Abstract
This article studies the partial vanishing viscosity limit of the 2D Boussinesq system in a bounded domain with a slip boundary condition. The result is proved globally in time by a logarithmic Sobolev inequality.
2010 MSC: 35Q30; 76D03; 76D05; 76D07.
Keywords:
Boussinesq system; inviscid limit; slip 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 Boussinesq system in Ω × (0, ∞):
where u, π, and θ denote unknown velocity vector field, pressure scalar and temperature of the fluid. ϵ > 0 is the heat conductivity coefficient and e_{2}:= (0, 1)^{t}. ω:= curlu:= ∂_{1}u_{2 } ∂_{2}u_{1 }is the vorticity.
The aim of this article is to study the partial vanishing viscosity limit ϵ → 0. When Ω:= ℝ^{2}, the problem has been solved by Chae [1]. When θ = 0, the Boussinesq system reduces to the wellknown NavierStokes equations. The investigation of the inviscid limit of solutions of the NavierStokes equations is a classical issue. We refer to the articles [27] when Ω is a bounded domain. However, the methods in [16] could not be used here directly. We will use a wellknown logarithmic Sobolev inequality in [8,9] to complete our proof. We will prove:
Theorem 1.1. Let u_{0 }∈ H^{3}, divu_{0 }= 0 in Ω, u_{0}·n = 0, curlu_{0 }= 0 on ∂Ω and
for any T > 0, which implies
Here (u, θ) is the unique solution of the problem (1.1)(1.5) with ϵ = 0.
2 Proof of Theorem 1.1
Since (1.7) follows easily from (1.6) by the AubinLions compactness principle, we only need to prove the a priori estimates (1.6). From now on we will drop the subscript e and throughout this section C will be a constant independent of ϵ > 0.
First, we recall the following two lemmas in [810].
Lemma 2.1. ([8,9]) There holds
for any u ∈ H^{3}(Ω) with divu = 0 in Ω and u · n = 0 on ∂Ω.
Lemma 2.2. ([10]) For any u ∈ W^{s,p }with divu = 0 in Ω and u · n = 0 on ∂Ω, there holds
for any s > 1 and p ∈ (1, ∞).
By the maximum principle, it follows from (1.2), (1.3), and (1.4) that
Testing (1.3) by θ, using (1.2), (1.3), and (1.4), we see that
which gives
Testing (1.1) by u, using (1.2), (1.4), and (2.1), we find that
which gives
Here we used the wellknown inequality:
Applying curl to (1.1), using (1.2), we get
Testing (2.4) by ω^{p2}ω (p > 2), using (1.2), (1.4), and (2.1), we obtain
which gives
(2.4) can be rewritten as
with f_{1}: = θ  u_{1}ω, f_{2}:= u_{2}ω.
Using (2.1), (2.5) and the L^{∞}estimate of the heat equation, we reach the key estimate
Let τ be any unit tangential vector of ∂Ω, using (1.4), we infer that
on ∂Ω × (0, ∞).
It follows from (1.3), (1.4), and (2.7) that
Applying Δ to (1.3), testing by Δθ, using (1.2), (1.4), and (2.8), we derive
Now using the GagliardoNirenberg inequalities
we have
Similarly to (2.7) and (2.8), if follows from (2.4) and (1.4) that
Applying Δ to (2.4), testing by Δω, using (1.2), (1.4), (2.13), (2.10), and Lemma 2.2, we reach
which yields
Combining (2.11) and (2.14), using the Gronwall inequality, we conclude that
It follows from (1.1), (1.3), (2.15), and (2.16) that
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 partially supported by the Zhejiang Innovation Project (Grant No. T200905), the ZJNSF (Grant No. R6090109), and the NSFC (Grant No. 11171154).
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