Well-posedness and global attractors for a non-isothermal viscous relaxation of nonlocal Cahn-Hilliard equations

We investigate a non-isothermal viscous relaxation of some nonlocal Cahn-Hilliard equations. This perturbation problem generates a family of solution operators, exhibiting dissipation and conservation. The solution operators admit a family of compact global attractors that are bounded in a more regular phase-space.

In this article we consider the following problems: for α > 0, δ > 0, and ε > 0 the relaxation Problem P α,ε is, given T > 0 and (φ 0 , θ 0 ) tr , find (φ + , θ + ) tr satisfying φ + t = ∆µ + in Ω × (0, T ) (1.1) The main focus of this article is to examine the the asymptotic behavior of solutions to Problem P α,ε , via global attractors, and the regularity of these attractors. For ease of presentation, throughout we assume there is δ 0 > 0 so that δ ∈ (0, δ 0 ], and also (α, ε) ∈ (0, 1] × (0, 1]. Let us now give some preliminary words on the motivation for using nonlocal diffusion. First, in [2, Equation (0.2)] the nonlocal diffusion terms aφ − J * φ appear as, Ω J(x − y) (φ(x, t) − φ(y, t)) dy, i.e. a(x) = J * 1. Heuristically, this integral term "takes into account the individuals arriving at or leaving position x from other places." In this setting, the term a(x) ≥ 0 is a factor of how many individuals arrive at position x. Since the integration only takes place over Ω, individuals are not entering nor exiting the domain. Hence, this representation is faithful to the desired mass conservation law we typically associate with Neumann boundary conditions. Although Neumann boundary conditions for the chemical potential µ make sense from the physical point of view of mass conservation, it is not necessarily true that the interface between the two phases is always orthogonal to the boundary, which is implied by the boundary condition ∂ n φ = 0 which commonly appears in the literature. This is partially alleviated by using nonlocal diffusion on φ.
Finally, we now mention that (cf. [19]) the term −δφ t could be thought of as the linearization d dt G(φ) for some appropriate function G. In this case the internal energy is nonlinear in the order parameter θ; i.e., e := θ + G(φ).
The first goal of this article concerns determining the global well-posedness of the model problem Problem P α,ε . Second, we wish to determine the asymptotic behavior of the solutions to Problem P α,ε up to the existence of global attractors (or universal attractors) for appropriate α and ε.
The main points of this article are as follows: • For Problem P α,ε we establish (global) well-posedness of weak solutions using minimal assumptions on the nonlinear term F . • The weak solutions generate a strongly continuous one-parameter family of solution operators; i.e., a semigroup, which in turn admits a bounded absorbing set and certain compactness properties. Consequently the associated dynamical system is gradient. • The semigroup also admits a global attractor. We show the global attractor is bounded in a more regular space with √ αµ ∈ L ∞ (0, ∞; H 2 (Ω)). Each of these properties hold for every α ∈ (0, 1] and ε ∈ (0, 1]. The next section provides the functional framework behind Problem P α,ε .

Preliminaries
Now we detail some preliminaries that will be applied to both problems. To begin, define the spaces H := L 2 (Ω) and V := H 1 (Ω) with norms denoted by, · and · V , respectively. Otherwise, we write the norm of the Banach space X with · X . The inner-product in H is denoted by (·, ·). Denote the dual space of V by V ′ , and the dual paring in V ′ × V is denoted by ·, · . For every ψ ∈ V ′ , we denote by ψ the average of ψ over Ω, that is, where |Ω| is the Lebesgue measure of Ω. Throughout, we denote byψ := ψ − ψ and for future reference, observe ψ = ψ − ψ = 0. We will refer to the following norm in V ′ , which is equivalent to the usual one, Define the space L 2 0 (Ω) := {φ ∈ L 2 (Ω) : φ = 0}. Let A N = −∆ : L 2 0 (Ω) → L 2 0 (Ω) with domain D(A N ) = {ψ ∈ H 2 (Ω) : ∂ n ψ = 0 on Γ} denote the "Neumann-Laplace" operator. Of course the operator A N generates a bounded analytic semigroup, denoted e −AN t , and the operator is nonnegative and selfadjoint on It is well known that the restriction A N |V0 maps V 0 to V ′ 0 isomorphically, and the inverse map N = A −1 N : Additionally, these maps satisfy the relations, for all u ∈ V 0 and v, w ∈ V ′ 0 , The Sobolev space V is endowed with the norm, Denote by λ Ω > 0 the constant in the Poincaré-Wirtinger inequality, Whence, for c Ω := max{λ Ω , 1}, there holds, for all ψ ∈ V, For each m ≥ 0, α > 0, and ε > 0 define the following energy phase-space for Problem P α,ε , which is Hilbert when endowed with the α, ε-dependent norm whose square is given by, When we are concerned with the dynamical system associated with Problem P α,ε , we will utilize the following metric space We also define the more regular phase-space for Problem P α,ε , The following assumptions on J and F are based on [7,11]: (H3): There exists c 1 > 1 2 J L 1 (R 3 ) and c 2 ∈ R such that, for all s ∈ R, F (s) ≥ c 1 s 2 − c 2 .
