PHASE TRANSITIONS IN A COUPLED VISCOELASTIC SYSTEM WITH PERIODIC INITIAL-BOUNDARY CONDITION: (I) EXISTENCE AND UNIFORM BOUNDEDNESS

. This paper focuses on the phase transitions of a 2 × 2 system of mixed type for viscosity-capillarity with periodic initial-boundary condition in a viscoelastic material. By the Liapunov functional method, we prove the existence, uniqueness, regularity and uniform boundedness of the solution. The results are correct even for large initial data.

1. Introduction and main theorem. This work is concerned with the viscouscapillarity system of mixed type in the viscoelastic material dynamics (resp. the compressible van der Waals fluids): We study the coupled system with the initial condition and the 2L-periodic boundary condition where L > 0 is a given constant. Note that from the compatibility condition, we have v 0 (x) = v 0 (x + 2L), u 0 (x) = u 0 (x + 2L). (4) Here v(x, t) is the strain (resp. specific volume), u(x, t) the velocity, ε 1 > 0 and ε 2 > 0 the viscous constants, σ(v) the stress function (resp. pressure function), which is assumed to be sufficiently smooth and non-monotonic. The stationary Let then The solution to system (1) with different types of initial or initial-boundary values has been widely studied, see [1]- [9], [11]- [16], [18]- [30], [33] and the references therein. Among them, Eden et al [6] studied (1) with the periodic initial-boundary value for the van der Waals model with pressure function: p(v) = −σ(v) = Rθ v−b − a v 2 , and proved the global existence and boundedness of the solution in the weak sense. To prevent the pressure to blow up, namely, to guarantee the solution v(x, t) > b, they required that the initial data v 0 and u 0 are sufficiently small and that the nonconvexity of the state function p(v) is not too strong. Under the assumption of small initial data, by using the Galerkin approximation method, it is standard to show the existence, uniqueness and boundedness of the solution (v, u)(x, t). The role of the smallness of the initial data is essential for obtaining the global existence as well as the stability. See also the stability of traveling waves with small initial perturbation in [16], [18]- [23]. However, for any large initial data, the uniform boundedness of the solution with respect to the given initial datum usually are not expected, even for the global existence of the solution. Hence, the present study on the solution with large initial data is of significant interest to researchers in the mathematics and physics community. In this paper, by considering the nonlinearity of the form σ(v) = v 3 − v, we show that, even for any large initial data, the global solution (v, u)(x, t) to (1)-(3) in the strong sense exists uniquely and is uniformly bounded. This result improves the previous work [6]. To obtain the uniform boundedness of the solution for a given large initial data, we find that the standard energy method together with the Liapunov functional cannot be directly applied to the original equations (1), because σ ′ (v) changes signs in different regions (hyperbolic and elliptic). In order to overcome this difficulty, we first transform the original periodic IBVP (1)-(3) into a new system (see (16) and (20)), and then show the uniform boundedness of the solution by selecting a suitable Liapunov functional for the new system.
The existence of the stationary solutions to (1)-(3) have been studied in our previous paper [24] by use of the energy method in [31] and the transversality arguments in [10] for Cahan-Hilliard equation. We proved that when the product of the viscosities ε 1 ε 2 is large, there exists only one trivial solution (no phase transition), and when ε 1 ε 2 is small, there exists some non-trivial solutions, in which phase transitions occur through the three phases periodically. Furthermore, we found that the number of non-trivial solutions depends on the size of m 0 . However, an interesting and important continuation on this topic are to study the stability of these stationary solutions, i.e., the convergence of the solution of (1)-(3). This will be reported in our following paper [25]. In order to show the convergence by the compactness arguments and the energy method, it is essential for us firstly to establish the existence, the uniqueness, and in particular, the uniform boundedness of the solution (v, u)(x, t) of (1)-(3), which is the goal and the contribution of the present study.
