1687-2770-2011-541687-2770 Research <p>Vanishing heat conductivity limit for the 2D Cahn-Hilliard-Boussinesq system</p> JiangZaihongjzhong@zjnu.cn FanJishanfanjishan@njfu.com.cn

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, People's Republic of China

Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, People's Republic of China

 Boundary Value Problems 1687-2770 2011 2011 1 54 http://www.boundaryvalueproblems.com/content/2011/1/54 10.1186/1687-2770-2011-54 181020112212201122122011 2011Jiang 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. Cahn-Hilliard-Boussinesq inviscid limit non-slip boundary condition

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.

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 ,

div u = 0 ,

∂ t θ + u ⋅ ∇ θ = ε Δ θ ,

∂ t ϕ + u ⋅ ∇ ϕ = Δ μ ,

- Δ ϕ + f ′ ( ϕ ) = μ ,

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

( u , θ , ϕ ) ( x , 0 ) = ( u 0 , θ 0 , ϕ 0 ) ( x ) , x ∈ Ω ,

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 e 2 : = (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 ; Ḃ ∞ , ∞ 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 ; Ḃ ∞ , ∞ 0 ) . Here Ḃ ∞ , ∞ 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 , ϕ 0 ∈ H 4, div u 0 = 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 ,

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 .

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 .

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 .

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 ,

∥ u ∥ L ∞ ( 0 , T ; L 2 ) + ∥ u ∥ L 2 ( 0 , T ; H 1 ) ≤ C ,

∥ ∇ μ ∥ L 2 ( 0 , T ; L 2 ) ≤ C .

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 ,

which leads to

∥ ϕ ∥ L 2 ( 0 , T ; H 2 ) ≤ C .

We will use the following Gagliardo-Nirenberg inequality:

∥ ϕ ∥ L ∞ 2 ≤ C ∥ ϕ ∥ L 6 ∥ ϕ ∥ H 2 .

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 ,

which yields

∥ ϕ ∥ L 2 ( 0 , T ; H 3 ) ≤ C ,

∥ ϕ ∥ L 4 ( 0 , T ; L ∞ ) ≤ C ,

∥ ∇ ϕ ∥ L 2 ( 0 , T ; L ∞ ) ≤ C .

Testing (1.4) by Δ2 ϕ, 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 .

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 .

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 H 2-theory of the Stokes system:

∥ u ∥ H 2 + ∥ π ∥ H 1 ≤ C ∥ - Δ u + ∇ π ∥ L 2 .

Similarly to (2.13), we have

∥ ∂ t u ∥ L 2 ( 0 , T ; L 2 ) ≤ C .

(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 ,

for any 2 < p < ∞.

(2.16) gives

∥ ∇ u ∥ L 2 ( 0 , T ; L ∞ ) ≤ C .

It follows from (1.3) and (1.6) that

Δ θ = 0 o n ∂ Ω × ( 0 , ∞ ) .

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 .

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

∥ ∂ t θ ∥ L ∞ ( 0 , T ; L 2 ) ≤ C .

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 .

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 .

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 .

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.

Acknowledgements

This study was supported by the NSFC (No. 11171154) and NSFC (Grant No. 11101376).

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