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# Vanishing heat conductivity limit for the 2D Cahn-Hilliard-Boussinesq system

Zaihong Jiang1* and Jishan Fan2

Author Affiliations

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

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Boundary Value Problems 2011, 2011:54  doi:10.1186/1687-2770-2011-54

 Received: 18 October 2011 Accepted: 22 December 2011 Published: 22 December 2011

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.

### Abstract

This article studies the vanishing heat conductivity limit for the 2D Cahn-Hilliard-boussinesq system in a bounded domain with non-slip boundary condition. The result has been proved globally in time.

2010 MSC: 35Q30; 76D03; 76D05; 76D07.

##### Keywords:
Cahn-Hilliard-Boussinesq; inviscid limit; non-slip boundary condition

### 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-Boussinesq system in Ω × (0, ∞) [1]:

t u + ( u ) u + π - Δ u = μ ϕ + θ e 2 , (1.1)

div u = 0 , (1.2)

t θ + u θ = ε Δ θ , (1.3)

t ϕ + u ϕ = Δ μ , (1.4)

- Δ ϕ + f ( ϕ ) = μ , (1.5)

u = 0 , θ = 0 , ϕ n = μ n = 0 o n Ω × ( 0 , ) , (1.6)

( u , θ , ϕ ) ( x , 0 ) = ( u 0 , θ 0 , ϕ 0 ) ( 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. ε > 0 is the heat conductivity coefficient and e2 : = (0, 1)t. μ is a chemical potential and f ( ϕ ) : = 1 4 ( ϕ 2 - 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] Zhou and Fan proved a regularity criterion ω = ˙ c u r l u L 1 ( 0 , T ; , 0 ) 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 L 1 ( 0 , T ; , 0 ) . Here , 0 denotes the homogeneous Besov space. Chae [4] studied the vanishing viscosity limit ε → 0 when Ω = ℝ2. The aim of this article is to prove a similar result. We will prove that

Theorem 1.1. Let ( u 0 , θ 0 ) H 0 1 H 2 , ϕ0 H4, div u0 = 0 in Ω and ϕ 0 n = μ 0 n = 0 on ∂Ω. Then, there exists a positive constant C independent of ε such that

u ε L ( 0 , T ; H 2 ) C , θ ε L ( 0 , T ; H 2 ) C , ϕ ε L ( 0 , T ; H 4 ) C , t ( u ϵ , θ ε , ϕ ϵ ) L 2 ( 0 , T ; L 2 ) C , (1.8)

for any T > 0, which implies

( u ε , θ ε , ϕ ε ) ( u , θ , ϕ ) s t r o n g l y i n L 2 ( 0 , T ; H 1 ) w h e n ε 0 . (1.9)

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

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

θ L ( 0 , T ; L ) θ 0 L C . (2.1)

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

1 2 d d t θ 2 d x + ε θ 2 d x = 0 ,

whence

ε θ L 2 ( 0 , T ; H 1 ) C . (2.2)

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 d t 1 2 u 2 + 1 2 ϕ 2 + f ( ϕ ) d x + u 2 + μ 2 d x = θ e 2 u d x θ L 2 u L 2 C u L 2 ,

which gives

ϕ L ( 0 , T ; H 1 ) C , (2.3)

u L ( 0 , T ; L 2 ) + u L 2 ( 0 , T ; H 1 ) C , (2.4)

μ L 2 ( 0 , T ; L 2 ) C . (2.5)

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

1 2 d d t ϕ 2 d x + Δ ϕ 2 d x = ( ϕ 3 - ϕ ) Δ ϕ d x = - 3 ϕ 2 ϕ 2 d x - ϕ Δ ϕ d x - ϕ Δ ϕ d x 1 2 Δ ϕ 2 d x + 1 2 ϕ 2 d x ,

ϕ L 2 ( 0 , T ; H 2 ) C . (2.6)

We will use the following Gagliardo-Nirenberg inequality:

