Well-posedness for modified higher-order anisotropic Cahn

Abstract

Our aim in this paper is to study a modified higher-order (in space) phase field crystal model taking anisotropy into account. In particular, we deduce a priori estimates to prove well-posedness results and the dissipativity of the semigroup.

Keywords

Modified phase field crystal equation higher-order models anisotropy a priori estimates dissipative semigroup

1. Introduction

We study in this paper the modified higher-order anisotropic Cahn–Hilliard equations which read, for $k \in N$ , $k ⩾ 2$ , $x \in Ω \subset R^{d}$ ( $d = 1, 2, 3$ ), $\begin{matrix} (1.1) & β \partial_{t t} u + \partial_{t} u - Δ \sum_{i = 1}^{k} {(- 1)}^{i} \sum_{| α | = i} a_{α} D^{2 α} u - Δ f (u) = 0, \end{matrix}$ where Ω is assumed to be a bounded and regular domain occupied by the system with boundary Γ; u is the order parameter, for instance, the density of atoms; f is the derivative of a double-well potential F, and $β > 0$ is a relaxation time. The Cahn–Hilliard equation ([5,6,39]), which describes important features of binary alloys in phase separation processes, such as spinodal decomposition and coarsening, $\begin{matrix} (1.2) & \frac{\partial u}{\partial t} + Δ^{2} u - Δ f (u) = 0, \end{matrix}$ is considered as an $H^{- 1}$ -gradient flow of the so-called Ginzburg–Landau (see [23,24]) free energy, $\begin{matrix} (1.3) & Ψ_{GL} = \int_{Ω} (\frac{1}{2} | \nabla u |^{2} + F (u)) d x . \end{matrix}$ In (1.3), the term $| \nabla u |^{2}$ models short-ranged interactions. It is however interesting to note that such a term is obtained by truncation of higher-order ones (see [6]); it can also be seen as a first-order approximation of a nonlocal term accounting for long-ranged interactions (see [21] and [22]).

Concerning the inertial term (namely, the hyperbolic term: $β \partial_{t t} u$ ), in fact, a hyperbolic relaxation of the one-dimensional Cahn–Hilliard equation has been proposed in [15], in order to model rapid spinodal decompositions in a binary alloy. Furthermore, S. Gatti et al. provided in [20] a detailed analysis of the longterm properties of the solutions for a hyperbolic relaxation of the one-dimensional Cahn–Hilliard equation in the singular limit when the relaxation parameter goes to zero.

We notice that, when $k = 1$ , without consideration of anisotropy, the equation becomes $\begin{matrix} (1.4) & β \partial_{t t} u + \partial_{t} u - Δ [Δ^{2} u + 2 Δ u + f (u)] = 0, \end{matrix}$ which is a so-called modified phase field crystal equation (abbr., MPFC) and was proposed in [43] by P. Stefanovic et al. (see also in [44]). The MPFC equation incorporates both fast elastic relaxation and slower mass diffusion which has achieved to distinguish between the elastic relaxation and diffusion time scales. In [27] and [28], M. Grasselli and H. Wu proved the well-posedness and established the existence of an exponential attractor for the MPFC equation (1.4) endowed with periodic boundary conditions. Additionally, in [26], M. Grasselli and M. Pierre proposed a space semi-discrete and a fully discrete finite element scheme for the MPFC model and established their convergence to equilibrium both theoretically and numerically. We refer the readers to [48,49] for more numerical methods to solve the MPFC model and [17–19,29,51] for the theoretical and numerical study on the phase field model without a relaxation.

We further studied (1.1) in [52], in which we assumed the well-posedness of solutions, the numerical approximations for a hyperbolic relaxation of the higher-order anisotropic generalized Cahn–Hilliard models which was inspired by the work of M. Grasselli and M. Pierre in [26], employing the finite element and spectral methods. In this article, we will focus on the well-posedness of the hyperbolic equations and their solutions, more precisely, the existence, uniqueness and regularity.

Considering the anisotropic phenomenon, recently, G. Caginalp and E. Esenturk proposed in [4] (see also [8]) higher-order phase-field models in order to account for anisotropic interfaces (see also [7,30,42,45,47] and [50] for other approaches which, however, do not provide an explicit way to compute the anisotropy). More precisely, these authors proposed the following modified free energy, in which we omit the temperature: $\begin{matrix} (1.5) & Ψ_{HOGL} = \int_{Ω} (\frac{1}{2} \sum_{i = 1}^{k} \sum_{| α | = i} a_{α} {| D^{α} u |}^{2} + F (u)) d x, k \in N, \end{matrix}$ where, for $α = (k_{1}, k_{2}, k_{3}) \in {(N \cup {0})}^{3}$ , $\begin{matrix} | α | = k_{1} + k_{2} + k_{3} \end{matrix}$ and, for $α \neq (0, 0, 0)$ , $\begin{matrix} D^{α} = \frac{\partial^{| α |}}{\partial x_{1}^{k_{1}} \partial x_{2}^{k_{2}} \partial x_{3}^{k_{3}}} \end{matrix}$ (we agree that $D^{(0, 0, 0)} v = v$ ).

The corresponding higher-order anisotropic Cahn–Hilliard equation (it also corresponds to the case when $β = 0$ in equation (1.1)) then reads $\begin{matrix} (1.6) & \frac{\partial u}{\partial t} - Δ \sum_{i = 1}^{k} {(- 1)}^{i} \sum_{| α | = i} a_{α} D^{2 α} u - Δ f (u) = 0 . \end{matrix}$

We studied in [9] the corresponding higher-order isotropic models, namely, $\begin{matrix} (1.7) & \frac{\partial u}{\partial t} - Δ P (- Δ) u - Δ f (u) = 0, \end{matrix}$ where $\begin{matrix} P (s) = \sum_{i = 1}^{k} a_{i} s^{i}, a_{k} > 0, k ⩾ 1, s \in R . \end{matrix}$ We also refer the readers to [31–35] (see also [40,41]) the study on higher-order Cahn–Hilliard type equations.The anisotropic model (1.6) was analysed in [11], and the generalized anisotropic model which contains a continuous function $g (x, u)$ was studied in [12], in both of which, numerical simulations were performed to illustrate the anisotropic effects.

