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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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Citation and License

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

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/54


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

© 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.

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]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M1">View MathML</a>

(1.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M2">View MathML</a>

(1.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M3">View MathML</a>

(1.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M4">View MathML</a>

(1.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M5">View MathML</a>

(1.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M6">View MathML</a>

(1.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M7">View MathML</a>

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M8">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M9">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M10">View MathML</a>. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M11">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M12">View MathML</a>, ϕ0 H4, div u0 = 0 in Ω and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M13">View MathML</a> on ∂Ω. Then, there exists a positive constant C independent of ε such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M14">View MathML</a>

(1.8)

for any T > 0, which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M15">View MathML</a>

(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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M16">View MathML</a>

(2.1)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M17">View MathML</a>

whence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M18">View MathML</a>

(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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M19">View MathML</a>

which gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M20">View MathML</a>

(2.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M21">View MathML</a>

(2.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M22">View MathML</a>

(2.5)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M23">View MathML</a>

which leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M24">View MathML</a>

(2.6)

We will use the following Gagliardo-Nirenberg inequality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M25">View MathML</a>

(2.7)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M26">View MathML</a>

(2.8)

which yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M27">View MathML</a>

(2.9)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M28">View MathML</a>

(2.10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M29">View MathML</a>

(2.11)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M30">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M31">View MathML</a>

(2.12)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M32">View MathML</a>

which yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M33">View MathML</a>

(2.13)

Here, we have used the Gagliardo-Nirenberg inequalities:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M34">View MathML</a>

and the H2-theory of the Stokes system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M35">View MathML</a>

(2.14)

Similarly to (2.13), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M36">View MathML</a>

(2.15)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M37">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M38">View MathML</a>

(2.16)

for any 2 < p < ∞.

(2.16) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M39">View MathML</a>

(2.17)

It follows from (1.3) and (1.6) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M40">View MathML</a>

(2.18)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M41">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M42">View MathML</a>

(2.19)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M43">View MathML</a>

(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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M44">View MathML</a>

which gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M45">View MathML</a>

(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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M46">View MathML</a>

(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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M47">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M48">View MathML</a>

(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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/54/mathml/M49">View MathML</a>

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

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  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 OpenURL

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