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

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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 Introduction Let Ω ⊆ R2 be a bounded, simply connected domain with smooth boundary ∂ Ω, and n is the unit outward normal vector to ∂ Ω. We consider the following Cahn-Hilliard- 1
Boussinesq 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 the ﬂuid and the order parameter, respectively. > 0 is the heat conductivity coeﬃcient 1 and e2 := (0, 1)t . µ is a chemical potential and f (φ) := 4 (φ2 − 1)2 is the double well potential. When φ = 0, (1.1), (1.2) and (1.3) is the well-known Boussinesq system. In [2] . ˙0 Zhou and Fan proved a regularity criterion ω = curlu ∈ L1 (0, T ; B∞,∞ ) for the 3D Boussinesq system with partial viscosity. Later, in [3] Zhou and Fan studied the Cauchy problem of certain Boussinesq−α equations in n dimensions with n = 2 or ˙ ˙ 3. We establish regularity for the solution under u ∈ L1 (0, T ; B 0 ). Here B 0 ∞,∞ ∞,∞ denotes the homogeneous Besov space. Chae [4] studied the vanishing viscosity limit → 0 when Ω = R2 . The aim of this article is to prove a similar result. We will prove that Theorem 1.1. Let (u0 , θ0 ) ∈ H0 ∩ H 2 , φ0 ∈ H 4 , div u0 = 0 in Ω and ∂φ0 = ∂µ0 1 =0 ∂n ∂n on ∂ Ω. Then, there exists a positive constant C independent of such that u ≤ C, θ L∞ (0,T ;H 2 ) ≤ C, L∞ (0,T ;H 2 ) (1.8) φ L∞ (0,T ;H 4 ) ≤ C, ∂t (u , θ , φ ) L2 (0,T ;L2 ) ≤ C, for any T > 0, which implies (u , θ , φ ) → (u, θ, φ) strongly in L2 (0, T ; H 1 ) when → 0. (1.9) Here, (u, θ, φ) is the solution of the problem (1.1)–(1.7) with = 0. 2 Proof of Theorem 1.1 Since (1.9) follows easily from (1.8) by the Aubin-Lions compactness principle, we only need to prove the a priori estimates (1.8). From now on, we will drop the subscript and throughout this section C will be a constant independent of . First, by the maximum principle, it follows from (1.2), (1.3), and (1.6) that θ ≤ θ0 ≤ C. (2.1) L∞ L∞ (0,T ;L∞ ) 2
Testing (1.3) by θ, using (1.2) and (1.6), we see that 1d θ2 dx + | θ |2 d x = 0 , 2 dt whence √ θ ≤ C. (2.2) L2 (0,T ;H 1 ) Testing (1.1) and (1.4) by u and µ, respectively, using (1.2), (1.6), (2.1), and summing up the result, we ﬁnd that d 12 1 u + | φ|2 + f (φ)dx + | u|2 + | µ|2 dx dt 2 2 = θe2 udx ≤ θ u ≤C u L2 , L2 L2 which gives φ ≤ C, (2.3) L∞ (0,T ;H 1 ) u +u ≤ C, (2.4) L∞ (0,T ;L2 ) L2 (0,T ;H 1 ) µ ≤ C. (2.5) L2 (0,T ;L2 ) Testing (1.4) by φ, using (1.2), (1.5) and (1.6), we infer that 1d φ2 dx + |∆φ|2 dx = (φ3 − φ)∆φdx 2 dt φ2 | φ|2 dx − = −3 φ∆φdx ≤ − φ∆φdx 1 1 |∆φ|2 dx + φ2 dx, ≤ 2 2 which leads to φ ≤ C. (2.6) L2 (0,T ;H 2 ) We will use the following Gagliardo-Nirenberg inequality: 2 φ ≤C φ φ H2 . (2.7) L6 L∞ 3