Taking the nonlinearity into account, the authors in [10] have considered (1.7) (see also in [13,14,16,25,34,36,38] for other equations) endowed with a thermodynamically relevant potential F which is associated to the mean-field model: $\begin{matrix} (1.8) & \begin{matrix} F (s) & = \frac{θ_{c}}{2} (1 - s^{2}) + \frac{θ}{2} ((1 - s) ln (\frac{1 - s}{2}) + (1 + s) ln (\frac{1 + s}{2})), \\ s \in (- 1, 1), 0 < θ < θ_{c}, \end{matrix} \end{matrix}$ i.e., $\begin{matrix} (1.9) & f (s) = - θ_{c} s + \frac{θ}{2} ln \frac{1 + s}{1 - s}, \end{matrix}$ where θ and $θ_{c}$ are proportional to the absolute temperature and a critical temperature, respectively. However, in this paper (and also in [26–28] and [52], etc.), we consider a polynomial type nonlinearity, for example, $\begin{matrix} (1.10) & F (s) = \frac{1}{4} {(s^{2} - 1)}^{2}, \end{matrix}$ i.e., $\begin{matrix} (1.11) & f (s) = F^{'} (s) = s^{3} - s . \end{matrix}$

Our interests in this paper is to study the well-posedness of a hyperbolic relaxation of the higher-order anisotropic Cahn–Hilliard equation. In Section 2, detailed notations on operators, spaces and parameters are provided, then in Section 3, the exact problem is addressed and after which, in Section 4, a priori estimates are derived in detail. In Section 5, the existence and uniqueness of weak solution are given and proved, as well as the dissipativity of semigroup.

2. Prelimilary

For a real Banach space X, we set $‖ \cdot ‖_{X}$ as the norm of X and $⟨ \cdot, \cdot ⟩$ as the duality product between X and the topological dual of X. Generally, $‖ \cdot ‖$ denotes the $L^{2}$ -norm and $‖ \cdot ‖_{- 1} = ‖ {(- Δ)}^{\frac{1}{2}} \cdot ‖$ , where ${(- Δ)}^{- 1}$ denotes the inverse minus Laplace operator associated with Dirichlet boundary conditions. Moreover, for $v, w \in H^{- 1} (Ω)$ , ${((v, w))}_{- 1} = (({(- Δ)}^{- \frac{1}{2}} v, {(- Δ)}^{- \frac{1}{2}} w))$ , where $((\cdot, \cdot))$ denotes the usual $L^{2}$ -scalar product.

Assuming that $k ⩾ 2$ ( $k \in N$ ), and $a_{α} > 0$ ( $| α | = k$ ), we define the elliptic operator $A_{k}$ as $\begin{matrix} (2.1) & {⟨ A_{k} v, w ⟩}_{H^{- k} (Ω), H_{0}^{k} (Ω)} = \sum_{| α | = k} a_{α} ((D^{α} v, D^{α} w)), v, w \in H_{0}^{k} (Ω), \end{matrix}$ where $H^{- k} (Ω)$ is the topological dual of $H_{0}^{k} (Ω)$ . For arbitrary functions $v, w \in H_{0}^{k} (Ω)$ , we note that $\begin{matrix} (v, w) \mapsto \sum_{| α | = k} a_{α} ((D^{α} v, D^{α} w)) \end{matrix}$ is bilinear, symmetric, continuous and coercive, so that $\begin{matrix} (2.2) & A_{k} : H_{0}^{k} (Ω) \mapsto H^{- k} (Ω) \end{matrix}$ is well defined. It then follows from the elliptic regularity results for linear elliptic operators of order $2 k$ (see [1,2] and [3]) that $A_{k}$ is a strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain $D (A_{k}) = H^{2 k} (Ω) \cap H_{0}^{k} (Ω)$ , where, for $v \in D (A_{k})$ , $\begin{matrix} A_{k} v = {(- 1)}^{k} \sum_{| α | = k} a_{α} D^{2 α} v . \end{matrix}$ We further note that $D (A_{k}^{\frac{1}{2}}) = H_{0}^{k} (Ω)$ and, for $(v, w) \in D {(A_{k}^{\frac{1}{2}})}^{2}$ , $\begin{matrix} ((A_{k}^{\frac{1}{2}} v, A_{k}^{\frac{1}{2}} w)) = \sum_{| α | = k} a_{α} ((D^{α} v, D^{α} w)) . \end{matrix}$ We then note that (see, e.g., [46]) $‖ A_{k} \cdot ‖$ (resp., $‖ A_{k}^{\frac{1}{2}} \cdot ‖$ ) is equivalent to the usual $H^{2 k}$ -norm (resp., $H^{k}$ -norm) on $D (A_{k})$ (resp., $D (A_{k}^{\frac{1}{2}})$ ).

