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.

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 Well-posedness of the classical solution for the Kuramto–Sivashinsky equation with anisotropy effects

2021 ◽  
Vol 72 (2) ◽  
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Surface Tension ◽  
Classical Solutions ◽  
Well Posedness ◽  
Crystal Surfaces ◽  
Spatiotemporal Evolution ◽  
Hilliard Equation ◽  
Anisotropic Surface ◽  
The Cauchy Problem ◽  
Cahn Hilliard Equation ◽  
Sivashinsky Equation

AbstractThe Kuramto–Sivashinsky equation with anisotropy effects 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. Written in terms of the step slope, it can be represented in a form similar to a convective Cahn–Hilliard equation. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem, associated with this equation.

Cauchy Problem for Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

2003 ◽  
Vol 8 (1) ◽  
pp. 61-75
Author(s):  
V. Litovchenko
Keyword(s):  
Functional Analysis ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Initial Function ◽  
Well Posedness ◽  
Fractional Differentiation ◽  
Basic Tool ◽  
Conjugate Space ◽  
Analysis Technique ◽  
The Cauchy Problem

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.

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