Báo cáo sinh học: Vanishing heat conductivity limit for the 2D Cahn-Hilliard-Boussinesq system

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Báo cáo sinh học: " Vanishing heat conductivity limit for the 2D Cahn-Hilliard-Boussinesq system"Boundary Value Problems This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Vanishing heat conductivity limit for the 2D Cahn-Hilliard-Boussinesq system Boundary Value Problems 2011, 2011:54 doi:10.1186/1687-2770-2011-54 Zaihong Jiang (jzhong@zjnu.cn) Jishan Fan (fanjishan@njfu.com.cn) ISSN 1687-2770 Article type Research Submission date 18 October 2011 Acceptance date 22 December 2011 Publication date 22 December 2011 Article URL http://www.boundaryvalueproblems.com/content/2011/1/54 This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). For information about publishing your research in Boundary Value Problems go to http://www.boundaryvalueproblems.com/authors/instructions/ For information about other SpringerOpen publications go to http://www.springeropen.com © 2011 Jiang and Fan ; licensee Springer.This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Vanishing heat conductivity limit for the 2D Cahn-Hilliard-Boussinesq system Zaihong Jiang∗1 and Jishan Fan2 1 Department of Mathematics, Zhejiang Normal University, Jinhua 321004, People’s Republic of China 2 Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, People’s Republic of China ∗ Corresponding author: jzhong@zjnu.cn Email address: fanjishan@njfu.com.cn Abstract This article studies the vanishing heat conductivity limit for the 2D Cahn- Hilliard-boussinesq system in a bounded domain with non-slip boundary condi- tion. The result has been proved globally in time. 2010 MSC: 35Q30; 76D03; 76D05; 76D07. Keywords: Cahn–Hilliard–Boussinesq; inviscid limit; non-slip boundary con- dition.1 IntroductionLet Ω ⊆ R2 be a bounded, simply connected domain with smooth boundary ∂ Ω, andn is the unit outward normal vector to ∂ Ω. We consider the following Cahn-Hilliard- 1Boussinesq system in Ω × (0, ∞) [1]: ∂t u + ( u · )u + π − ∆u = µ φ + θe2 , (1.1) div u = 0, (1.2) ∂t θ + u · θ = ∆θ, (1.3) ∂t φ + u · φ = ∆µ, (1.4) −∆φ + f (φ) = µ, (1.5) ∂φ ∂µ u = 0, θ = 0 , = = 0 on ∂ Ω × (0, ∞), (1.6) ∂n ∂n (u, θ, φ)(x, 0) = (u0 , θ0 , φ0 )(x), x ∈ Ω, (1.7)where u, π, θ and φ denote unknown velocity field, pressure scalar, temperature of thefluid and the order parameter, respectively. > 0 is the heat conductivity coefficient 1and e2 := (0, 1)t . µ is a chemical potential and f (φ) := 4 (φ2 − 1)2 is the double wellpotential. When φ = 0, (1.1), (1.2) and (1.3) is the well-known Boussinesq system. In [2] . ˙0Zhou and Fan proved a regularity criterion ω = curlu ∈ L1 (0, T ; B∞,∞ ) for the 3DBoussinesq system with partial viscosity. Later, in [3] Zhou and Fan studied theCauchy problem of certain Boussinesq−α equations in n dimensions with n = 2 or ...