(H4): There exists c 3 > 0, c 4 ≥ 0, and p ∈ (1, 2] such that, for all s ∈ R, (H5): There exist c 5 , c 6 > 0, and q > 0 such that, for all s ∈ R, Let us make some remarks and report some important consequences of these assumptions. From [5,Remark 2]: assumption (H2) implies that the potential F is a quadratic perturbation of a (strictly) convex function. Indeed, if we set a * := a L ∞ (Ω) , then F can be represented as with G ∈ C 2 (R) being strictly convex, since G ′′ ≥ c 0 . With (H3), for each m ≥ 0 there are constants c 7 , c 8 , c 9 , c 10 > 0 (with c 8 and c 9 depending on m and F ) such that, and |F (s)| − c 10 ≤ |F ′ (s)|(1 + |s|). (2.8) The last inequality appears in [12, page 8]. With the positivity condition (H3), it follows that, for all s ∈ R, (2.9) A word of notation: In many calculations, functional notation indicating dependence on the variable t is dropped; for example, we will write ψ in place of ψ(t). Throughout the article, C > 0 will denote a generic constant, while Q : R d + → R + will denote a generic increasing function in each of the d components. Unless explicitly stated, all of these generic terms will be independent of the parameters α, δ, ε, T, and m. Finally, throughout we will use the following abbreviations c J := J L 1 (Ω) and d J := ∇J L 1 (Ω) . (2.10) 3. The relaxation Problem P α,ε 3.1. Global well-posedness of Problem P α,ε .
In addition, upon setting, for every ϕ, ϑ ∈ V, there holds, for almost all t ∈ (0, T ), Also, there holds, We say that ζ = (φ, θ) tr is a global weak solution of Problem P α,ε if it is a weak solution on [0, T ], for any T > 0. The initial conditions (3.10) hold in the L 2 -sense; i.e., for every ϕ, ϑ ∈ V, hold.
Theorem 3.2. Assume (H1)-(H5) hold with p ∈ ( 6 5 , 2] and q ≥ 1 2 . For any ζ 0 = (φ 0 , θ 0 ) tr ∈ H × H with F (φ 0 ) ∈ L 1 (Ω), there exists a global weak solution ζ = (φ, θ) tr to Problem P α,ε in the sense of Definition 3.1 satisfying the additional regularity, for any T > 0, Proof. We follow the proofs of [5, Theorem 1] and [22, Theorem 2.1]. The proof proceeds in several steps. The existence proof begins with a Faedo-Galerkin approximation procedure to which we later pass to the limit. We first assume that φ 0 ∈ D(A N ) and θ 0 ∈ H. (The first assumption will be used to show that there is a sequence {φ 0n } ∞ n=1 such that φ 0n → φ 0 in H 2 (Ω) as well as L ∞ (Ω), which will be important in light of the fact that F (φ 0n ) is of arbitrary polynomial growth per assumptions (H1)-(H5).) The existence of a weak solution for φ 0 ∈ H with F (φ 0 ) ∈ L 1 (Ω) will follow from a density argument and by exploiting the fact that the potential F is a quadratic perturbation of a convex function (cf. equation (2.5)).