It follows from (2.6), (2.7), (2.5), (2.3) and (1.5) that T | ∆φ|2 dxdt 0 T | (f (φ) − µ)|2 dxdt = 0 T T | µ|2 dxdt + C | (φ3 − φ)|2 dxdt ≤C 0 0 T φ4 | φ|2 dxdt ≤C +C 0 T 2 4 ≤C +C φ φ L∞ dt L∞ (0,T ;L2 ) 0 T 2 2 ≤C +C φ φ H 2 dt L6 0 T 2 2 ≤C +C φ φ H 2 dt ≤ C, (2.8) L∞ (0,T ;H 1 ) 0 which yields φ ≤ C, (2.9) L2 (0,T ;H 3 ) φ ≤ C, (2.10) L4 (0,T ;L∞ ) φ ≤ C. (2.11) L2 (0,T ;L∞ ) Testing (1.4) by ∆2 φ, using (1.5), (2.4), (2.3), (2.10) and (2.11), we derive 1d |∆φ|2 dx + |∆2 φ|2 dx 2 dt φ · ∆2 φdx + ∆(φ3 − φ) · ∆2 φdx =− u· ∆2 φ + ∆(φ3 − φ) ∆2 φ ≤u φ L∞ L2 L2 L2 L2 2 ≤C φ ∆φ L∞ L2 2 ∆2 φ +C ( φ ∆φ +φ φ φ + ∆φ L2 ) L∞ L∞ L2 L2 L2 L∞ ∆2 φ ≤C φ L∞ L2 2 ∆2 φ +C ( φ ∆φ L2 + φ H 2 φ L∞ + ∆φ L2 ) L2 L∞ 122 ≤ ∆ φ L2 + C φ 2 ∞ + C φ 4 ∞ ∆φ 2 2 L L L 2 +C φ 2 ∞ φ 2 2 + C ∆φ 2 2 , L H L which implies φ +φ ≤ C. (2.12) L∞ (0,T ;H 2 ) L2 (0,T ;H 4 ) 4
Testing (1.1) by −∆u + π , using (1.2), (1.6), (2.12), (2.1) and (2.4), we reach 1d | u|2 d x + π )2 dx (−∆u + 2 dt = (µ φ + θe2 − u · u)(−∆u + π )dx ≤( µ φ +θ L2 + u u L4 ) − ∆u + π L∞ L2 L4 L2 1/2 1/2 1/2 1/2 ≤ C( φ +1+ u u · u L2 ∆u L2 ) − ∆u + π L∞ L2 L2 L2 1 2 4 2 ≤C φ +C +C u + − ∆u + π L2 , L∞ L2 2 which yields u +u ≤ C. (2.13) L∞ (0,T ;H 1 ) L2 (0,T ;H 2 ) Here, we have used the Gagliardo-Nirenberg inequalities: 2 u ≤C u u L2 , L2 L4 2 u ≤C u u H2 , L2 L4 and the H 2 -theory of the Stokes system: u +π ≤ C − ∆u + π L2 . (2.14) H2 H1 Similarly to (2.13), we have ∂t u ≤ C. (2.15) L2 (0,T ;L2 ) (1.1), (1.2), (1.6) and (1.7) can be rewritten as   ∂t u − ∆u + π = g := µ φ + θe2 − u · u, in Ω × (0, ∞),  u = 0, on ∂ Ω × (0, ∞),   u(x, 0) = u0 (x). Using (2.12), (2.1), (2.13), and the regularity theory of Stokes system, we have ∂t u +u ≤C g L2 (0,T ;Lp ) L2 (0,T ;W 2,p ) L2 (0,T ;Lp ) ≤C µ φ +C θ L∞ (0,T ;Lp ) L∞ (0,T ;L∞ ) L2 (0,T ;L∞ ) +C u u ≤ C, (2.16) L∞ (0,T ;L2p ) L2 (0,T ;L2p ) for any 2 < p < ∞. (2.16) gives u ≤ C. (2.17) L2 (0,T ;L∞ ) It follows from (1.3) and (1.6) that ∆θ = 0 on ∂ Ω × (0, ∞). (2.18) 5