ϕ L 2 C ϕ L 6 ϕ H 2 . (2.7)

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

0 T Δ ϕ 2 d x d t = 0 T ( f ( ϕ ) - μ ) 2 d x d t C 0 T μ 2 d x d t + C 0 T ( ϕ 3 - ϕ ) 2 d x d t C + C 0 T ϕ 4 ϕ 2 d x d t C + C ϕ L ( 0 , T ; L 2 ) 2 0 T ϕ L 4 d t C + C 0 T ϕ L 6 2 ϕ H 2 2 d t C + C ϕ L ( 0 , T ; H 1 ) 2 0 T ϕ H 2 2 d t C , (2.8)

which yields

ϕ L 2 ( 0 , T ; H 3 ) C , (2.9)

ϕ L 4 ( 0 , T ; L ) C , (2.10)

ϕ L 2 ( 0 , T ; L ) C . (2.11)

Testing (1.4) by Δ2ϕ, using (1.5), (2.4), (2.3), (2.10) and (2.11), we derive

1 2 d d t Δ ϕ 2 d x + Δ 2 ϕ 2 d x = - u ϕ Δ 2 ϕ d x + Δ ( ϕ 3 - ϕ ) Δ 2 ϕ d x u L 2 ϕ L Δ 2 ϕ L 2 + Δ ( ϕ 3 - ϕ ) L 2 Δ 2 ϕ L 2 C ϕ L Δ 2 ϕ L 2 + C ( ϕ L 2 Δ ϕ L 2 + ϕ L ϕ L ϕ L 2 + Δ ϕ L 2 ) Δ 2 ϕ L 2 C ϕ L Δ 2 ϕ L 2 + C ( ϕ L 2 Δ ϕ L 2 + ϕ H 2 ϕ L + Δ ϕ L 2 ) Δ 2 ϕ L 2 1 2 Δ 2 ϕ L 2 2 + C ϕ L 2 + C ϕ L 4 Δ ϕ L 2 2 + C ϕ L 2 ϕ H 2 2 + C Δ ϕ L 2 2 ,

which implies

ϕ L ( 0 , T ; H 2 ) + ϕ L 2 ( 0 , T ; H 4 ) C . (2.12)

Testing (1.1) by -Δu + ∇π, using (1.2), (1.6), (2.12), (2.1) and (2.4), we reach

1 2 d d t u 2 d x + ( - Δ u + π ) 2 d x = ( μ ϕ + θ e 2 - u u ) ( - Δ u + π ) d x ( μ L 2 ϕ L + θ L 2 + u L 4 u L 4 ) - Δ u + π L 2 C ( ϕ L + 1 + u L 2 1 2 u L 2 1 2 u L 2 1 2 Δ u L 2 1 2 ) - Δ u + π L 2 C ϕ L 2 + C + C u L 2 4 + 1 2 - Δ u + π L 2 2 ,

which yields

u L ( 0 , T ; H 1 ) + u L 2 ( 0 , T ; H 2 ) C . (2.13)

Here, we have used the Gagliardo-Nirenberg inequalities:

u L 4 2 C u L 2 u L 2 , u L 4 2 C u L 2 u H 2 ,

and the H2-theory of the Stokes system:

u H 2 + π H 1 C - Δ u + π L 2 . (2.14)

Similarly to (2.13), we have

t u L 2 ( 0 , T ; L 2 ) C . (2.15)

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

t u - Δ u + π = g : = μ ϕ + θ e 2 - u u , i n Ω × ( 0 , ) , u = 0 , o n Ω × ( 0 , ) , u ( x , 0)  =  u 0 ( x ) .

