Partial vanishing viscosity limit for the 2D Boussinesq system with a slip boundary condition
© Jin et al; licensee Springer. 2012
Received: 12 November 2011
Accepted: 15 February 2012
Published: 15 February 2012
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.
KeywordsBoussinesq system inviscid limit slip boundary condition
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:
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.
for any s > 1 and p ∈ (1, ∞).
with f1: = θ - u1ω, f2:= -u2ω.
on ∂Ω × (0, ∞).
This completes the proof.
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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