Similarly, we can define the linear operator ${\bar{A}}_{k} = - Δ A_{k}$ , $\begin{matrix} {\bar{A}}_{k} : H_{0}^{k + 1} (Ω) \mapsto H^{- k - 1} (Ω) \end{matrix}$ which is a strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain $D ({\bar{A}}_{k}) = H^{2 k + 2} (Ω) \cap H_{0}^{k + 1} (Ω)$ , where, for $v \in D ({\bar{A}}_{k})$ , $\begin{matrix} {\bar{A}}_{k} v = {(- 1)}^{k + 1} Δ \sum_{| α | = k} a_{α} D^{2 α} v . \end{matrix}$ Additionally, $D ({\bar{A}}_{k}^{\frac{1}{2}}) = H_{0}^{k + 1} (Ω)$ and, for $(v, w) \in D {({\bar{A}}_{k}^{\frac{1}{2}})}^{2}$ , $\begin{matrix} (({\bar{A}}_{k}^{\frac{1}{2}} v, {\bar{A}}_{k}^{\frac{1}{2}} w)) = \sum_{| α | = k} a_{α} ((\nabla D^{α} v, \nabla D^{α} w)) . \end{matrix}$ Besides, $‖ {\bar{A}}_{k} \cdot ‖$ (resp., $‖ {\bar{A}}_{k}^{\frac{1}{2}} \cdot ‖$ ) is equivalent to the usual $H^{2 k + 2}$ -norm (resp., $H^{k + 1}$ -norm) on $D ({\bar{A}}_{k})$ (resp., $D ({\bar{A}}_{k}^{\frac{1}{2}})$ ).

We finally define the linear operator ${\tilde{A}}_{k} = {(- Δ)}^{- 1} A_{k}$ , $\begin{matrix} {\tilde{A}}_{k} : H_{0}^{k - 1} (Ω) \mapsto H^{- k + 1} (Ω); \end{matrix}$ as $- Δ$ and $A_{k}$ commute, we can note that ${(- Δ)}^{- 1}$ and $A_{k}$ commute, so that ${\tilde{A}}_{k} = A_{k} {(- Δ)}^{- 1}$ . Furthermore, according to [11], ${\tilde{A}}_{k}$ is strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain $D ({\tilde{A}}_{k}) = H^{2 k - 2} (Ω) \cap H_{0}^{k - 1} (Ω)$ , where, for $v \in D ({\tilde{A}}_{k})$ , $\begin{matrix} {\tilde{A}}_{k} v = {(- 1)}^{k} \sum_{| α | = k} a_{α} D^{2 α} {(- Δ)}^{- 1} v . \end{matrix}$ Furthermore, $D ({\tilde{A}}_{k}^{\frac{1}{2}}) = H_{0}^{k - 1} (Ω)$ and, for $(v, w) \in D {({\tilde{A}}_{k}^{\frac{1}{2}})}^{2}$ , $\begin{matrix} (({\tilde{A}}_{k}^{\frac{1}{2}} v, {\tilde{A}}_{k}^{\frac{1}{2}} w)) = \sum_{| α | = k} a_{α} ((D^{α} {(- Δ)}^{- \frac{1}{2}} v, D^{α} {(- Δ)}^{- \frac{1}{2}} w)) . \end{matrix}$ Besides, $‖ {\tilde{A}}_{k} \cdot ‖$ (resp., $‖ {\tilde{A}}_{k}^{\frac{1}{2}} \cdot ‖$ ) is equivalent to the usual $H^{2 k - 2}$ -norm (resp., $H^{k - 1}$ -norm) on $D ({\tilde{A}}_{k})$ (resp., $D ({\tilde{A}}_{k}^{\frac{1}{2}})$ ).

In what follows, the same letters c, $c^{'}$ , $c^{″}$ denote (generally positive) constants which may vary from line to line. Similarly, the same letter Q denotes (positive) monotone increasing and continuous function which may vary from line to line. Furthermore, the boundary conditions are Dirichlet boundary condition for a sufficiently regular boundary.

3. Setting of the problem

We consider the following modified higher-order anisotropic phase field crystal equation, for $k \in N$ , $k ⩾ 2$ , $x \in Ω \subset R^{d}$ , $t \in (0, + \infty)$ , $\begin{array}{l} (3.1) & β u_{t t} + u_{t} - Δ \sum_{i = 1}^{k} {(- 1)}^{i} \sum_{| α | = i} a_{α} D^{2 α} u - Δ f (u) = 0, \\ (3.2) & u |_{t = 0} = u_{0} (x), u_{t} |_{t = 0} = v_{0} (x), \end{array}$ where the nonlinearity $f (s)$ is the derivative of a double-well potential F. We consider $F (s)$ in this article as an approximation to the thermodynamically relevant potential which is a logarithmic function (see [6,13]). Typically, $\begin{matrix} (3.3) & f (s) = F^{'} (s) = s^{3} - s . \end{matrix}$ More generally, we assume that $\begin{array}{l} (3.4) & f \in C^{2} (R), f (0) = 0, \\ (3.5) & f^{'} ⩾ - c_{0}, c_{0} ⩾ 0, \\ (3.6) & f (s) s ⩾ c_{1} F (s) - c_{2} ⩾ - c_{3}, c_{1} > 0, c_{2}, c_{3} ⩾ 0, s \in R, \\ (3.7) & F (s) ⩾ c_{4} s^{4} - c_{5}, c_{4} > 0, c_{5} ⩾ 0, s \in R . \end{array}$ It can be verified that f defined by (3.3) satisfies all the assumptions from (3.4) to (3.7).

4. A priori dissipative estimates

We rewrite problem (3.1) as $\begin{matrix} (4.1) & β u_{t t} + u_{t} - Δ (A_{k} u + B_{k} u + f (u)) = 0, \end{matrix}$ where $\begin{matrix} (4.2) & B_{k} u : = \sum_{i = 1}^{k - 1} {(- 1)}^{i} \sum_{| α | = i} a_{α} D^{2 α} u . \end{matrix}$ Then we have

