0, ∞), and the initial data (ρ 0 , u 0 , χ 0 ) satisfies (u 0 , χ 0 ) ∈ H 1 0 ∩ H 2 and the following compatibility condition where g ∈ L 2 and h ∈ H 1 . Then the strong solution (ρ, u, χ) obtained in Theorem 1 becomes a classical solution and satisfies, for any 0 < T < +∞, A few remarks are listed in order: Remark 1 Compared with the previous results [2], ours are more general. First, the viscosity ν(ρ) depends on the density; Second, we remove the compatibility condition in obtaining the strong solution in Theorem 1, and Theorem 2 is established under (12), which has improved Theorem 2 in [2]; Third, the density ρ is uniformly bounded for all time and the large time behavior of u is also obtained, see (11) for details.
Remark 2 Similar to [2], we have to use the no-slip boundary condition on χ to deal with the term ρ 2 u x χ 2 t dx in (29) because u x L ∞ is not time-integrable when we establish the time-independent lower order estimates, see (26) below.
Remark 3 The concentration χ is uniformly (in time) bounded with higher order estimates in (10), without any decay as t → +∞, perhaps because it appears in the hyperbolicity in (3) rather than in the parabolicity in (24).
We now make some comments on the analysis of this paper. To obtain the results stated in Theorems 1 and 2, which mainly establish the time-independent lower order estimates and the time-dependent higher order ones, the method used in [2] is not suitable here, due to the all time-dependent a priori estimates. Moreover, it is difficult to obtain the large time behavior of solutions (11). Here, it is noted that we borrow some ideas from [13], where they discussed the global large classical solutions to the compressible Navier-Stokes equations. The key uniform upper bound of the density is obtained by Zlotnik's inequality, which is also successfully used to system (1)-(7) (see Lemma 4). Furthermore, the key time-independent L 2 -norm of u x is bounded by the material derivative u t +uu x (see Lemma 5). With the lower order estimates obtained in Lemmas 3-6, the time-dependent higher order estimates on (ρ, u, χ) are obtained by standard energy estimates and the properties of one dimension.
The paper is organized as follows. In the next section, we deduce the desired estimates globally in time. By using the a priori estimates obtained in Sect. 2, we complete the proofs of Theorems 1 and 2 in Sect. 3.

A priori estimates
In this section, we establish some necessary a priori estimates of the solutions to (1)-(7) to extend the local solution to a global one, which is guaranteed by the following Lemma 1, whose proof can be obtained by similar arguments as those in [5].
Before starting the a priori estimates, we list Zlotnik's inequality which could be found in [20] and will be used to establish the uniform upper bound of the density.

Lemma 2 Let the function y satisfy
for all 0 ≤ t 1 < t 2 ≤ T with some nonnegative constants N 0 and N 1 , then whereζ is a constant such that g(ζ ) ≤ -N 1 for ζ ≥ζ .

A priori estimates (I): Lower order estimates
We emphasize that, in this subsection, C denotes some positive constant, which may be changed line by line and depends only onν, δ, γ and the initial data (ρ 0 , u 0 , χ 0 ), but without the lower bound of the initial density ρ 0 and the length of T. First of all, we have the following basic energy estimates.
Proof This lemma can be obtained by standard energy estimates. Multiplying (2), (3) by u and μ, respectively, by integrating by parts and by using (1) and (4), we obtain (14), the details can be found in [2].
Due to the basic energy inequality (14), we first consider the uniform upper bound of the density ρ, which does not depend on the length of time T.
Proof To prove this lemma, we borrow some some ideas of [13]. First of all, integrating (2) over (0, x), we obtain ∂ ∂t which implies that Combining (16) with (17), it follows from (14) that where we have used the following fact (due to (1)) Now, we focus on the estimates of the last term on the right-hand side of (19). First, by (14) and Hölder's inequality, we easily obtain Next, we also have Finally, it follows from Lemma 2, Zlotnik's inequality, and (19) that which together with (5) shows (15). This completes the proof.
Next, we focus on L 2 -estimates about ρχ t , u x , and χ xx , which are the key estimates for the proofs of the main theorems.

A priori estimates (II): higher order estimates
In this subsection, we derive the higher-order estimates of the smooth solution (ρ, u, χ) to system (1)- (7). Particularly, in this subsection the constant C may depend on the initial data (ρ 0 , u 0 , χ 0 ), γ , δ, andν. Almost the a priori estimates obtained in this subsection could be obtained by similar arguments as in [2], we estimate them here to make the paper selfcontained and satisfy the new assumptions on the initial data and compatibility condition (12).

Lemma 10
Proof Multiplying (28) by χ tt , integrating the resultant equality by parts, we obtain which together with the compatibility condition (12) yields by using the fact that Furthermore, (56) together with (51) leads to To proceed, differentiating (28) with respect to t, multiplying the resultant equation by χ tt , and integrating by parts, we obtain Multiplying the above inequality by t and using Gronwall's inequality, we obtain where we have used (46), (47), and (56). Then, differentiating (2) with respect to t leads to Multiplying the above equation by u tt and integrating the resultant equality by parts, we obtain Now, we focus on the estimates of the terms on the right-hand side of (60). First, due to (36) and (47), we obtain It follows from (46) and (47) that Next, it follows from (1) and integration by parts that where we have used (15), (36), (46). Moreover, by (46), one easily shows that Furthermore, J 5 is estimated as Similarly, we deduce that Substituting all the above estimates into (60), we obtain Multiplying the above inequality by t and adding (58) to the resultant inequality, then integrating it over (0, t), after choosing ε small enough, and using (46), (47), (56), Gronwall's inequality, we show that due to Moreover, due to (50), (51), (56), and (62), we obtain Furthermore, it follows from (59) that which together with (62) leads to Similarly, due to (28), one has which together with (57), (62), (63), (64), and (56) shows (55). This completes the proof.

Proofs of the main theorems
In this section, based on the a priori estimates derived in Sect. 2, we extend the local classical solution obtained in Lemma 1 to a global one.
Next, we will extend the local existence time T 0 of the strong solution to be infinity and therefore prove the global existence result. Let T * be the maximal time of existence for the strong solution, thus, T * ≥ T 0 . For any 0 < τ < T ≤ T * with T finite, we obtain Let ρ * , u * , χ * (ρ, u, χ) T * , x = lim t→T * (ρ, u, χ)(t, x), it follows from (65) and (66) that (ρ * , u * , χ * ) satisfies the initial condition stated in Theorem 1. Therefore, we take (ρ * , u * , χ * ) as the initial data at T * and then use the local result, Lemma 1, to extend the strong solution beyond the maximum existence time T * . This contradicts the assumption on T * . We finally show that T * could be infinity and complete the proof of the global existence of the strong solution.
It remains to prove (11), process of which is similar to that in in [13], we sketch it here for completeness. Due to integration by parts, we have This completes the proof.
Proof of Theorem 2 With the higher-order estimates in Lemmas 8-10 at hand, the proof of Theorem 2 is similar to that of Theorem 1, and so it is omitted here for simplicity.