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 1Department of Mathematics, Zhejiang Normal University, Jinhua 321004, People’s Republic of China 2Department 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]:

(1.1)

∂tu + (u · ∇)u + ∇π − ∆u = µ∇φ + θe2, div u = 0, (1.2)

(1.3)

(1.4)

∂tθ + u · ∇θ = (cid:178)∆θ, ∂tφ + u · ∇φ = ∆µ, −∆φ + f (cid:48)(φ) = µ, (1.5)

u = 0, θ = 0, = = 0 on ∂Ω × (0, ∞), (1.6) ∂φ ∂n ∂µ ∂n (1.7) (u, θ, φ)(x, 0) = (u0, θ0, φ0)(x), x ∈ Ω,

where u, π, θ and φ denote unknown velocity field, pressure scalar, temperature of the fluid and the order parameter, respectively. (cid:178) > 0 is the heat conductivity coefficient and e2 := (0, 1)t. µ is a chemical potential and f (φ) := 1 4 (φ2 − 1)2 is the double well potential.

∞,∞). Here ˙B0

. = curlu ∈ L1(0, T ; ˙B0

∂n = ∂µ0

∂n = 0

0 ∩ H 2, φ0 ∈ H 4, div u0 = 0 in Ω and ∂φ0 Theorem 1.1. Let (u0, θ0) ∈ H 1 on ∂Ω. Then, there exists a positive constant C independent of (cid:178) such that

When φ = 0, (1.1), (1.2) and (1.3) is the well-known Boussinesq system. In [2] Zhou and Fan proved a regularity criterion ω ∞,∞) 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 ; ˙B0 ∞,∞ denotes the homogeneous Besov space. Chae [4] studied the vanishing viscosity limit (cid:178) → 0 when Ω = R2. The aim of this article is to prove a similar result. We will prove that

(1.8) (cid:107)u(cid:178)(cid:107)L∞(0,T ;H 2) ≤ C, (cid:107)θ(cid:178)(cid:107)L∞(0,T ;H 2) ≤ C, (cid:107)φ(cid:178)(cid:107)L∞(0,T ;H 4) ≤ C, (cid:107)∂t(u(cid:178), θ(cid:178), φ(cid:178))(cid:107)L2(0,T ;L2) ≤ C,

for any T > 0, which implies

(1.9) (u(cid:178), θ(cid:178), φ(cid:178)) → (u, θ, φ) strongly in L2(0, T ; H 1) when (cid:178) → 0.

Here, (u, θ, φ) is the solution of the problem (1.1)–(1.7) with (cid:178) = 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 (cid:178) and throughout this section C will be a constant independent of (cid:178).

First, by the maximum principle, it follows from (1.2), (1.3), and (1.6) that

(2.1) (cid:107)θ(cid:107)L∞(0,T ;L∞) ≤ (cid:107)θ0(cid:107)L∞ ≤ C.

2

Testing (1.3) by θ, using (1.2) and (1.6), we see that

(cid:90) (cid:90)

θ2dx + (cid:178) |∇θ|2dx = 0, 1 2 d dt

whence √ (2.2) (cid:178)(cid:107)θ(cid:107)L2(0,T ;H 1) ≤ C.

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

(cid:90) (cid:90)

u2 + |∇φ|2 + f (φ)dx + |∇u|2 + |∇µ|2dx 1 2 1 2 d dt (cid:90)

= θe2udx ≤ (cid:107)θ(cid:107)L2(cid:107)u(cid:107)L2 ≤ C(cid:107)u(cid:107)L2,

which gives

(2.3)

(2.4)

(2.5) (cid:107)φ(cid:107)L∞(0,T ;H 1) ≤ C, (cid:107)u(cid:107)L∞(0,T ;L2) + (cid:107)u(cid:107)L2(0,T ;H 1) ≤ C, (cid:107)∇µ(cid:107)L2(0,T ;L2) ≤ C.

