H 1 solutions for a Kuramoto–Sinelshchikov–Cahn–Hilliard type equation

The Kuramoto–Sinelshchikov–Cahn–Hilliard equation models the spinodal decomposition of phase separating systems in an external ﬁeld, the spatiotemporal evolution of the morphology of steps on crystal surfaces and the growth of thermodynamically unstable crystal surfaces with strongly anisotropic surface tension. In this paper, we prove the well-posedness of the Cauchy problem, associated with this equation.

(1.2) (1.1) occurs in many branches of mechanics and physics. For example, taking ν = q = h = m = α = 0, (1.1) reads which is known as the convective Cahn-Hilliard equation. It models the spinodal decomposition of phase separating systems in an external field [35,62,84], the spatiotemporal evolution of the morphology of steps on crystal surfaces [73], and the growth of thermodynamically unstable crystal surfaces with strongly anisotropic surface tension [39][40][41]43,66], where the constant κ, r are the driving forces. For instance, in the case of a growing crystal surface with strongly anisotropic surface tension, the function u represents is the surface slope, while the constants κ, r are the growth driving force proportional to the difference between the bulk chemical potentials of the solid and fluid phases.
(1.3) was also obtained by Watson [81] as a small-slope approximation of the crystal-growth model obtained in [32].
From a mathematical point of view, the coarsening dynamics for (1.3) has been studied in the limit 0 < κ 1, r = τ = 0 in [35,41] and analytically in [82]. In [1], a numerical scheme is studied for (1.3), while the existence of the periodic solution are analyzed in [33,53]. in [62,69], the existence of exact solutions for (1.3) and the its viscous form have been investigated. In [27], the authors proved the well-posedness of the classical solution of (1.1), under assumption u 0 ∈ H (R), ∈ {2, 3, 4}. (1.4) Moreover, [41] shows that, taking r = τ = 0, when κ tends to ∞, (1.3) reduces to the Kuramoto-Sivashinsky equation (see (1.9)). Physically, it means that, with the growth of the driving force, there must be a transition from the coarsening dynamics to a chaotic spatiotemporal behavior. Taking κ = r = τ = 0 in (1.3), we obtain that (1.5) which is known as the Cahn-Hilliard equation. It describes the spinodal decomposition in phase-separating systems [10,11]. It also describes the coarsening dynamics of the faceting of thermodynamically unstable surfaces [46,77]. Krekhov [50] shows that (1.5) can be an effective tool in technological applications to design nanostructured materials.
From a mathematical point of view, in [2], the the existence of some extremely slowly evolving solutions for (1.5) is proven, considering an boundary domain, while, in [8,37], the problem of a global attractor is studied, In [27], the well-posedness of the classical solution of under Assumption (1.4) is proven. In [42,85], numerical schemes for (1.5) are analyzed, while, in [80], an approximate analytical solution is studied.
Observe that (1.5) is has been much studied, as shown in the papers [9,34,86] and their references.
, we obtain the following equation: which is known as the Korteweg-de Vries equation [49]. It has a very wide range of applications, such as magnetic fluid waves, ion sound waves, and longitudinal astigmatic waves. From a mathematical point of view, in [15,17,47], the Cauchy problem for (1.6) is studied, while in [51], the author reviewed the travelling wave solutions for (1.6). Moreover, in [18,61,74], the convergence of the solution of (1.6) to the unique entropy one of the Burgers equation is proven. Taking which is known as the modified Korteweg-de Vries equation. [3,4,19,[58][59][60] show that (1.7) is a non-slowly-varying envelope approximation model that describes the physics of few-cycle-pulse optical solitons. In [15,47], the Cauchy problem for (1.7) is studied, while, in [20,74], the convergence of the solution of (1.7) to the unique entropy solution of the following scalar conservation law ∂ t u + q∂ x u 3 = 0. (1.8) (1.9) arises in interesting physical situations, for example as a model for long waves on a viscous fluid owing down an inclined plane [79] and to derive drift waves in a plasma [31]. (1.9) was derived also independently by Kuramoto [54][55][56] as a model for phase turbulence in reaction-diffusion systems and by Sivashinsky [76] as a model for plane flame propagation, describing the combined influence of diffusion and thermal conduction of the gas on the stability of a plane flame front.
(1.9) also describes incipient instabilities in a variety of physical and chemical systems [13,44,57]. Moreover, (1.9), which is also known as the Benney-Lin equation [6,63], was derived by Kuramoto in the study of phase turbulence in the Belousov-Zhabotinsky reaction [64].
The dynamical properties and the existence of exact solutions for (1.9) have been investigated in [36,48,52,70,71,83]. In [5,12,38], the control problem for (1.9) with periodic boundary conditions, and on a bounded interval are studied, respectively. In [14], the problem of global exponential stabilization of (1.9) with periodic boundary conditions is analyzed. In [45], it is proposed a generalization of optimal control theory for (1.9), while in [65] the problem of global boundary control of (1.9) is considered. In [72], the existence of solitonic solutions for (1.9) is proven. In [7,21,78], the wellposedness of the Cauchy problem for (1.9) is proven, using the energy space technique, a priori estimates together with an application of the Cauchy-Kovalevskaya and the fixed point method, respectively. In particular, in [21], the well-posedness of (1.1) is proven assuming In [28,67,68], the initial-boundary value problem for (1.9) is studied, using a priori estimates together with an application of the Cauchy-Kovalevskaya and the energy space technique, respectively. Finally, following [22,61,74], in [23], the convergence of the solution of (1.9) to the unique entropy one of the Burgers equation is proven.
The main result of this paper is the following theorem.
Moreover, if u 1 and u 2 are two solutions of (1.1), we have that for some suitable C(T ) > 0, and every 0 ≤ t ≤ T .
Observe that Theorem 1.1 holds also when τ = δ = 0, which corresponds the Kuramoto-Sivashinsky equation. Moreover, even if the equation is of the fourth order, the proof of Theorem 1.1 is based on the Aubin-Lions Lemma due to the functional setting (see [26,29,30,75]). The paper is organized as follows. In Sect. 2, we prove several a priori estimates on a vanishing viscosity approximation of (1.1). Those play a key role in the proof of our main result, that is given in Sect. 3.

Vanishing viscosity approximation
Our existence argument is based on passing to the limit in a vanishing viscosity approximation of (1.1).
Fix a small number 0 < ε < 1 and let u ε = u ε (t, x) be the unique classical solution of the following problem [24,25]: x ∈ R, where C 0 is a positive constant, independent on ε.
Let us prove some a priori estimates on u ε . 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 begin by proving the following result.
Proof Let 0 ≤ t ≤ T . We begin by observing that Multiplying (2.1) by 2u ε , thanks to (2.5), an integration on R gives Therefore, we have that Due to the Young inequality, Consequently, by (2.6), we have that Observe that Therefore, by the Young inequality, . By (2.2) and the the Gronwall Lemma, we get .

Proof of Theorem 1.1
This section is devoted to the proof of Theorem 1.1. We begin by proving the following lemma.
Funding Open access funding provided by Politecnico di Bari within the CRUI-CARE Agreement.
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