A Note on the Solutions for a Higher-Order Convective Cahn–Hilliard-Type Equation

: The higher-order convective Cahn-Hilliard equation describes the evolution of crystal surfaces faceting through surface electromigration, the growing surface faceting, and the evolution of dynamics of phase transitions in ternary oil-water-surfactant systems. In this paper, we study the H 3 solutions of the Cauchy problem and prove, under different assumptions on the constants appearing in the equation and on the mean of the initial datum, that they are well-posed.

On the initial datum, we assume Inspired by [1][2][3][4][5][6][7][8][9][10][11][12], in light of (7), we define the following function: on which we assume The equation in (1) is derived in [13] to model the evolution of a crystalline surface with small slopes that undergoes faceting. The unknown u gives the surface slope, the constant κ is proportional to the atomic flux deposition strength and the convective term κ∂ x u 2 arises from the deposited atoms normal impingement. The sixth-order linear term ∂ 6 x u regularizes the equation, taking into account the surface curvature and the anisotropy of the surface energy under the surface diffusion.
From a mathematical point of view, the existence and uniqueness of weak solutions of (1) with periodic boundary conditions is proven in [14], under the assumptions κ > 0 and γ = 0. In the same setting, a similar result is proven in two space dimensions in [15]. In [16], the authors derived the stationary solutions of (1), again assuming κ > 0 and γ = 0. In [17], the existence of a global-in-time attractor is studied, while the well-posedness of the classical solutions of (1) is proven in [18], requiring (7)- (9), and γ = 0. In this paper, we will prove that, if (2) or (3) hold, we have the well-posedness of the classical solutions of (1) assuming (6), while if (5) holds, we have the well-posedness of (1) assuming (7)- (9).
Taking κ = 0, (1) becomes which is a Cahn-Hilliard type equation [19][20][21]. It was deduced in [22] to describe the evolution of crystal surfaces faceting through surface electromigration. It also describes the phase transition development in ternary oil-water-surfactant systems. One part of the surfactant is hydrophilic, and the other one (termed amphiphile) is lipophilic. Oil, water, and microemulsion (i.e., a homogeneous, isotropic mixture of oil and water) can coexist in equilibrium. The unknown u, in (10), gives the local difference between oil and water concentrations. From a mathematical point of view, in [23] the initial-boundary-value problem for (10) is analyzed, under appropriate assumptions on γ, β, α, δ. In [24], the authors analyze the existence of a global-in-time attractor. The existence of weak solutions for the initial-boundary-value problem for (10) is proven in the case of degenerate mobility in [25]. Finally, in [18], the well-posedness of the classical solution of the Cauchy problem of (10) is proven, assuming (7)- (9), with γ = 0. In this paper, we will show that the classical solutions of the Cauchy problem of (10) are well-posed, assuming (6), if γ ≤ 0 and α > 0, while in the general case, we will prove the same result assuming (7)- (9).
Observe that in [13], it is proven that, as κ → ∞, (1) reduces which is known as the Kuramoto-Sivashinsky equation (see [26][27][28]). In Section 4, we will prove the well-posedness of the Cauchy problem for (11), assuming (6). When β = δ = 0 and α = f 2 = 0, (1) reads (12) appears in several physical situations; for example, it models long waves on a viscous fluid flowing down an inclined plane [29] and drift waves in a plasma [30]. (12) was also independently deduced by Kuramoto [27,31,32] to describe the phase turbulence in reaction-diffusion systems, and by Sivashinsky [28] to describe plane flame propagation, taking into account the combined influence of diffusion and thermal conduction of the gas on the stability of a plane flame front. Equation (12) can be used to study incipient instabilities in several physical and chemical systems [33][34][35]. Moreover, (12) is also termed the Benney-Lin equation [36,37], and was deduced by Kuramoto as a model for phase turbulence in the Belousov-Zhabotinsky reaction [38].
The existence and the dynamical properties of the exact solutions for (12) can be found in [39][40][41][42][43][44]. The control problem for (12) with periodic boundary conditions, and on a bounded interval, are studied in [45][46][47]. The problem of global-in-time exponential stabilization of (12) with periodic boundary conditions is analyzed in [48]. A generalization of the optimal control theory for (12) is proposed in [49], while the f global boundary controllability of (12) is considered in [50]. The existence of solitonic solutions for (12) is proven in [51]. The well-posedness of the Cauchy problem for (12) is proven in [52][53][54], using the energy space technique, a priori estimates together with an application of the Cauchy-Kovalevskaya and the fixed point method, respectively. Instead, the initial-boundary value problem for (1) is studied, using a priori estimates together with an application of the Cauchy-Kovalevskaya, and the energy space technique in [55][56][57]. Inspired by [58][59][60], the convergence of the solution of (12) to the unique entropy one of the Burgers equation is proven in [61].
Finally, due to its general structure, we conjecture that (1) can have a possible application in machine learning (see [62,63]).

Results and Organization of the Paper
In this paper, we improve and complete the results of [14,[16][17][18] working with H 3 initial data and having general assumptions on the constants appearing in (1). The main result of this paper is the following theorem. We prove the global-in-time existence, uniqueness, and stability of the solutions of the Cauchy problem (1). Theorem 1. Fix T > 0. Assuming one of the following (i) (2) and (6), (ii) (3) and (6), (iii) (4) and (6), (iv) (5) and (7), and (9), there exists a unique solution u of (1) such that u ∈ H 1 ((0, T) × R) ∩ L ∞ (0, T; H 3 (R)). (13) In particular, under the Assumptions (7) and (9), we have that Moreover, if u 1 and u 2 are two solutions of (1), we have that for some suitable C(T) > 0, and every 0 ≤ t ≤ T.
The well-posedness of (1) is guaranteed for a short time by the Cauchy-Kowaleskaya Theorem [64]. The solutions are indeed global, thanks to suitable a priori estimates.
The paper is organized as follows. In Section 3 we prove Theorem 1, assuming (i) or (ii). In Section 4 we prove Theorem 1, assuming (iii). In Section 5 we prove Theorem 1, assuming (iv).