Lemma 4.1.
Suppose $(u, u_{t})$ is a regular solution to problem ( 3.1 )–( 3.2 ). Then the following dissipative estimate holds $\begin{matrix} (4.3) & ‖ u ‖_{H^{k} (Ω)}^{2} + ‖ u_{t} ‖_{- 1}^{2} + \int_{0}^{t} e^{- c^{'} (t - s)} {‖ u_{t} (s) ‖}_{- 1}^{2} d s ⩽ e^{- c^{'} t} Q (‖ u_{0} ‖_{H^{k} (Ω)}, ‖ v_{0} ‖_{- 1}) + c^{″} . \end{matrix}$ Moreover, if we further assume that f is of class $C^{k + 1}$ , the following dissipative estimate holds $\begin{matrix} (4.4) & ‖ u_{t} ‖_{H^{k - 1} (Ω)}^{2} + ‖ u ‖_{H^{2 k} (Ω)}^{2} ⩽ c e^{- c^{'} t} Q (‖ u_{0} ‖_{H^{2 k} (Ω)}, ‖ v_{0} ‖_{H^{k - 1} (Ω)}) + c^{″} . \end{matrix}$
Proof.
Multiplying (4.1) by ${(- Δ)}^{- 1} u_{t}$ , and integrating over Ω, we obtain $\begin{matrix} (4.5) & \frac{1}{2} \frac{d}{d t} (β ‖ u_{t} ‖_{- 1}^{2} + {‖ A_{k}^{\frac{1}{2}} u ‖}^{2} + B_{k}^{\frac{1}{2}} [u] + 2 \int_{Ω} F (u) d x) + ‖ u_{t} ‖_{- 1}^{2} = 0, \end{matrix}$ where $\begin{matrix} (4.6) & B_{k}^{\frac{1}{2}} [u] : = \sum_{i = 1}^{k - 1} \sum_{| α | = i} a_{α} {‖ D^{α} u ‖}^{2} . \end{matrix}$ We can note that, owing to the interpolation inequality $\begin{matrix} (4.7) & \begin{matrix} ‖ v ‖_{H^{i} (Ω)} ⩽ c (i) ‖ v ‖_{H^{m} (Ω)}^{\frac{i}{m}} ‖ v ‖^{1 - \frac{i}{m}}, \\ v \in H^{m} (Ω), i \in {1, \dots, m - 1}, m \in N, m ⩾ 2, \end{matrix} \end{matrix}$ there holds $\begin{matrix} (4.8) & | B_{k}^{\frac{1}{2}} [u] | ⩽ \frac{1}{2} {‖ A_{k}^{\frac{1}{2}} u ‖}^{2} + c ‖ u ‖^{2} . \end{matrix}$

Multiplying (4.1) by ${(- Δ)}^{- 1} u$ , we obtain $\begin{matrix} (4.9) & \frac{d}{d t} (β {((u_{t}, u))}_{- 1} + \frac{1}{2} ‖ u ‖_{- 1}^{2}) - β ‖ u_{t} ‖_{- 1}^{2} + {‖ A_{k}^{\frac{1}{2}} u ‖}^{2} + B_{k}^{\frac{1}{2}} [u] + ((f (u), u)) = 0 . \end{matrix}$ Multiplying (4.9) by a sufficiently small coefficient η (in fact, normally, we take $η \in (0, \frac{1}{2 β})$ ), then summing the resulting equation and (4.5), we get the following differential inequality $\begin{matrix} (4.10) & \frac{d}{d t} E_{1} (t) + χ_{1} (t) = 0, \end{matrix}$ where $\begin{matrix} (4.11) & \begin{matrix} E_{1} (t) = η β {((u_{t}, u))}_{- 1} + \frac{1}{2} E_{1} (u) + \frac{1}{2} β ‖ u_{t} ‖_{- 1}^{2} + \frac{1}{2} η ‖ u ‖_{- 1}^{2}, \\ χ_{1} (t) = (1 - η β) ‖ u_{t} ‖_{- 1}^{2} + η ({‖ A_{k}^{\frac{1}{2}} u ‖}^{2} + B_{k}^{\frac{1}{2}} [u] + ((f (u), u))), \\ E_{1} (u) = {‖ A_{k}^{\frac{1}{2}} u ‖}^{2} + B_{k}^{\frac{1}{2}} [u] + 2 \int_{Ω} F (u) d x, \end{matrix} \end{matrix}$ and $E_{1}$ satisfies, owing to the Young inequality $\begin{matrix} (4.12) & c ‖ u ‖^{2} ⩽ \frac{c_{4}}{2} ‖ u ‖_{L^{4}}^{4} + c^{'}, \end{matrix}$ (3.7) and (4.8), that $\begin{matrix} (4.13) & E_{1} ⩾ c (‖ u ‖_{H^{k} (Ω)}^{2} + \int_{Ω} F (u) d x) - c^{'}, c > 0 . \end{matrix}$ It follows from the Cauchy–Schwarz inequality that $\begin{matrix} (4.14) & η β | {((u_{t}, u))}_{- 1} | ⩽ \frac{β}{4} ‖ u_{t} ‖_{- 1}^{2} + η^{2} β ‖ u ‖_{- 1}^{2} . \end{matrix}$ We also note that, according to an embedding theorem, $\begin{matrix} (4.15) & ‖ u ‖_{- 1}^{2} ⩽ c ‖ u ‖_{H^{k} (Ω)}^{2} . \end{matrix}$ Thus, for $η \in (0, \frac{1}{2 β})$ small enough, we have $\begin{matrix} (4.16) & Q (‖ u_{t} ‖_{- 1}, E_{1} (u)) ⩾ E_{1} (t) ⩾ \frac{1}{2} E_{1} (t) + \frac{β}{4} ‖ u_{t} ‖_{- 1}^{2} + (\frac{η}{2} - η^{2} β) ‖ u ‖_{- 1}^{2}, \end{matrix}$ where, $Q (w, z)$ is a positive monotone increasing function with respect to w and z. Furthermore, owing to the assumption (3.6), it is obvious that $\begin{matrix} (4.17) & ((f (u), u)) ⩾ c \int_{Ω} F (u) d x - c^{'} | Ω |, \end{matrix}$ with which, according to the definition of $E_{1} (t)$ and $χ_{1} (t)$ , for a sufficient small coefficient $c > 0$ , we can obtain $\begin{matrix} (4.18) & \frac{1}{2} χ_{1} (t) ⩾ c E_{1} (t) + c^{'} ‖ u_{t} ‖_{- 1}^{2} - c^{″} . \end{matrix}$

