The Cauchy problem for the degenerate convective Cahn-Hilliard equation
type: Item
instance of: scholarly article
title: The Cauchy problem for the degenerate convective Cahn-Hilliard equation
zbMATH Open document ID: 1407.35099
DOI: 10.1216/RMJ-2018-48-8-2595
author: Aibo Liu
author: Chein-Shan Liu
published in: Rocky Mountain Journal of Mathematics
publication date: 2019-01-10T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmjm/1546138823
review text: The authors investigate the Cauchy problem associated with the following degenerate convective Cahn-Hilliard equation \[ \begin{cases} u_t+\Delta_{x'}^2u-\Delta_{x'}A(u)-{\mathbf r}\cdot \nabla B(u)=0, \;\;x'=(x_2,\dots,x_n)\in \mathbb{R}^{n-1}, \;t>0,& \\ u(x,0)=u_0(x),& \end{cases} \] where \(A=A(u)\), \(B=B(u)\) are sufficiently smooth functions, \({\mathbf r}=(r_1,\dots,r_n) \in \mathbb{R}^n\), \(\nabla =(\partial_{x_1},\dots, \partial_{x_n})\), \(\Delta_{x'}=\sum_{i=2}^n\partial_{x_i}^2\). In addition, \(A(u)={\mathcal{O}}(|u|^{\theta +1})\), \(B(u)={\mathcal{O}}(|u|^{\theta +1})\) with the same growth property and \(\theta \geq 1\) is an integer. They prove existence of a unique global classical solution for small \(u_0\) by establishing uniform estimates of the solution. Decay estimates are also addressed.
Mathematics Subject Classification ID: 35K25
Mathematics Subject Classification ID: 35A09
Mathematics Subject Classification ID: 35K59
zbMATH DE Number: 6999276
zbMATH Keywords: global classical solution
zbMATH Keywords: a priori decay estimates
zbMATH Keywords: long-short wave method
zbMATH Keywords: frequency decomposition method
reviewed by: Gheorghe Moroşanu
MaRDI profile type: Publication
cites work: Q5318420
cites work: Global classical solutions to the Cauchy problem of conservation laws with degenerate diffusion
cites work: Global existence of solutions to the initial-boundary value problem of conservation law with degenerate diffusion term
cites work: A note on the optimal temporal decay estimates of solutions to the Cahn-Hilliard equation
cites work: The convective Cahn-Hilliard equation
cites work: 3D convective Cahn-Hilliard equation
cites work: Existence of solutions for two types of generalized versions of the Cahn-Hilliard equation.
cites work: Q4003851
cites work: On the Convective Cahn–Hilliard Equation
cites work: On the convective Cahn-Hilliard equation with degenerate mobility
cites work: Optimal decay rate of solutions for Cahn-Hilliard equation with inertial term in multi-dimensions
cites work: Periodic Stationary Patterns Governed by a Convective Cahn--Hilliard Equation
cites work: Optimal control for the convective Cahn-Hilliard equation in 2D case
OpenAlex ID: W2907447045
Wikidata QID: Q128688719
description: scientific article; zbMATH DE number 6999276