On higher-order anisotropic perturbed Caginalp phase field systems

Our aim in this paper is to study the existence and uniqueness of solution for hyperbolic relaxations of higher-order anisotropic Caginalp phase field systems with homogeous Dirichlet boundary conditions with regular potentials.


∂T ∂t
− ∆T = − ∂u ∂t (4) called conserved system (in the sense that, when endowed with Neumann boundary conditions, the spatial average of u is conserved). In this context, u is the order parameter, T is the relative temperature (defined as T =T − T E , whereT is the absolute temperature and T E is the equilibrium melting temperature) and f is the derivative of a double-well potential F (a typical choice is F (s) = 1 4 (s 2 − 1) 2 , hence the usual cubic nonlinear term f (s) = s 3 − s). Furthermore, we have set all physical parameters equal to one. These systems have been introduced to model phase transition phenomena, such as melting-solidication phenomena, and have been much studied from a mathematical point of view. We refer the reader to, e.g., [3,4,13]. Both systems are based on the (total Ginzburg-Landau) free energy (5) Ψ GL = where Ω is the domain occupied by the system (we assume here that it is a bounded and regular domain of R 3 , with boundary Γ), and the enthalpy (6) H = u + T.
As far as the evolution for the order parameter is concerned, one postulates the relaxation dynamics (with relaxation parameter set equal to one) for the nonconserved model, and for the conserved one, where D Du denotes a variational derivative with respect to u, which yields (1) and (2). Then, we have the energy equation where q is the heat flux. Assuming finally the usual Fourier law for heat conduction, we obtain (3) or (4). In (5), the term |∇u| 2 models short-ranged interactions. It is, however, interesting to note that such a term is obtained by truncation of higher-order ones (see [10]); it can also be seen as a first-order approximation of a nonlocal term accounting for long-ranged interactions (see [14,15]).
In the late 1960 s, several authors proposed a heat conduction theory based on two temperatures (see [11,24]). More precisely, one now considers the conductive temperature T and the thermodynamic temperature α. In particular, for simple materials, these two temperatures are shown to coincide. However, for non-simple materials, they differ and are related as follows: In that case, the free energy reads, in terms of the (relative) thermodynamic temperature α, and (8) yields, in view of (12), the following evolution equation for the order parameter Furthermore, the enthalpy now reads which yields, owing to (9), the energy equation Finally, the heat flux is given, in the type III theory with two temperatures, by (see [16,22]) is the conductive thermal displacement. Noting that T = ∂θ ∂t , we finally deduce from (13), the following variant of the Caginalp phase-field system (18) ∂u ∂t G. Caginalp and E. Esenturk recently proposed in [7] (see also [12]) higher-order phase field models in order to account for anisotropic interfaces (see also [17] for other approaches which, however, do not provide an explicit way to compute the anisotropy). More precisely, these authors proposed the following modified (total) free energy: and, for β = (0, 0, 0), (we agree that D (0,0,0) w = w).
As far as the conserved case is concerned, the above generalized free energy yields, proceeding as above, the following evolution equation for the order parameter u: In particular, for k = 1 (anisotropic conserved Caginalp phase-field system), we have an equation of the form and, for k = 2 (fourth-order anisotropic conserved Caginalp phase-field system), we have an equation of the form L. Cherfils, A. Miranville and S. Peng studied in [18] the corresponding higherorder isotropic equation (without the coupling with the temperature), namely, the equation endowed with the Dirichlet/Navier boundary conditions Our aim in this paper is to study the perturbed model of the higher-order anisotropic equation (21), coupled with the temperature equation (19). The perturbation is expressed by the presence of the term (−∆) ∂ 2 u ∂t 2 + ∂u ∂t in the equation (21). Then, we have an hyperbolic relaxation of the viscous Cahn-Hilliard equation. When = 0, the system has already been studied without the effects of anisotropy (see [22]), and with effects of anisotropy (see [23]). In particular, we obtain the existence and uniqueness of solutions.