As a result, combining the above inequalities (4.10)–(4.18), we can rewrite (4.10) as $\begin{matrix} (4.19) & \frac{d}{d t} E_{1} (t) + c (E_{1} (t) + ‖ u_{t} ‖_{- 1}^{2}) ⩽ c^{'}, \end{matrix}$ where c depends on η and $E_{1}$ satisfies $\begin{matrix} (4.20) & E_{1} ⩾ c (‖ u ‖_{H^{k} (Ω)}^{2} + ‖ u_{t} ‖_{- 1}^{2} + \int_{Ω} F (u) d x) - c^{'}, c > 0, \end{matrix}$ where c depends on β. We note that, with the continuity of F and the interpolation inequality, we have $| \int_{Ω} F (u) d x | ⩽ Q (‖ u ‖_{H^{k} (Ω)})$ . Combining (4.19) and (4.20), and Gronwall’s lemma, we obtain $\begin{matrix} (4.21) & ‖ u_{t} ‖_{- 1}^{2} + {‖ u (t) ‖}_{H^{k} (Ω)}^{2} ⩽ c e^{- c^{'} t} Q (‖ u_{0} ‖_{H^{k} (Ω)}, ‖ v_{0} ‖_{- 1}) + c^{″}, c^{'} > 0, t ⩾ 0, \end{matrix}$ and $\begin{matrix} (4.22) & \int_{t}^{t + r} {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} d s ⩽ c e^{- c^{'} t} Q (‖ u_{0} ‖_{H^{k} (Ω)}, ‖ v_{0} ‖_{- 1}) + c^{″}, r > 0 given . \end{matrix}$ In addition, applying again Gronwall’s lemma on (4.19), we obtain, for $c^{'} > 0$ , $t ⩾ 0$ , $\begin{matrix} (4.23) & E_{1} (t) + \int_{0}^{t} e^{- c^{'} (t - s)} {‖ u_{t} (s) ‖}_{- 1}^{2} d s ⩽ E_{1} (0) e^{- c^{'} t} + c^{″}, \end{matrix}$ more precisely, with the definition of $E_{1} (t)$ , there holds that $\begin{matrix} ‖ u ‖_{H^{k} (Ω)}^{2} + ‖ u_{t} ‖_{- 1}^{2} + \int_{0}^{t} e^{- c^{'} (t - s)} {‖ u_{t} (s) ‖}_{- 1}^{2} d s ⩽ e^{- c^{'} t} Q (‖ u_{0} ‖_{H^{k} (Ω)}, ‖ v_{0} ‖_{- 1}) + c^{″} . \end{matrix}$

We then multiply (4.1) by $u_{t}$ and integrate over Ω. It gives: $\begin{matrix} \frac{1}{2} \frac{d}{d t} (β ‖ u_{t} ‖^{2} + {‖ {\bar{A}}_{k}^{\frac{1}{2}} u ‖}^{2} + {\bar{B}}_{k}^{\frac{1}{2}} [u]) + ‖ u_{t} ‖^{2} ⩽ ‖ Δ f (u) ‖ ‖ u_{t} ‖, \end{matrix}$ with $\begin{matrix} {\bar{B}}_{k}^{\frac{1}{2}} [u] = \sum_{i = 1}^{k - 1} \sum_{| α | = i} a_{α} {‖ \nabla D^{α} u ‖}^{2} and | {\bar{B}}_{k}^{\frac{1}{2}} [u] | ⩽ c ‖ u ‖_{H^{k} (Ω)}^{2} . \end{matrix}$ Noting that f is of class $C^{2}$ , so that for $k ⩾ 2$ , $\begin{matrix} (4.24) & ‖ Δ f (u) ‖ ⩽ Q (‖ u ‖_{H^{k} (Ω)}), \end{matrix}$ we get from (4.21) that, for $c^{'} > 0$ , $\begin{matrix} (4.25) & \frac{d}{d t} (β ‖ u_{t} ‖^{2} + {‖ {\bar{A}}_{k}^{\frac{1}{2}} u ‖}^{2} + {\bar{B}}_{k}^{\frac{1}{2}} [u]) + ‖ u_{t} ‖^{2} ⩽ e^{- c^{'} t} Q (‖ u_{0} ‖_{H^{k} (Ω)}, ‖ v_{0} ‖_{- 1}) + c^{″} . \end{matrix}$

We then test (4.1) by u, and integrate over Ω, owing to (4.21) and (4.24), applying Cauchy–Schwarz inequality, to get $\begin{matrix} (4.26) & \begin{matrix} \frac{d}{d t} (β ((u_{t}, u)) + \frac{1}{2} ‖ u ‖^{2}) - β ‖ u_{t} ‖^{2} + {‖ {\bar{A}}_{k}^{\frac{1}{2}} u ‖}^{2} + {\bar{B}}_{k}^{\frac{1}{2}} [u] \\ ⩽ e^{- c^{'} t} Q (‖ u_{0} ‖_{H^{k} (Ω)}, ‖ v_{0} ‖_{- 1}) + c^{″} + ‖ u ‖^{2} . \end{matrix} \end{matrix}$