Proof of Theorem 1 Assuming (i) or (ii)
In this section, we prove Theorem 1, assuming (i) or (ii). For the sake of notational simplicity, define and then (1) reads Since the short time well-posedness of (17) is guaranteed by the Cauchy-Kowaleskaya Theorem [64], here we need to prove some suitable global a priori estimates.
From now on, we denote with C 0 the constants which depend only on the initial data, and with C(T), the constants which depend also on T.
We continue by proving an L 2 − estimate, which is independent on κ, a, b.
for every 0 ≤ t ≤ T.
Proof. Let 0 ≤ t ≤ T. We begin by observing that Multiplying (17) by 2u, an integration on R and several integrations by part give Due to the Young inequality, we can estimate the right-hand side of (33), as follows: Consequently, (33) becomes Observe that, by (32), Using (20), It follows from (34) and (36) that Since
We continue with an a priori estimate in the space L 2 (0, ∞; H 4 (R)).
We continue with an a priori estimate in the space L 2 (0, ∞; H 5 (R)).
Proof. Let 0 ≤ t ≤ T. We begin by observing that, by (32), we have that Multiplying (17) by 2∂ 4 x u, thanks to (46), an integration on R gives Due to Lemma 1 and the Young inequality, where D 2 is a positive constant, which will be specified later. Therefore, by (47), , and taking .
Proof. Let 0 ≤ t ≤ T. We begin by observing that Due to the Young inequality, It follows from (50) that By (44), we have that .
We continue with an a priori estimate in the space L 2 (0, ∞; H 6 (R)). Lemma 6. Fix T > 0 and assume (2) or (3). There exists a constant C(T) > 0, independent on κ, a, b, such that In particular, we have that where C(T) is independent on κ, a, b.
Proof. Let 0 ≤ t ≤ T. We begin by observing that, by (46), Multiplying (17) by −2∂ 6 x u, thanks to (46), an integration on R gives Therefore, we have that Due to Lemma 1, (44), (45) and the Young inequality, where D 3 is a positive constant, which will be specified later. It follows from (54) that .
We continue with an a priori estimate in the space H 1 ((0, ∞) × R).
We are finally ready to prove Theorem 1, assuming (i) or (ii).

Proof of Theorem 1 assuming (i) or (ii).
The well-posedness of (1) is guaranteed for a short time by the Cauchy-Kowaleskaya Theorem [64]. Thanks to the a priori estimates proved in Lemmas 1-7, we have that the global-in-time existence of a is the solution of (1) that satisfies (13). The stability estimates (15) can be proved using the same arguments of [18] (Theorem 1).

Proof of Theorem 1 Assuming (iii)
In this section, we prove Theorem 1 assuming (iii). Due to (4), here, (1) becomes The argument of this section is analogous to that of the previous one. We deduce the local-in-time well-posedness from the Cauchy-Kowaleskaya Theorem [64], and we improve the local-in-time existence to the global-in-time one, proving some suitable a priori estimates on u.

Lemma 8.
Fix T > 0. There exists a constant C(T) > 0, such that for every 0 ≤ t ≤ T. In particular, (29) holds. Moreover, we have that for every 0 ≤ t ≤ T.
Proof. Let 0 ≤ t ≤ T. Multiplying (58) by 2u, an integration on R gives Since, using the Young inequality, , we can pass from (61) to Observe that Therefore, by the Young inequality, where D 4 is a positive constant, which will be specified later. Observe again that Consequently, by the Young inequality, where D 5 is a positive constant, which will be specified later. Il follows from (63) and (64) that . Choosing we have that , that is It follows from (62) and (66) that Choosing we have that d dt By the the Gronwall Lemma and (6), we get which gives (59). We prove (29). Thanks to (59), (66) and (67), .
It follows from (73) that .
We are finally ready to prove Theorem 1 assuming (iii).

Proof of Theorem 1 assuming (iii).
The well-posedness of (1) is guaranteed for a short time by the Cauchy-Kowaleskaya Theorem [64]. Thanks to the a priori estimates proved in Lemmas 8-12, we have that the global-in-time existence of a is solution of (1) that satisfies (13). The stability estimates (15) can be proved using the same arguments of [18] (Theorem 1).

Proof of Theorem 1 Assuming (iv)
In this final section, we prove Theorem 1 assuming (iv). The argument is again analogous to the one of the previous sectionss. We deduce the local-in-time well-posedness from the Cauchy-Kowaleskaya Theorem [64], and we improve the local-in-time existence to the global-in-time one proving some suitable a priori estimates on u.
We begin with the zero mean estimate.

Remark 1. In light of
Moreover, again by (14), we have that We continue by proving some energy estimates on the function P.
for every 0 ≤ t ≤ T.
Due to the Young inequality, .

Conclusions
This paper is dedicated to the well-posedness of a solution to the Cauchy problem for a higher-order convective Cahn-Hilliard equation. Such an equation models the evolution of crystal surfaces faceting through surface electromigration, the growing surface faceting, and the evolution of dynamics of phase transitions in ternary oil-water-surfactant systems. The well-posedness of (1) is proved for a short time by the Cauchy-Kowaleskaya Theorem [64]. The global-in-time well-posedness is thus proved, proving several a priori estimates.
Author Contributions: All authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.