Partial vanishing viscosity limit for the 2D Boussinesq system with a slip boundary condition

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.

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 [1]. 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.

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.

First, we recall the following two lemmas in [8-10].

Lemma 2.1. ([8,9]) There holds

for any u ∈ H³(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 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

which gives

(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

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.

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).

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

2. Beirão da Veiga, H, Crispo, F: Sharp inviscid limit results under Navier type boundary conditions. An Lp Theory, J MathFluid Mech. 12, 397–411 (2010)

3. Beirão da Veiga, H, Crispo, F: Concerning the W^k,p-inviscid limit for 3-D flows under a slip boundary condition. J Math Fluid Mech. 13, 117–135 (2011). Publisher Full Text

4. Clopeau, T, Mikelić, A, Robert, R: On the vanishing viscosity limit for the 2D incompressible Navier-Stokes equations with the friction type boundary conditions. Nonlin-earity. 11, 1625–1636 (1998). Publisher Full Text

5. Iftimie, D, Planas, G: Inviscid limits for the Navier-Stokes equations with Navier friction boundary conditions. Nonlinearity. 19, 899–918 (2006). Publisher Full Text

6. Xiao, YL, Xin, ZP: On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition. Commun Pure Appl Math. 60, 1027–1055 (2007). Publisher Full Text

7. Crispo, F: On the zero-viscosity limit for 3D Navier-Stokes equations under slip boundary conditions. Riv Math Univ Parma (N.S.). 1, 205–217 (2010)

8. Ferrari, AB: On the blow-up of solutions of 3-D Euler equations in a bounded domain. Commun Math Phys. 155, 277–294 (1993). Publisher Full Text

9. Shirota, T, Yanagisawa, T: A continuation principle for the 3D Euler equations for incompressible fluids in a bounded domain. Proc Japan Acad Ser. A69, 77–82 (1993)

10. Bourguignon, JP, Brezis, H: Remarks on the Euler equation. J Funct Anal. 15, 341–363 (1974). Publisher Full Text