Multiplying (4.26) by a sufficient small positive coefficient $η^{'} \in (0, \frac{1}{2 β})$ , and summing the resulting inequality with (4.19) and (4.25), it leads to $\begin{matrix} (4.27) & \begin{matrix} \frac{d}{d t} Y_{1} (t) + D_{1} (t) & ⩽ e^{- c^{'} t} Q (‖ u_{0} ‖_{H^{k} (Ω)}, ‖ v_{0} ‖_{- 1}) + c^{″} + ‖ u ‖^{2} \\ ⩽ e^{- c^{'} t} Q (‖ u_{0} ‖_{H^{k} (Ω)}, ‖ v_{0} ‖_{- 1}) + c^{″}, \end{matrix} \end{matrix}$ where $\begin{matrix} (4.28) & \begin{matrix} Y_{1} (t) = E_{1} + \frac{β}{2} ‖ u_{t} ‖^{2} + \frac{1}{2} {‖ {\bar{A}}_{k}^{\frac{1}{2}} u ‖}^{2} + \frac{1}{2} {\bar{B}}_{k}^{\frac{1}{2}} [u] + η^{'} β ((u_{t}, u)) + \frac{η^{'}}{2} ‖ u ‖^{2}, \\ D_{1} (t) = η^{'} {‖ {\bar{A}}_{k}^{\frac{1}{2}} u ‖}^{2} + η^{'} {\bar{B}}_{k}^{\frac{1}{2}} [u] + (1 - η^{'} β) ‖ u_{t} ‖^{2} + c E_{1} + c ‖ u_{t} ‖_{- 1}^{2} . \end{matrix} \end{matrix}$

Proceeding as above, according to (3.7), the interpolation and Young inequalities (4.7) and (4.12), it follows that, $\begin{matrix} (4.29) & Q (‖ u_{t} ‖, ‖ u ‖_{H^{k + 1} (Ω)}) ⩾ Y_{1} (t) ⩾ \frac{β}{4} ‖ u_{t} ‖^{2} + \frac{1}{4} {‖ {\bar{A}}_{k}^{1 / 2} u ‖}^{2} - c, \end{matrix}$ with which, therefore, we can deduce from the expressions of $Y_{1}$ and $D_{1}$ that there holds $\begin{matrix} (4.30) & \frac{d}{d t} Y_{1} (t) + c_{0} Y_{1} (t) ⩽ c e^{- c^{'} t} Q (‖ u_{0} ‖_{H^{k} (Ω)}, ‖ v_{0} ‖_{- 1}) + c^{″} . \end{matrix}$ Owing to Gronwall’s lemma and (4.29), we thus obtain $\begin{matrix} (4.31) & \begin{matrix} Y_{1} (t) & ⩽ Y_{1} (0) e^{- c_{0} t} + \int_{0}^{t} e^{- c_{0} (t - s)} (c e^{- c^{'} s} Q (‖ u_{0} ‖_{H^{k} (Ω)}, ‖ v_{0} ‖_{- 1}) + c^{″}) d s \\ ⩽ e^{- c^{'} t} Q (‖ u_{0} ‖_{H^{k + 1} (Ω)}, ‖ v_{0} ‖) + c^{″}, \end{matrix} \end{matrix}$ so that $\begin{matrix} (4.32) & ‖ u_{t} ‖^{2} + ‖ u ‖_{H^{k + 1} (Ω)}^{2} ⩽ e^{- c^{'} t} Q (‖ u_{0} ‖_{H^{k + 1} (Ω)}, ‖ v_{0} ‖) + c^{″} . \end{matrix}$

Multiplying then (4.1) by ${\tilde{A}}_{k} u$ , and integrating over Ω, owing to the interpolation inequality (4.7), we have $\begin{matrix} \frac{d}{d t} (β (({\tilde{A}}_{k}^{\frac{1}{2}} u_{t}, {\tilde{A}}_{k}^{\frac{1}{2}} u)) + \frac{1}{2} {‖ {\tilde{A}}_{k}^{\frac{1}{2}} u ‖}^{2}) - β {‖ {\tilde{A}}_{k}^{\frac{1}{2}} u_{t} ‖}^{2} + ‖ A_{k} u ‖^{2} + ((B_{k} u, A_{k} u)) ⩽ | ((f (u), A_{k} u)) | . \end{matrix}$ It then follows from the continuity of f and F, and the continuous embedding $H^{k} \subset C (\overline{Ω})$ for $k ⩾ 2$ , and (4.21), that $\begin{matrix} (4.33) & \begin{matrix} \frac{d}{d t} (β (({\tilde{A}}_{k}^{\frac{1}{2}} u_{t}, {\tilde{A}}_{k}^{\frac{1}{2}} u)) + \frac{1}{2} {‖ {\tilde{A}}_{k}^{\frac{1}{2}} u ‖}^{2}) + c ‖ A_{k} u ‖^{2} - β {‖ {\tilde{A}}_{k}^{\frac{1}{2}} u_{t} ‖}^{2} + ((B_{k} u, A_{k} u)) \\ ⩽ e^{- c^{'} t} Q (‖ u_{0} ‖_{H^{k} (Ω)}, ‖ v_{0} ‖_{- 1}) + c^{″} . \end{matrix} \end{matrix}$ Multiplying (4.1) by ${\tilde{A}}_{k} u_{t}$ , integrating over Ω and by parts, and noting that the two operators $A_{k}$ and $B_{k}$ commute, we obtain $\begin{matrix} (4.34) & \frac{1}{2} \frac{d}{d t} (β {‖ {\tilde{A}}_{k}^{\frac{1}{2}} u_{t} ‖}^{2} + ‖ A_{k} u ‖^{2} + ((B_{k} u, A_{k} u))) + {‖ {\tilde{A}}_{k}^{\frac{1}{2}} u_{t} ‖}^{2} ⩽ | (({\bar{A}}_{k}^{\frac{1}{2}} f (u), {\tilde{A}}_{k}^{\frac{1}{2}} u_{t})) | . \end{matrix}$