Testing (1.4) by φ, using (1.2), (1.5) and (1.6), we infer that

(cid:90) (cid:90) (cid:90)

φ2dx + |∆φ|2dx = (φ3 − φ)∆φdx 1 2 d dt (cid:90) (cid:90) (cid:90)

φ∆φdx ≤ − φ∆φdx = −3 (cid:90) φ2|∇φ|2dx − (cid:90)

|∆φ|2dx + φ2dx, ≤ 1 2 1 2

which leads to

(2.6) (cid:107)φ(cid:107)L2(0,T ;H 2) ≤ C.

We will use the following Gagliardo-Nirenberg inequality:

L∞ ≤ C(cid:107)φ(cid:107)L6(cid:107)φ(cid:107)H 2.

(cid:107)φ(cid:107)2 (2.7)

3

It follows from (2.6), (2.7), (2.5), (2.3) and (1.5) that

(cid:90) (cid:90) T |∇∆φ|2dxdt

0 (cid:90) T

(cid:90)

0

=

0

0

(cid:90) |∇(f (cid:48)(φ) − µ)|2dxdt (cid:90) T (cid:90) (cid:90) T ≤ C |∇µ|2dxdt + C |∇(φ3 − φ)|2dxdt

0

(cid:90) (cid:90) T ≤ C + C

L∞dt

L∞(0,T ;L2)

0

φ4|∇φ|2dxdt (cid:90) T ≤ C + C(cid:107)∇φ(cid:107)2 (cid:107)φ(cid:107)4

L6(cid:107)φ(cid:107)2

0

H 2dt (cid:90) T

(cid:90) T ≤ C + C (cid:107)φ(cid:107)2

H 2dt ≤ C,

L∞(0,T ;H 1)

0

(cid:107)φ(cid:107)2 ≤ C + C(cid:107)φ(cid:107)2 (2.8)

which yields

(2.9)

(2.10)

(2.11) (cid:107)φ(cid:107)L2(0,T ;H 3) ≤ C, (cid:107)φ(cid:107)L4(0,T ;L∞) ≤ C, (cid:107)∇φ(cid:107)L2(0,T ;L∞) ≤ C.

Testing (1.4) by ∆2φ, using (1.5), (2.4), (2.3), (2.10) and (2.11), we derive

(cid:90) (cid:90)

|∆φ|2dx + |∆2φ|2dx 1 2 d dt (cid:90) (cid:90)

= − u · ∇φ · ∆2φdx + ∆(φ3 − φ) · ∆2φdx

≤ (cid:107)u(cid:107)L2(cid:107)∇φ(cid:107)L∞(cid:107)∆2φ(cid:107)L2 + (cid:107)∆(φ3 − φ)(cid:107)L2(cid:107)∆2φ(cid:107)L2 ≤ C(cid:107)∇φ(cid:107)L∞(cid:107)∆2φ(cid:107)L2

L∞(cid:107)∆φ(cid:107)L2 + (cid:107)φ(cid:107)L∞(cid:107)∇φ(cid:107)L∞(cid:107)∇φ(cid:107)L2 + (cid:107)∆φ(cid:107)L2)(cid:107)∆2φ(cid:107)L2

+C((cid:107)φ(cid:107)2

≤ C(cid:107)∇φ(cid:107)L∞(cid:107)∆2φ(cid:107)L2

L∞ + C(cid:107)φ(cid:107)4

L∞(cid:107)∆φ(cid:107)2 L2

≤ (cid:107)∆2φ(cid:107)2

L∞(cid:107)∆φ(cid:107)L2 + (cid:107)φ(cid:107)H 2(cid:107)∇φ(cid:107)L∞ + (cid:107)∆φ(cid:107)L2)(cid:107)∆2φ(cid:107)L2 L2 + C(cid:107)∇φ(cid:107)2 L∞(cid:107)φ(cid:107)2

L2,

H 2 + C(cid:107)∆φ(cid:107)2

+C((cid:107)φ(cid:107)2 1 2 +C(cid:107)∇φ(cid:107)2

which implies

(2.12) (cid:107)φ(cid:107)L∞(0,T ;H 2) + (cid:107)φ(cid:107)L2(0,T ;H 4) ≤ C.

