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

Giuseppe Maria Coclite,Lorenzo Di Ruvo

doi:10.3390/math8101835

Giuseppe Maria Coclite, Lorenzo Di Ruvo

Open Access

https://doi.org/10.3390/math8101835

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Abstract

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 H3 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.

Highlights

In this paper, we study the well-posedness of the Cauchy problem:
We prove the global-in-time existence, uniqueness, and stability of the solutions of the Cauchy problem (1)
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)

Summary

Introduction

In [18], the well-posedness of the classical solution of the Cauchy problem of (10) is proven, assuming (7)–(9), with γ = 0. We prove the global-in-time existence, uniqueness, and stability of the solutions of the Cauchy problem (1). Multiplying (27), by −β2∂2xu + δ2u3 + b2u, arguing as in the previous case, an integration on R gives d dt β2 2 Due to the Young inequality, we can estimate the right-hand side of (33), as follows: 2δ2 |∂2xu||∂2x

Results

Conclusion