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Vanishing heat conductivity limit for the 2D Cahn-Hilliard-Boussinesq system
Boundary Value Problemsvolume 2011, Article number: 54 (2011)
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, ∞) :
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  Zhou and Fan proved a regularity criterion for the 3D Boussinesq system with partial viscosity. Later, in  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  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
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
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
Testing (1.4) by Δ2ϕ, using (1.5), (2.4), (2.3), (2.10) and (2.11), we derive
Testing (1.1) by -Δu + ∇π, using (1.2), (1.6), (2.12), (2.1) and (2.4), we reach
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 < ∞.
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
It follows from (1.3), (1.6), (2.19) and (2.13) that
Taking ∂ t to (1.4) and (1.5), testing by ∂ t ϕ, using (1.2), (1.6), (2.12), and (2.15), we have
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 ∂ t u, using (1.2), (1.6), (2.17), (2.22), (2.21) and (1.5), we conclude that
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.
Boyer Franck: Mathematical study of multi-phase flow under shear through order parameter formulation. Asymptot Anal 1999, 20: 175-212.
Fan Jishan, Zhou Yong: A note on regularity criterion for the 3D Boussinesq system with partial viscosity. Appl Math Lett 2009, 22: 802-805. 10.1016/j.aml.2008.06.041
Zhou Yong, Fan Jishan: On the Cauchy problems for certain Boussinesq- α equations. Proc R Soc Edinburgh Sect A 2010, 140: 319-327. 10.1017/S0308210509000122
Chae Dongho: Global regularity for the 2D Boussinesq equations with partial viscosity terms. Adv Math 2006, 203: 497-513. 10.1016/j.aim.2005.05.001
This study was supported by the NSFC (No. 11171154) and NSFC (Grant No. 11101376).
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.