Step 1. (Construction and boundedness of approximate maximal solutions) Recall that the linear operator A N + I is positive and self-adjont on H. Then we have a complete system of eigenfunctions : ∂ n χ = 0 on Γ}. We know by spectral theory that the eigenvalues may be ordered and counted according to their multiplicities in order to form a (real) diverging sequence. The set of respective eigenvectors, {ψ i } ∞ i=1 , forms an orthogonal basis in V , which we may assume is orthonormal in H.
By construction, clearly Ψ ∞ is dense in D(A N ). Then, for any fixed T > 0 and n ∈ N, we will seek functions of the form that solve the following approximating problems for any δ 0 > 0, δ ∈ (0, δ 0 ], (α, ε) ∈ (0, 1] × (0, 1], and for all t ∈ [0, T ], for every ϕ, ϑ ∈ Ψ n , and where φ 0n = P n φ 0 and θ 0n = P n θ 0 ; P n being the n-dimensional projection of H onto Ψ n . Throughout the remainder of the proof we set M 0 := φ 0 and N 0 := θ 0 . The functions a i and b i are assumed to be (at least) C 2 ((0, T )). It is also worth noting that (3.12) and (3.13), also hold for the discretized functions φ n and θ n .
To show the existence of at least one solution to (3.21)-(3.26), we now suppose that n is fixed and we take ϕ = φ k and ϑ = θ k for some 1 ≤ k ≤ n. Then substituting the discretized functions (3.20) into (3.21)-(3.26), we arrive at a system of n ODEs in the unknowns a k = a k (t) and b k = b k (t) on Ψ n . Since J ∈ W 1,1 (R 3 ) and F ∈ C 2,1 loc (R), we may apply Cauchy's/Carathéodory's theorem for ODEs to find that there is T n ∈ (0, T ) such that a k , b k ∈ C 2 ((0, T n )), for 1 ≤ k ≤ n, and (3.21)-(3.22) hold in the classical sense for all t ∈ [0, T n ]. Since F ′ ∈ C 1 (R), this argument shows the existence of a unique maximal solution to the projected problem (3.21)-(3.26).
Now we need to derive some a priori estimates to apply to the approximate maximal solutions to show that T n = +∞, for every n ≥ 1, and that the corresponding sequences φ n , θ n and µ n are bounded in some appropriate function spaces. To begin, we take ϕ = µ n as a test function in (3.21) and ϑ = θ n as a test function in (3.22), to obtain and where ρ n := P n ρ(·, φ n ) = µ n + P n (J * φ n ) − αφ ′ n + δθ n . Hence, Combining (3.18) (with the discritized functions), (3.27)-(3.30) yields the differential identity, Estimating the two products in (3.31), we find and Observe that with the aid of hypothesis (H3), there holds Now, combining (3.31)-(3.33) and integrating the resulting inequality with respect to t over (0, T n ) and applying (3.34) to the result produces, Using the basic estimate P n ψ ≤ ψ we find, where (and with (H3)), (3.37) and where the extra term appearing on the right-hand side of (3.36) is to make the V norm for θ n on the left-hand side. Since the right-hand side of (3.36) is independent of n and t, we deduce, by means of a Grönwall inequality, that T n = +∞, for every n ≥ 1, i.e., the projected problem (3.21)-(3.26) has a unique global in time solution as T > 0 is arbitrary, and (3.36) is satisfied for every t ≥ 0. Furthermore, from (3.36), we obtain the following estimates for any given 0 < T < +∞, we observe the two basic estimates hold for every η > 0, Together, these two yield and with (3.38), (3.40) and (3.41) we deduce (3.44). Now we seek a uniform bound for µ n in L 2 (0, T ; H) so that we may bound µ n uniformly in L 2 (0, T ; V ) (by virtue of (2.2)). A simple estimate with (2.9) shows, (3.45) The desired bound now follows because of the uniform bounds in (3.38), and (3.41)-(3.43). Thus, we have shown Moreover, directly from (3.46) and the discretized equation (3.48) Next we obtain a bound for √ αφ n . Indeed, we take ϕ = φ n in (3.21) to obtain, So we estimate (recall (2.10)) and we use the basic estimate, Utilizing the bounds (3.38) and (3.41) following (3.36), and the definition of the V norm (2.2), we integrate (3.52) with respect to t over (0, T ) to find, We use the above results to bound θ ′ n . Let us choose ϑ = θ ′ n in (3.22), which yields 1 2 Integration over (0, T ) and the bounds (3.39) and (3.43) shows us that, (3.55) Finally, we provide a bound for {ρ(·, φ n )}. Using (H4) again (see (2.9)), we easily find, for any p ∈ (1, 2], Employing (3.38) and (3.43), it follows from (3.56) that This concludes Step 1.