Applying ∆ to (1.3), testing by ∆θ, using (1.2), (1.6), (2.16), (2.17) and (2.18), we obtain 1d |∆θ|2 dx + | ∆θ|2 dx 2 dt =− (∆(u · θ) − u ∆θ)∆θdx ≤ C ( ∆u θ + u ∆θ L2 ) ∆θ L∞ L4 L4 L2 2 ≤ C ( ∆u + u L∞ ) ∆θ L2 , L4 which implies √ θ + θ ≤ C. (2.19) L∞ (0,T ;H 2 ) L2 (0,T ;H 3 ) It follows from (1.3), (1.6), (2.19) and (2.13) that ∂t θ ≤ C. (2.20) L∞ (0,T ;L2 ) Taking ∂t to (1.4) and (1.5), testing by ∂t φ, using (1.2), (1.6), (2.12), and (2.15), we have 1d |∂t φ|2 dx + |∆∂t φ|2 dx 2 dt ∆(3φ2 ∂t φ − ∂t φ) · ∂t φdx =− ∂t u · φ · ∂t φdx + (3φ2 ∂t φ − ∂t φ)∆∂t φdx =− ∂t u · φ · ∂t φdx + + ( 3φ 2 ∞ + 1) ∂t φ L2 ∆∂t φ ≤ ∂t u φ ∂t φ L∞ L2 L2 L2 L 1 + ∆∂t φ 2 2 + C ∂t φ 2 2 , ≤ ∂t u φ ∂t φ L∞ L2 L2 L L 2 which gives ∂t φ + ∂t φ ≤ C. (2.21) L∞ (0,T ;L2 ) L2 (0,T ;H 2 ) By the regularity theory of elliptic equation, it follows from (1.4), (1.5), (1.6), (2.21), (2.13) and (2.12) that φ ≤ C ∆φ ≤ C µ − f (φ) L∞ (0,T ;H 4 ) L∞ (0,T ;H 2 ) L∞ (0,T ;H 2 ) ≤C µ + C f (φ) L∞ (0,T ;H 2 ) L∞ (0,T ;H 2 ) ≤ C ∆µ + C f (φ) L∞ (0,T ;L2 ) L∞ (0,T ;H 2 ) ≤ C ∂t φ + u · φ + C f (φ) L∞ (0,T ;L2 ) L∞ (0,T ;H 2 ) ≤ C ∂t φ +C u φ L∞ (0,T ;L2 ) L∞ (0,T ;L4 ) L∞ (0,T ;L4 ) +C f (φ) ≤ C. (2.22) L∞ (0,T ;H 2 ) 6
Taking ∂t to (1.1), testing by ∂t u, using (1.2), (1.6), (2.17), (2.22), (2.21) and (1.5), we conclude that 1d |∂t u|2 dx + | ∂t u|2 dx 2 dt =− ∂t u · u · ∂t udx + (∂t µ · φ+µ· ∂t φ + ∂t θe2 )∂t udx 2 ≤ u ∂t u + ( ∂t u φ +µ ∂t φ + ∂t θ L2 ) ∂t u L∞ L∞ L∞ L2 L2 L2 L2 2 3 ≤ u ∂t u + C ( ∆∂t φ + ∂t (φ − φ) + ∂t φ + 1) ∂t u L2 , L∞ L2 L2 L2 L2 which implies ∂t u + ∂t u ≤ C. (2.23) L∞ (0,T ;L2 ) L2 (0,T ;H 1 ) Using (2.23), (2.22), (2.1), (2.13), (1.1), (1.2), (1.6) and the H 2 -theory of the Stokes system, we arrive at u L∞ (0,T ;H 2 ) ≤ C. This completes the proof. Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors read and approved the ﬁnal manuscript. Acknowledgment This study was supported by the NSFC (No. 11171154) and NSFC (Grant No. 11101376). References [1] Boyer, Franck: Mathematical study of multi-phase ﬂow under shear through order parameter formulation. Asymptot. Anal. 20, 175–212 (1999) [2] Fan, Jishan; Zhou, Yong: A note on regularity criterion for the 3D Boussinesq system with partial viscosity. Appl. Math. Lett. 22, 802–C805 (2009) [3] Zhou, Yong; Fan, Jishan: On the Cauchy problems for certain Boussinesq-α equations. Proc. R. Soc. Edinburgh Sect. A 140, 319–C327 (2010) 7
[4] Chae, Dongho: Global regularity for the 2D Boussinesq equations with partial viscosity terms. Adv. Math. 203, 497–513 (2006). 8