Abstract
This article studies the vanishing heat conductivity limit for the 2D CahnHilliardboussinesq system in a bounded domain with nonslip boundary condition. The result has been proved globally in time.
2010 MSC: 35Q30; 76D03; 76D05; 76D07.
Keywords:
CahnHilliardBoussinesq; inviscid limit; nonslip boundary condition1 Introduction
Let Ω ⊆ ℝ^{2 }be a bounded, simply connected domain with smooth boundary ∂Ω, and n is the unit outward normal vector to ∂Ω. We consider the following CahnHilliardBoussinesq 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 e_{2 }: = (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 wellknown 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 Boussinesqα 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 = ℝ^{2}. The aim of this article is to prove a similar result. We will prove that
Theorem 1.1. Let , ϕ_{0 }∈ H^{4}, div u_{0 }= 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 AubinLions 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 GagliardoNirenberg inequality:
It follows from (2.6), (2.7), (2.5), (2.3) and (1.5) that
which yields
Testing (1.4) by Δ^{2}ϕ, 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 GagliardoNirenberg inequalities:
and the H^{2}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 ∂_{t}ϕ, 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 ∂_{t}u, 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 H^{2}theory of the Stokes system, we arrive at
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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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

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

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