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 condition
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 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 e2:= (0, 1)t. ω:= curlu:= ∂1u2 - ∂2u1 is the vorticity.
The aim of this article is to study the partial vanishing viscosity limit ϵ → 0. When Ω:= ℝ2, the problem has been solved by Chae . When θ = 0, the Boussinesq system reduces to the well-known Navier-Stokes equations. The investigation of the inviscid limit of solutions of the Navier-Stokes equations is a classical issue. We refer to the articles [2-7] when Ω is a bounded domain. However, the methods in [1-6] could not be used here directly. We will use a well-known logarithmic Sobolev inequality in [8,9] to complete our proof. We will prove:
Theorem 1.1. Let u0 ∈ H3, divu0 = 0 in Ω, u0·n = 0, curlu0 = 0 on ∂Ω and . Then there exists a positive constant C independent of ϵ such that
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 Aubin-Lions 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.
for any u ∈ H3(Ω) with divu = 0 in Ω and u · n = 0 on ∂Ω.
Lemma 2.2. () For any u ∈ Ws,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
Testing (1.1) by u, using (1.2), (1.4), and (2.1), we find that
Here we used the well-known inequality:
Applying curl to (1.1), using (1.2), we get
Testing (2.4) by |ω|p-2ω (p > 2), using (1.2), (1.4), and (2.1), we obtain
(2.4) can be rewritten as
with f1: = θ - u1ω, f2:= -u2ω.
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 Gagliardo-Nirenberg inequalities
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
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.
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
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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