scholarly journals On the Cauchy problem of a sixth-order Cahn–Hilliard equation arising in oil-water-surfactant mixtures

2020 ◽  
pp. 1-24 ◽  
Author(s):  
Xiaopeng Zhao
Keyword(s):  
Cauchy Problem ◽  
Sixth Order ◽  
Surfactant Mixtures ◽  
Hilliard Equation ◽  
The Cauchy Problem ◽  
Oil Water ◽  
Cahn Hilliard Equation

scholarly journals A Class of Sixth Order Viscous Cahn-Hilliard Equation with Willmore Regularization in ℝ3

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1865
Author(s):  
Xiaopeng Zhao ◽  
Ning Duan
Keyword(s):  
Cauchy Problem ◽  
Splitting Method ◽  
Linear Term ◽  
Sixth Order ◽  
Well Posedness ◽  
Time Behavior ◽  
Order Linear ◽  
Hilliard Equation ◽  
The Cauchy Problem ◽  
Cahn Hilliard Equation

The main purpose of this paper is to study the Cauchy problem of sixth order viscous Cahn–Hilliard equation with Willmore regularization. Because of the existence of the nonlinear Willmore regularization and complex structures, it is difficult to obtain the suitable a priori estimates in order to prove the well-posedness results, and the large time behavior of solutions cannot be shown using the usual Fourier splitting method. In order to overcome the above two difficulties, we borrow a fourth-order linear term and a second-order linear term from the related term, rewrite the equation in a new form, and introduce the negative Sobolev norm estimates. Subsequently, we investigate the local well-posedness, global well-posedness, and decay rate of strong solutions for the Cauchy problem of such an equation in R3, respectively.

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

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1835
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Type Equation ◽  
Higher Order ◽  
Crystal Surfaces ◽  
Hilliard Equation ◽  
The Cauchy Problem ◽  
The Mean ◽  
Oil Water ◽  
Cahn Hilliard Equation ◽  
Well Posed ◽  
Dynamics Of Phase Transitions

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.

Global well-posedness and temporal decay estimates to the fractional Cahn–Hilliard equation in ℝ N \mathbb{R}^{N}

2019 ◽  
Vol 31 (3) ◽  
pp. 803-814
Author(s):  
Ning Duan ◽  
Xiaopeng Zhao
Keyword(s):  
Cauchy Problem ◽  
Decay Rate ◽  
Weak Solutions ◽  
Well Posedness ◽  
Decay Estimates ◽  
Hilliard Equation ◽  
Temporal Decay ◽  
The Cauchy Problem ◽  
Higher Order Derivative ◽  
Cahn Hilliard Equation

AbstractThis paper is devoted to study the global well-posedness of solutions for the Cauchy problem of the fractional Cahn–Hilliard equation in{\mathbb{R}^{N}}({N\in\mathbb{N}^{+}}), provided that the initial datum is sufficiently small. In addition, the{L^{p}}-norm ({1\leq p\leq\infty}) temporal decay rate for weak solutions and the higher-order derivative of solutions are also studied.

scholarly journals $$H^1$$ solutions for a Kuramoto–Sinelshchikov–Cahn–Hilliard type equation

2021 ◽  
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Surface Tension ◽  
Cauchy Problem ◽  
Type Equation ◽  
Well Posedness ◽  
Crystal Surfaces ◽  
Spatiotemporal Evolution ◽  
Hilliard Equation ◽  
Anisotropic Surface ◽  
The Cauchy Problem ◽  
Cahn Hilliard Equation

AbstractThe 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.

scholarly journals Weak Solutions for a Sixth Order Cahn-Hilliard Type Equation with Degenerate Mobility

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Aibo Liu ◽  
Changchun Liu
Keyword(s):  
Existence Result ◽  
Type Equation ◽  
Initial Boundary ◽  
Sixth Order ◽  
Surfactant Mixtures ◽  
Initial Boundary Problem ◽  
Degenerate Mobility ◽  
Oil Water ◽  
Three Space ◽  
Diffusional Mobility

We study an initial-boundary problem for a sixth order Cahn-Hilliard type equation, which arises in oil-water-surfactant mixtures. An existence result for the problem with a concentration dependent diffusional mobility in three space dimensions is presented.

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