Notation. Before stating our main results, we introduce the following notations. Throughout the paper, C > 0 denotes a generic constant, while C i > 0 (i = 0, 1, 2, · · · ) represents a specific constant, R = (−∞, ∞). Since solutions (v, u)(x, t) of (1)-(3) are periodic, we introduce spaces of periodic functions. Letting p denote the period, we introduce the Hilbert space L 2 per (R) of locally square integrable functions that are periodic with period p, The periodic spaces L ∞ per (R) and L k per (R), where k is a positive integer, are similarly defined. Let T > 0 be a number and B be a We now state our main result.
2. Existence, uniqueness and uniform boundedness. In this section, we prove the global existence and uniqueness of the solution (v, u) to system (1)-(3). It is important to note that there is no restriction on the amplitude of the initial data. The uniform boundedness of the solution (v, u)(x, t) plays a key role in the study, and it leads to the global existence, in particular, the convergence to certain steady-state solution (see [24,25]). First, by using the fixed-point iteration, we can prove the local existence. The detail of the proof is omitted.
we then have the following well-known alternative result (for example, see [17,32]).
Now we are going to prove T max = ∞. The most important step is to show the boundedness of the solution (v, u)(x, t). In fact, we will prove that not only the solution is bounded but it also is uniformly bounded for any given initial datum.
It has been already shown that, the solution of (1)- (3) is bounded when the initial datum is sufficiently small (see [5,6,16], [18]- [23]). But for large initial datum, it is not trivial to get such a uniform boundedness directly from the equations (1) because σ ′ (v) changes signs in different regions. To overcome the difficulty, we transform the original system, so that we could find a suitable Liapunov functional for the equivalent new system and the uniform boundedness can then be proved.
Then the system (1)-(3) can be rewritten as We now technically set up It is easy to verify We have also the periodic condition In fact, noting the zero-average ofv in (11), i.e., x+2L xv (y, t)dy = 0, we have Similarly, ψ(x + 2L, t) = ψ(x, t) can be proved. whereσ where c(t) and d(t) are integral constants Proof. Integrating the equations of (16) with respect to x over [0, 2L], and noting the periodicity, we have
Thus, we finally have Differentiating the second equation of (20) with respect to x and substituting the first equation of (20) to this resultant equation, we obtain a scalar equation on φ Before proving the uniform boundedness of the solution, we introduce the following Poincaré inequality.
Lemma 2.4. Let φ(x, t) be the solution of (21). It holds for all t ≥ 0 that Proof. Consider the following eigenvalue problem The eigenvalues are and the corresponding eigenfunctions arẽ which satisfy where ṽ i,k ,ṽ j,l = 2L 0ṽ i,k (x)ṽ j,l (x)dx is the inner product of L 2 per . It is known that the sequence {ṽ i,k (x)} (i = 1, 2 and k = 1, 2, 3, · · · ) forms an orthonormal basis for the space Therefore, as a periodic function satisfying φ(x, t) = φ(x+2L, t) and per,0 , and can be expressed in the Fourier form where the coefficients A k (t) and B k (t) are determined by Making the inner products leads to and Since β k ≥ β 1 = π L for k = 1, 2, · · · , then (27) and (28) imply namely, This can be extended to Similarly, we can show for the periodic function ψ(x, t): Combining (30) and (31), we prove (22).

Lemma 2.5 (Key). It holds uniformly
where C is a positive constant independent of T max .
Proof. Define an energy functional, the so-called Liapunov functional where Differentiating E(t) with respect to t, and using integration by parts and the equation of (21), we have Integrating (34) with respect to t over [0, t] yields E(t) ≤ E(0) =: C 3 , i.e., Notice that and by the Cauchy-Schwartz inequality (ab ≤ ηa 2 + (1/4η)b 2 for any η > 0) then substituting (37) on (36), we obtain In the same way, we obtain Thus, (39) and (38) give Applying (40) into (35), we finally prove the boundedness of φ in the form where and (41), we then prove the uniform boundedness for φ in the form Furthermore, from the first equation of (20), i.e., φ t − ψ x = ε 1 φ xx , we can easily prove Thus, the proof of this Lemma is completed.
Lemma 2.6 (Key). It holds uniformly where C > 0 is a constant independent of T max .