Step 3. (Energy identity) To begin, let φ 0 ∈ D(A N ), θ 0 ∈ H and let ζ = (φ, θ) tr be the corresponding weak solution. Recall from (3.76), we have for almost any t ∈ (0, T ), φ n (t) → φ(t) strongly in H and a.e. in Ω. (3.84) Since F is measurable (see (H3)), Fatou's lemma implies Additionally, thanks to (3.78) and the fact that P n ∈ L(V, V ), then  .74), as well as the weak lower-semicontinuity of the norm, we arrive at the differential inequality We now show the energy equality (3.19) holds. The proof is based on the proof of [5, Corollary 2]. Here we require the regularity given in (H5). Indeed, take ϕ = µ in (3.8). Because of (3.4), we find the product φ t , µ must contain the dual pairing φ t , F ′ (φ) . It is here where we employ (2.5) where G is monotone increasing. Now define the functional G : H → R by Hence, Next we add in the identity obtained after taking ϑ = θ in (3.9) and apply (3.18) to find d dt Integrating this differential identity on (0, t) produces (3.19) as claimed. This concludes Step 3.
As before, we can now formalize the semi-dynamical system generated by Problem P α,ε .
Corollary 3.5. Let the assumptions of Theorem 3.2 be satisfied. We can define a strongly continuous semigroup (of solution operators) S α,ε = (S α,ε (t)) t≥0 , for each α > 0 and ε > 0, where ζ(t) = (φ(t), θ(t)) is the unique global weak solution to Problem P α,ε . Furthermore, as a consequence of (3.91), if we assume M 1 = M 2 and N 1 = N 2 , the semigroup S α,ε (t) : X α,ε m → X α,ε m is Lipschitz continuous on X α,ε m , uniformly in t on compact intervals. 3.2. Bounded absorbing sets for Problem P α,ε . We now give a dissipation estimate for Problem P α,ε from which we deduce the existence of an absorbing set. The idea of the estimate follows [12,Proposition 2]. It is here where we require the slight modification of hypothesis (H1).
Consequently, the set given by where Q(·, ·) is the function from (3.109), is a closed, bounded absorbing set in H α,ε m , positively invariant under the semigroup S α,ε .

3: the global attractor is unique maximal compact invariant subset in H α,ε m given by
Furthermore,

4:
The global attractor A α,ε is connected and given by the union of the unstable manifolds connecting the equilibria of S α,ε (t). 5: For each ζ 0 = (φ 0 , θ 0 ) tr ∈ H α,ε m , the set ω(ζ 0 ) is a connected compact invariant set, consisting of the fixed points of S α,ε (t).
With the existence of a bounded absorbing set set B 0 α,ε (in Lemma 3.6), the existence of a global attractor now depends on the precompactness of the semigroup of solution operators S α,ε . We begin by discussing the precompactness of the second component θ which follows from a straight forward result. Indeed, the next result refers to the instantaneous regularization of the "thermal" function θ. This result will also be useful later in Section 3.4. Lemma 3.10. Under the assumptions of Lemma 3.6, the global weak solutions to Problem P α,ε satisfy the following: for every τ > 0, (3.131) and, for all t ≥ τ, there hold the bounds, Proof. The result follows from a standard density argument (cf. e.g. [29, pp. 243-244]). We return to the beginning of the proof of Theorem 3.2 by letting θ 0 ∈ D(A N ) = {ψ ∈ H 2 (Ω) : ∂ n ψ = 0}, ϑ = −∆θ n , and T > 0. In place of (3.28), we find there holds Multiplying (3.134) by t to then integrate over (0, T ) yields, Here we integrate (3.125) on (0, T ) after omitting the positive terms ν 3 E + φ t 2 V ′ from the left-hand side to find the bounds When we combine (3.135)-(3.137) and choose any 0 < τ < T , we find, for all τ ≤ t < T , Moreover, for every τ > 0 and t ≥ τ such that τ ≤ t < T, thus, for the above bounds we also find θ n ⇀ θ weakly in L 2 (τ, T ; H 2 (Ω)). (3.142) In order to recover the result for θ 0 ∈ H, recall that D(A N ) is dense in H, so for any θ 0 ∈ H, there is a sequence (θ 0n ) ∞ n=1 ⊂ D(A N ) such that θ 0n → θ 0 in H. Therefore, for any θ 0 ∈ H and T > 0 we deduce (3.134)-(3.142) hold as well. Finally, the required bound (3.132) follows from (3.138), and (3.133) follows from (3.139). This completes the proof.