4

Testing (1.1) by −∆u + ∇π, using (1.2), (1.6), (2.12), (2.1) and (2.4), we reach

(cid:90) (cid:90)

|∇u|2dx + (−∆u + ∇π)2dx d dt 1 2 (cid:90)

L2 )(cid:107) − ∆u + ∇π(cid:107)L2

= (µ∇φ + θe2 − u · ∇u)(−∆u + ∇π)dx

L∞ + C + C(cid:107)∇u(cid:107)4

L2 (cid:107)∆u(cid:107)1/2 L2,

(cid:107) − ∆u + ∇π(cid:107)2 ≤ C(cid:107)∇φ(cid:107)2 ≤ ((cid:107)µ(cid:107)L2(cid:107)∇φ(cid:107)L∞ + (cid:107)θ(cid:107)L2 + (cid:107)u(cid:107)L4(cid:107)∇u(cid:107)L4)(cid:107) − ∆u + ∇π(cid:107)L2 L2 · (cid:107)∇u(cid:107)1/2 L2 (cid:107)∇u(cid:107)1/2 ≤ C((cid:107)∇φ(cid:107)L∞ + 1 + (cid:107)u(cid:107)1/2 1 L2 + 2

which yields

(2.13) (cid:107)u(cid:107)L∞(0,T ;H 1) + (cid:107)u(cid:107)L2(0,T ;H 2) ≤ C.

L4 ≤ C(cid:107)u(cid:107)L2(cid:107)∇u(cid:107)L2,

Here, we have used the Gagliardo-Nirenberg inequalities:

L4 ≤ C(cid:107)∇u(cid:107)L2(cid:107)u(cid:107)H 2,

(cid:107)u(cid:107)2 (cid:107)∇u(cid:107)2

and the H 2-theory of the Stokes system:

(2.14) (cid:107)u(cid:107)H 2 + (cid:107)π(cid:107)H 1 ≤ C(cid:107) − ∆u + ∇π(cid:107)L2.

Similarly to (2.13), we have

(2.15) (cid:107)∂tu(cid:107)L2(0,T ;L2) ≤ C.

(1.1), (1.2), (1.6) and (1.7) can be rewritten as

in Ω × (0, ∞),  

 ∂tu − ∆u + ∇π = g := µ∇φ + θe2 − u · ∇u, 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

(cid:107)∂tu(cid:107)L2(0,T ;Lp) + (cid:107)u(cid:107)L2(0,T ;W 2,p) ≤ C(cid:107)g(cid:107)L2(0,T ;Lp) ≤ C(cid:107)µ(cid:107)L2(0,T ;L∞)(cid:107)∇φ(cid:107)L∞(0,T ;Lp) + C(cid:107)θ(cid:107)L∞(0,T ;L∞)

(2.16) +C(cid:107)u(cid:107)L∞(0,T ;L2p)(cid:107)∇u(cid:107)L2(0,T ;L2p) ≤ C,

for any 2 < p < ∞. (2.16) gives

(2.17) (cid:107)∇u(cid:107)L2(0,T ;L∞) ≤ C.

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 (cid:90) (cid:90)

|∆θ|2dx + (cid:178) |∇∆θ|2dx 1 2 d dt (cid:90)

= − (∆(u · ∇θ) − u∇∆θ)∆θdx

L2,

≤ C((cid:107)∆u(cid:107)L4(cid:107)∇θ(cid:107)L4 + (cid:107)∇u(cid:107)L∞(cid:107)∆θ(cid:107)L2)(cid:107)∆θ(cid:107)L2 ≤ C((cid:107)∆u(cid:107)L4 + (cid:107)∇u(cid:107)L∞)(cid:107)∆θ(cid:107)2

which implies √ (2.19) (cid:107)θ(cid:107)L∞(0,T ;H 2) + (cid:178)(cid:107)θ(cid:107)L2(0,T ;H 3) ≤ C.

It follows from (1.3), (1.6), (2.19) and (2.13) that

(2.20) (cid:107)∂tθ(cid:107)L∞(0,T ;L2) ≤ C.