We further assume that f is of class $C^{k + 1}$ , which yields $‖ {\bar{A}}_{k}^{\frac{1}{2}} f (u) ‖^{2} ⩽ Q (‖ u ‖_{H^{k + 1} (Ω)})$ . Owing to (4.32) and Cauchy–Schwarz inequality, there holds that $\begin{matrix} (4.35) & \begin{matrix} \frac{1}{2} \frac{d}{d t} (β {‖ {\tilde{A}}_{k}^{\frac{1}{2}} u_{t} ‖}^{2} + ‖ A_{k} u ‖^{2} + ((B_{k} u, A_{k} u))) + \frac{1}{2} {‖ {\tilde{A}}_{k}^{\frac{1}{2}} u_{t} ‖}^{2} \\ ⩽ e^{- c^{'} t} Q (‖ u_{0} ‖_{H^{k + 1} (Ω)}, ‖ v_{0} ‖) + c^{″} . \end{matrix} \end{matrix}$

Multiplying (4.33) by $η^{″} \in (0, \frac{1}{2 β})$ and summing the resulting inequality with (4.19) and (4.35), we get $\begin{matrix} (4.36) & \frac{d}{d t} Y_{2} (t) + D_{2} (t) ⩽ c e^{- c^{'} t} Q (‖ u_{0} ‖_{H^{k + 1} (Ω)}, ‖ v_{0} ‖) + c^{″}, c, c^{'} > 0, t > 0, \end{matrix}$ where $\begin{matrix} (4.37) & \begin{matrix} Y_{2} (t) & = E_{1} + η^{″} β (({\tilde{A}}_{k}^{\frac{1}{2}} u_{t}, {\tilde{A}}_{k}^{\frac{1}{2}} u)) + \frac{η^{″}}{2} {‖ {\tilde{A}}_{k}^{\frac{1}{2}} u ‖}^{2} + \frac{β}{2} {‖ {\tilde{A}}_{k}^{\frac{1}{2}} u_{t} ‖}^{2} \\ + \frac{1}{2} ‖ A_{k} u ‖^{2} + \frac{1}{2} ((B_{k} u, A_{k} u)), \\ D_{2} (t) & = (\frac{1}{2} - η^{″} β) {‖ {\tilde{A}}_{k}^{\frac{1}{2}} u_{t} ‖}^{2} + c η^{″} ‖ A_{k} u ‖^{2} + η^{″} ((B_{k} u, A_{k} u)) . \end{matrix} \end{matrix}$ Applying Cauchy–Schwarz, (4.7) and proceeding as above, it follows that, $\begin{matrix} | ((B_{k} u, A_{k} u)) | ⩽ \frac{1}{4} ‖ A_{k} u ‖^{2} + C ‖ u ‖^{2} . \end{matrix}$

Then, arguing as previously with (3.7) and (4.12) and for $η^{″}$ small enough, we have: $\begin{matrix} (4.38) & Q (‖ u_{t} ‖_{H^{k - 1} (Ω)}, ‖ u ‖_{H^{2 k} (Ω)}) ⩾ Y_{2} (t) ⩾ \frac{β}{4} {‖ {\tilde{A}}_{k}^{\frac{1}{2}} u_{t} ‖}^{2} + \frac{1}{4} ‖ A_{k} u ‖^{2} - c, \end{matrix}$ with which, we can deduce from the definition of $Y_{2} (t)$ and $D_{2} (t)$ that there holds $\begin{matrix} (4.39) & \frac{d}{d t} Y_{2} (t) + c_{0} Y_{2} (t) ⩽ c e^{- c^{'} t} Q (‖ u_{0} ‖_{H^{k + 1} (Ω)}, ‖ v_{0} ‖) + c^{″}, c, c^{'} > 0, t > 0 . \end{matrix}$ According to Gronwall’s lemma, the interpolation inequality and (4.32), we finally obtain (4.4), then the proof is complete. □

5. The dissipative semigroup

Based on the a priori estimates, we have the

Theorem 5.1.

For any initial data $(u_{0}, v_{0}) \in H_{0}^{k} (Ω) \times H^{- 1} (Ω)$ , problem ( 3.1 )–( 3.2 ) possesses a unique weak solution $(u, u_{t})$ , such that, for $\forall T > 0$ , u satisfies $\begin{matrix} u \in L^{\infty} (R^{+}; H_{0}^{k} (Ω)) and u_{t} \in L^{2} (0, T; H^{- 1} (Ω)) . \end{matrix}$

If we assume that $(u_{0}, v_{0}) \in (H^{k + 1} (Ω) \cap H_{0}^{k} (Ω)) \times L^{2} (Ω)$ , then we have, $\begin{matrix} u \in L^{\infty} (R^{+}; H^{k + 1} (Ω) \cap H_{0}^{k} (Ω)) and u_{t} \in L^{2} (0, T; L^{2} (Ω)) . \end{matrix}$

If we further assume that f is of class $C^{k + 1}$ , and $(u_{0}, v_{0}) \in (H^{2 k} (Ω) \cap H_{0}^{k} (Ω)) \times H^{k - 1} (Ω)$ , then $\begin{matrix} u \in L^{\infty} (R^{+}; H^{2 k} (Ω) \cap H_{0}^{k} (Ω)) and u_{t} \in L^{\infty} (0, T; H^{k - 1} (Ω)) . \end{matrix}$

Proof.

We can prove the existence and the regularities in (i), (ii), (iii) by applying, for instance, a standard Galerkin scheme and the a priori estimates which have been proved in the previous section.

