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

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.

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]:

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 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 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 . Here 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 , ϕ₀ ∈ H⁴, div u₀ = 0 in Ω and on ∂Ω. Then, there exists a positive constant C independent of ε such that

for any T > 0, which implies

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

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

whence

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

which gives

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

which leads to

We will use the following Gagliardo-Nirenberg inequality:

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

which yields

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

which implies

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

which yields

Here, we have used the Gagliardo-Nirenberg inequalities:

and the H²-theory of the Stokes system:

Similarly to (2.13), we have

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

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

for any 2 < p < ∞.

(2.16) gives

It follows from (1.3) and (1.6) that

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

which implies

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

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

which gives

By the regularity theory of elliptic equation, it follows from (1.4), (1.5), (1.6), (2.21), (2.13) and (2.12) that

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

which implies

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

This completes the proof.

The authors declare that they have no competing interests.

All authors read and approved the final manuscript.

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

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