The precompactness of the semigroup of solution operators S α,ε now depends on the precompactness of the first component. To this end we will show there is a t * > 0 such that the map S α,ε (t * ) is a so-called α-contraction on B 0 ; that is, there is a time t * > 0, a constant 0 < ν < 1 and a precompact pseudometric M * on B 0 , where B 0 is the bounded absorbing set from Lemma 3.6, such that for all ζ 1 , ζ 2 ∈ B 0 , S α,ε (t * )ζ 1 − S α,ε (t * )ζ 2 H0 ≤ ν ζ 1 − ζ 2 H0 + M * (ζ 1 , ζ 2 ). (3.143) Such a contraction is commonly used in connection with phase-field type equations as an alternative to establish the precompactness of a semigroup; for some particular recent results see, [14,25,30].
where C 1 > 0 depends on δ 0 , c J , and the embedding H ֒→ V ′ , C 2 > 0 depends on F , J, Ω, δ 0 , and c J , and where the constantν 1 is given in Proposition 3.4. Consequently, there is t * > 0 such that the operator S α,ε (t * ) is a strict contraction up to the precompact pseudometric on B 0 , in the sense of (3.143), given by where C * > 0 depends on t * andν 1 , but is independent of t, α, and ε. Furthermore, S α,ε is precompact on B 0 .
Proof. The proof is based on the proof of Proposition 3.4. Here we multiply (3.92)-(3.94) by, respectively, ,φ − φ andθ, then sum the resulting identities to yield, This time estimating the resulting products using assumption (H2) yields, where we recall the continuous embedding H ֒→ V ′ . We also write, and, After applying Grönwall's inequality to (3.152), we obtain, for all t ≥ 0, It is important to note that by (3.91), where C = C(F, J, Ω, δ 0 ) > 0. Moreover, with (3.106) again, 1 Clearly there is a t * > 0 so that e −ν4t * /2 < 1. Thus, the operator S α,ε (t * ) is a strict contraction up to the pseudometric M * defined by (3.145 Finally, with the compactness result for the second component given in Lemma 3.10, the operators S α,ε are precompact on H α,ε m . The proof is complete. Proof of Theorem 3.9. The precompactness of the solution operators S α,ε follows via the method of precompact pseudometrics (see Lemma 3.10 and Lemma 3.11). With the existence of a bounded absorbing set B α,ε 0 in H α,ε m (Lemma 3.6), the existence of a global attractor in H α,ε m is well-known and can be found in [27,3] for example. Additional characteristics of the attractor follow thanks to the gradient structure of Problem P α,ε (Remark 3.3). In particular, the first three claims in the statement of Theorem 3.9 are a direct result of the existence of the an absorbing set, a Lyapunov functional E ε , and the fact that the system (X α,ε m , S α,ε (t), E ε ) is gradient. The fourth property is a direct result [27, Theorem VII.4.1], and the fifth follows from [29,Theorem 6.3.2]. This concludes the proof.
3.4. Further uniform estimates and regularity properties for Problem P α,ε . Our next aim is to bound the global attractor in a more regular space by showing the existence of an absorbing set in V α,ε m . Once this is established, we will bound the (α-weighted) chemical potential √ αµ in H 2 (Ω), which also establishes a bound in L ∞ (Ω). Some of the results in this subsection require hypothesis (H5) with q ≥ 2, and hence the existence of a global attractor for Problem P α,ε . Lemma 3.12. Under the assumptions of Lemma 3.6, the set given by for some positive monotonically increasing function Q α ∼ α −1 , is a closed, bounded absorbing set in V α,ε m , positively invariant under the semigroup S α,ε .
Hence, the bound on the right-hand side of (3.183) is well defined. This establishes (3.170). This finishes the proof.