Setting of the problem
We consider the following initial and boundary value problem, for k ∈ N * (k 2): We assume that and > 0 ( ≤ 0 , 0 > 0) is a relaxation parameter or the perturbation parameter.
We consider the regular potential f (s) = which sastifies the following properties: [19]) We can more generally consider a polynomial potential of the form or even a regular potential f having a polynomial growth of the form a 2p+2 r 2p+2 , a 2p+2 > 0, p ≥ 1, at infinity.
We introduce the elliptic operator A k defined by where H −k (Ω) is the topological dual of H k 0 (Ω). Furthermore, ((., .)) denotes the usual L 2 -scalar product, with associated norm . ; more generally, we denote by . X the norm on the Banach space X. We can note that is bilinear, symmetric, continuous and coercive, so that It then follows from elliptic regularity results for linear elliptic operators of order 2k (see [1] and [2]) that A k is a strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain We further note that D(A We finally note that (see, e.g., [5] Similarly, we can define the linear operator A k = −∆A k , where which is a strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain Besides, A k . (resp., A note that, as −∆ and A k commute, then the same holds for (−∆) −1 and A k , so We have the (see [20,21]) Lemma 2.1. The operatorÃ k is a strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain Besides, Ã k . (resp., Ã Proof. (see [20,21]) We first note thatÃ k clearly is linear and unbounded. Then, since (−∆) −1 and A k commute, it easily follows thatÃ k is selfadjoint. Next, the domain ofÃ k is defined by where −∆f ∈ H 2 (Ω), it follows from the elliptic regularity results of [1,2] that v ∈ H 2k−2 (Ω), so that D(Ã k ) = H 2k−2 (Ω) ∩ H k−1 0 (Ω). Noting then thatÃ −1 k maps L 2 (Ω) onto H 2k−2 (Ω) and recalling that k ≥ 2, we deduce thatÃ k has compact inverse.
We now note that, considering the spectral properties of −∆ and A k (see, e.g., [25]) and recalling that these two operators commute, −∆ and A k have a spectral basis formed of common eigenvectors. This yields that, ∀s 1 , s 2 ∈ R, (−∆) s1 and A s2 k commute. Having this, we see thatÃ (Ω), and, for Finally, as far as the equivalences of norms are concerned, we can note that, for instance, the norm Ã k . is equivalent to the norm (−∆) − 1 2 . H k (Ω) and, thus, to the norm (−∆) k−1 2 . . Having this, we rewrite (24) as Throughout the paper, the same letters c, c and c denote (generally positive and independent of ) constants which may vary from line to line. Similarly, the same letter Q denotes (positive and independent of ) monotone increasing (with respect to each argument) and continuous functions which may vary from line to line.

A priori estimates
We multiply (34) by (−∆) −1 ∂u ∂t and (25) by ∂θ ∂t − ∆ ∂θ ∂t , sum the two resulting equalities and integrate over Ω and by parts. This gives d dt where is not necessarily nonnegative). We can note that, owing to the interpolation inequality . . , m − 1}, m ≥ 2, there holds (see [20,21]) This yields, employing (31), noting that, owing to Youngs inequality, , ∀ε > 0. We obtain a differential equality of the form We then multiply (34) by (−∆) −1 u and obtain, Since, owing to (31), we have Multiplying (25) by −∆θ and integrating over Ω, we have We sum (41), δ 1 (44) and δ 2 (45), where δ i > 0, i = 1, 2, is chosen small enough, so that ∂u It follows from (47) − (48) and Gronwalls lemma that where c, c and c are independent of and r > 0 given. We multiply (34) by (−∆) −1 ∂ 2 u ∂t 2 and integrate over Ω to obtain Owing to the interpolation inequality (37), we find where c is independent of . It follows from the continuity of f and the continuous embedding H k (Ω) ⊂ C(Ω) for k ≥ 2 that Thanks to (49), we have d dt where c, c and Q are independent of .