Now we assume that there are at least two pairs of solutions to the problem (3.1)–(3.2), $(u_{1}, v_{1})$ and $(u_{2}, v_{2})$ (where $v_{j} = \frac{\partial u_{j}}{\partial t}$ , $j = 1, 2$ ), respectively associated to the initial data $(u_{0, 1}, v_{0, 1})$ and $(u_{0, 2}, v_{0, 2})$ . Then we set $u = u_{1} - u_{2}$ , $v = v_{1} - v_{2}$ , $u_{0} = u_{0, 1} - u_{0, 2}$ and $v_{0} = v_{0, 1} - v_{0, 2}$ to have $\begin{array}{l} (5.1) & β u_{t t} + u_{t} - Δ A_{k} u - Δ B_{k} u - Δ (f (u_{1}) - f (u_{2})) = 0, \\ (5.2) & D^{α} u = 0, on Γ, | α | ⩽ k, \\ (5.3) & u |_{t = 0} = u_{0}, v |_{t = 0} = v_{0} . \end{array}$ Multiplying (5.1) by ${(- Δ)}^{- 1} u_{t}$ and integrating over Ω and by parts, we obtain, adding to both parts of the resulting equation the term $\frac{γ}{2} \frac{d}{d t} ‖ u ‖^{2}$ , $\begin{matrix} (5.4) & \begin{matrix} \frac{1}{2} \frac{d}{d t} (β ‖ u_{t} ‖_{- 1}^{2} + {‖ A_{k}^{1 / 2} u ‖}^{2} + B_{k}^{\frac{1}{2}} [u] + γ ‖ u ‖^{2}) + ‖ u_{t} ‖_{- 1}^{2} \\ ⩽ | ((f (u_{1}) - f (u_{2}), u_{t})) | + γ | ((u, u_{t})) | . \end{matrix} \end{matrix}$ Considering the right-hand side, we note that f is of class $C^{2}$ , then we have $\begin{matrix} \nabla (f (u_{1}) - f (u_{2})) & = f^{'} (u_{1}) \nabla u_{1} - f^{'} (u_{2}) \nabla u_{2} \\ = (f^{'} (u_{1}) - f^{'} (u_{2})) \nabla u_{1} + f^{'} (u_{2}) \nabla u \\ = f^{″} (ϵ_{u_{1}, u_{2}}) u \nabla u_{1} + f^{'} (u_{2}) \nabla u . \end{matrix}$ Noting that $f^{″} (ϵ_{u_{1}, u_{2}}), f^{'} (u_{2}) \in L^{\infty} (Ω)$ and $u, \nabla u_{1} \in L^{4} (Ω)$ , then $\begin{matrix} | ((f (u_{1}) - f (u_{2}), u_{t})) | + γ | ((u, u_{t})) | & ⩽ ‖ \nabla (f (u_{1}) - f (u_{2})) ‖ ‖ u_{t} ‖_{- 1} + γ ‖ \nabla u ‖ ‖ u_{t} ‖_{- 1} \\ ⩽ Q (‖ u_{0, 1} ‖_{H_{0}^{k} (Ω)}, ‖ u_{0, 2} ‖_{H_{0}^{k} (Ω)}, ‖ v_{0, 1} ‖_{H^{- 1} (Ω)}, \\ ‖ v_{0, 2} ‖_{H^{- 1} (Ω)}) (‖ u ‖_{L^{4} (Ω)} + ‖ u ‖_{H^{1} (Ω)}) ‖ u_{t} ‖_{- 1} \\ ⩽ c ‖ u ‖_{H^{k} (Ω)} ‖ u_{t} ‖_{- 1} . \end{matrix}$ Using (4.8) and choosing γ large enough, we deduce that $\begin{matrix} E & : = β ‖ u_{t} ‖_{- 1}^{2} + {‖ A_{k}^{1 / 2} u ‖}^{2} + B_{k}^{\frac{1}{2}} [u] + γ \int_{Ω} u^{2} d x \\ ⩾ β ‖ u_{t} ‖_{- 1}^{2} + \frac{1}{2} {‖ A_{k}^{1 / 2} u ‖}^{2} ⩾ β ‖ u_{t} ‖_{- 1}^{2} + c ‖ u ‖_{H^{k} (Ω)}^{2} . \end{matrix}$ Thus, owing to interpolation inequality and Cauchy–Schwarz inequality, we have $\begin{matrix} (5.5) & \frac{d}{d t} E + ‖ u_{t} ‖_{- 1}^{2} ⩽ c ‖ u ‖_{H^{k} (Ω)}^{2} ⩽ c^{'} E . \end{matrix}$ Applying the Gronwall’s lemma, we get $\begin{matrix} (5.6) & ‖ u_{t} ‖_{- 1}^{2} + ‖ u ‖_{H^{k} (Ω)}^{2} ⩽ c e^{c^{'} t} (‖ u_{0} ‖_{H^{1} (Ω)}^{2} + ‖ v_{0} ‖_{- 1}^{2}), \end{matrix}$ which yields that the solutions are continuously dependent on the initial condition, as well the uniqueness. □

Thus, we can define the family of solving operators: $\begin{matrix} S (t) : Φ \to Φ, (u_{0}, v_{0}) \mapsto (u (t), u_{t} (t)), \forall t ⩾ 0, \end{matrix}$ where $Φ = H_{0}^{k} (Ω) \times H^{- 1} (Ω)$ , and u is the solution given by Theorem 5.1. This family of solving operators indeed forms a continuous semigroup for the topology of $H_{0}^{k} (Ω) \times H^{- 1} (Ω)$ ( $\forall t ⩾ 0$ ). It thus follows from (4.3) and (4.21) (see [37,46]) that

Theorem 5.2.

The semigroup $S (t)$ is dissipative in Φ, in the sense that $S (t)$ possesses a bounded absorbing set $B_{1}$ which is bounded in Φ.

Remark 5.3.

Thus, we can proceed as in [27] and have the existence of the global attactor $A$ which is compact in Φ.

If we further assume that f is of class $C^{k + 1}$ and the initial data $(u_{0}, v_{0}) \in (H^{2 k} (Ω) \cap H_{0}^{k} (Ω)) \times H^{k - 1} (Ω)$ , it follows from (4.4) that the semigroup $S (t)$ is dissipative in $(H^{2 k} (Ω) \cap H_{0}^{k} (Ω)) \times H^{k - 1} (Ω)$ .

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Well-posedness for modified higher-order anisotropic Cahn–Hilliard equations

Abstract

Keywords

1. Introduction

2. Prelimilary

3. Setting of the problem

4. A priori dissipative estimates

References