Using (2.12), (2.1), (2.13), and the regularity theory of Stokes system, we have

t u L 2 ( 0 , T ; L p ) + u L 2 ( 0 , T ; W 2 , p ) C g L 2 ( 0 , T ; L p ) C μ L 2 ( 0 , T ; L ) ϕ L ( 0 , T ; L p ) + C θ L ( 0 , T ; L ) + C u L ( 0 , T ; L 2 p ) u L 2 ( 0 , T ; L 2 p ) C , (2.16)

for any 2 < p < ∞.

(2.16) gives

u L 2 ( 0 , T ; L ) C . (2.17)

It follows from (1.3) and (1.6) that

Δ θ = 0 o n Ω × ( 0 , ) . (2.18)

Applying Δ to (1.3), testing by Δθ, using (1.2), (1.6), (2.16), (2.17) and (2.18), we obtain

1 2 d d t Δ θ 2 d x + ε Δ θ 2 d x = - ( Δ ( u θ ) - u Δ θ ) Δ θ d x C ( Δ u L 4 θ L 4 + u L Δ θ L 2 ) Δ θ L 2 C ( Δ u L 4 + u L ) Δ θ L 2 2 ,

which implies

θ L ( 0 , T ; H 2 ) + ε θ L 2 ( 0 , T ; H 3 ) C . (2.19)

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

t θ L ( 0 , T ; L 2 ) C . (2.20)

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

1 2 d d t t ϕ 2 d x + Δ t ϕ 2 d x = - t u ϕ t ϕ d x + Δ ( 3 ϕ 2 t ϕ - t ϕ ) t ϕ d x = - t u ϕ t ϕ d x + ( 3 ϕ 2 t ϕ - t ϕ ) Δ t ϕ d x t u L 2 ϕ L t ϕ L 2 + ( 3 ϕ L 2 + 1 ) t ϕ L 2 Δ t ϕ L 2 t u L 2 ϕ L t ϕ L 2 + 1 2 Δ t ϕ L 2 2 + C t ϕ L 2 2 ,

which gives

t ϕ L ( 0 , T ; L 2 ) + t ϕ L 2 ( 0 , T ; H 2 ) C . (2.21)

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 L ( 0 , T ; H 4 ) C Δ ϕ L ( 0 , T ; H 2 ) C μ - f ( ϕ ) L ( 0 , T ; H 2 ) C μ L ( 0 , T ; H 2 ) + C f ( ϕ ) L ( 0 , T ; H 2 ) C Δ μ L ( 0 , T ; L 2 ) + C f ( ϕ ) L ( 0 , T ; H 2 ) C t ϕ + u ϕ L ( 0 , T ; L 2 ) + C f ( ϕ ) L ( 0 , T ; H 2 ) C t ϕ L ( 0 , T ; L 2 ) + C u L ( 0 , T ; L 4 ) ϕ L ( 0 , T ; L 4 ) + C f ( ϕ ) L ( 0 , T ; H 2 ) C . (2.22)

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

1 2 d d t t u 2 d x + t u 2 d x = - t u u t u d x + ( t μ ϕ + μ t ϕ + t θ e 2 ) t u d x u L t u L 2 2 + ( t u L 2 ϕ L + μ L t ϕ L 2 + t θ L 2 ) t u L 2 u L t u L 2 2 + C ( Δ t ϕ L 2 + t ( ϕ 3 - ϕ ) L 2 + t ϕ L 2 + 1 ) t u L 2 ,

which implies

t u L ( 0 , T ; L 2 ) + t u L 2 ( 0 , T ; H 1 ) C . (2.23)

Using (2.23), (2.22), (2.1), (2.13), (1.1), (1.2), (1.6) and the H2-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.

### Acknowledgements

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, 802–805 (2009). Publisher Full Text

3. Zhou, Yong, Fan, Jishan: On the Cauchy problems for certain Boussinesq-α equations. Proc R Soc Edinburgh Sect A. 140, 319–327 (2010). Publisher Full Text

4. Chae, Dongho: Global regularity for the 2D Boussinesq equations with partial viscosity terms. Adv Math. 203, 497–513 (2006). Publisher Full Text