Taking ∂t to (1.4) and (1.5), testing by ∂tφ, using (1.2), (1.6), (2.12), and (2.15), we have (cid:90) (cid:90)

|∂tφ|2dx + |∆∂tφ|2dx 1 2 d dt (cid:90) (cid:90)

= − ∂tu · ∇φ · ∂tφdx + ∆(3φ2∂tφ − ∂tφ) · ∂tφdx (cid:90) (cid:90)

= − ∂tu · ∇φ · ∂tφdx + (3φ2∂tφ − ∂tφ)∆∂tφdx

L∞ + 1)(cid:107)∂tφ(cid:107)L2(cid:107)∆∂tφ(cid:107)L2

≤ (cid:107)∂tu(cid:107)L2(cid:107)∇φ(cid:107)L∞(cid:107)∂tφ(cid:107)L2 + ((cid:107)3φ(cid:107)2

L2 + C(cid:107)∂tφ(cid:107)2

L2,

(cid:107)∆∂tφ(cid:107)2 ≤ (cid:107)∂tu(cid:107)L2(cid:107)∇φ(cid:107)L∞(cid:107)∂tφ(cid:107)L2 + 1 2

which gives

(2.21) (cid:107)∂tφ(cid:107)L∞(0,T ;L2) + (cid:107)∂tφ(cid:107)L2(0,T ;H 2) ≤ C.

By the regularity theory of elliptic equation, it follows from (1.4), (1.5), (1.6),

(2.21), (2.13) and (2.12) that

(cid:107)φ(cid:107)L∞(0,T ;H 4) ≤ C(cid:107)∆φ(cid:107)L∞(0,T ;H 2) ≤ C(cid:107)µ − f (cid:48)(φ)(cid:107)L∞(0,T ;H 2)

≤ C(cid:107)µ(cid:107)L∞(0,T ;H 2) + C(cid:107)f (cid:48)(φ)(cid:107)L∞(0,T ;H 2) ≤ C(cid:107)∆µ(cid:107)L∞(0,T ;L2) + C(cid:107)f (cid:48)(φ)(cid:107)L∞(0,T ;H 2) ≤ C(cid:107)∂tφ + u · ∇φ(cid:107)L∞(0,T ;L2) + C(cid:107)f (cid:48)(φ)(cid:107)L∞(0,T ;H 2) ≤ C(cid:107)∂tφ(cid:107)L∞(0,T ;L2) + C(cid:107)u(cid:107)L∞(0,T ;L4)(cid:107)∇φ(cid:107)L∞(0,T ;L4)

(2.22) +C(cid:107)f (cid:48)(φ)(cid:107)L∞(0,T ;H 2) ≤ C.

6

Taking ∂t to (1.1), testing by ∂tu, using (1.2), (1.6), (2.17), (2.22), (2.21) and (1.5), we conclude that (cid:90) (cid:90)

|∂tu|2dx + |∇∂tu|2dx 1 2 d dt (cid:90) (cid:90)

= − ∂tu · ∇u · ∂tudx + (∂tµ · ∇φ + µ · ∇∂tφ + ∂tθe2)∂tudx

L2 + ((cid:107)∂tu(cid:107)L2(cid:107)∇φ(cid:107)L∞ + (cid:107)µ(cid:107)L∞(cid:107)∇∂tφ(cid:107)L2 + (cid:107)∂tθ(cid:107)L2)(cid:107)∂tu(cid:107)L2 L2 + C((cid:107)∆∂tφ(cid:107)L2 + (cid:107)∂t(φ3 − φ)(cid:107)L2 + (cid:107)∇∂tφ(cid:107)L2 + 1)(cid:107)∂tu(cid:107)L2,

≤ (cid:107)∇u(cid:107)L∞(cid:107)∂tu(cid:107)2 ≤ (cid:107)∇u(cid:107)L∞(cid:107)∂tu(cid:107)2

which implies

(2.23) (cid:107)∂tu(cid:107)L∞(0,T ;L2) + (cid:107)∂tu(cid:107)L2(0,T ;H 1) ≤ C.

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

(cid:107)u(cid:107)L∞(0,T ;H 2) ≤ C.

This completes the proof.

(cid:164)

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Acknowledgment

11171154) and NSFC (Grant No.

This study was supported by the NSFC (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