On the Well-Posedness of A High Order Convective Cahn-Hilliard Type Equations

Giuseppe Maria Coclite,Lorenzo Di Ruvo

doi:10.3390/a13070170

Giuseppe Maria Coclite, Lorenzo Di Ruvo

Open Access

https://doi.org/10.3390/a13070170

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Abstract

High order convective Cahn-Hilliard type equations describe the faceting of a growing surface, or the dynamics of phase transitions in ternary oil-water-surfactant systems. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem, associated with this equation.

Highlights

We investigate the well-posedness of the following Cauchy problem: (
We considered the high order convective Cahn-Hilliard type equations that describe the faceting of a growing surface, or the dynamics of phase transitions in ternary oil-water-surfactant systems
The main result of this paper is the following theorem

Summary

Introduction

We investigate the well-posedness of the following Cauchy problem:. ∂t u + κ∂ x u2 − β2 ∂6x u + α∂4x u + δ2 ∂4x u3 = 0, t > 0, u(0, x ) = u0 ( x ), x ∈ R, x ∈ R,. Algorithms 2020, 13, 170 dependent regularization, and all other terms represent the anisotropy of the surface energy under surface diffusion These equations became popular objects of theoretical studies in the last decade, because they do play an important role in material modeling. The existence and uniqueness of weak solutions of (1) with periodic boundary conditions is proven in [23], with κ > 0. In [31] the existence of global attractors is analyzed and in [32] the existence of weak solutions for the initial -boundary-value problem for (6) with degenerate mobility is proven. The argument of Theorem 1 relies on deriving suitable a priori estimates together with an application of the Fixed Point. Standard arguments and the Fixed Point Theorem would give us the local in time existence of analytic solutions.

A Priori Estimates

Proof of Theorem 1

Findings

Conclusions