Open Access Research

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

Liangbing Jin1, Jishan Fan2, Gen Nakamura3 and Yong Zhou1*

Author affiliations

1 Department of Mathematics, Zhejiang Normal University, Jinhua 321004, P. R. China

2 Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, P.R. China

3 Department of Mathematics, Hokkaido University Sapporo 060-0810, Japan

For all author emails, please log on.

Citation and License

Boundary Value Problems 2012, 2012:20  doi:10.1186/1687-2770-2012-20


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/20


Received:12 November 2011
Accepted:15 February 2012
Published:15 February 2012

© 2012 Jin et al; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 condition

1 Introduction

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, ∞):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M1">View MathML</a>

(1.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M2">View MathML</a>

(1.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M3">View MathML</a>

(1.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M4">View MathML</a>

(1.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M5">View MathML</a>

(1.5)

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 , u0·n = 0, curlu0 = 0 on ∂Ω and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M6">View MathML</a>. Then there exists a positive constant C independent of ϵ such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M7">View MathML</a>

(1.6)

for any T > 0, which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M8">View MathML</a>

(1.7)

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.

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

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M9">View MathML</a>

for any u H3(Ω) with divu = 0 in and u · n = 0 on ∂Ω.

Lemma 2.2. ([10]) For any u Ws,p with divu = 0 in and u · n = 0 on ∂Ω, there holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M10">View MathML</a>

for any s > 1 and p ∈ (1, ∞).

By the maximum principle, it follows from (1.2), (1.3), and (1.4) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M11">View MathML</a>

(2.1)

Testing (1.3) by θ, using (1.2), (1.3), and (1.4), we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M12">View MathML</a>

which gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M13">View MathML</a>

(2.2)

Testing (1.1) by u, using (1.2), (1.4), and (2.1), we find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M14">View MathML</a>

which gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M15">View MathML</a>

(2.3)

Here we used the well-known inequality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M16">View MathML</a>

Applying curl to (1.1), using (1.2), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M17">View MathML</a>

(2.4)

Testing (2.4) by |ω|p-2ω (p > 2), using (1.2), (1.4), and (2.1), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M18">View MathML</a>

which gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M19">View MathML</a>

(2.5)

(2.4) can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M20">View MathML</a>

with f1: = θ - u1ω, f2:= -u2ω.

Using (2.1), (2.5) and the L-estimate of the heat equation, we reach the key estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M21">View MathML</a>

(2.6)

Let τ be any unit tangential vector of ∂Ω, using (1.4), we infer that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M22">View MathML</a>

(2.7)

on ∂Ω × (0, ∞).

It follows from (1.3), (1.4), and (2.7) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M23">View MathML</a>

(2.8)

Applying Δ to (1.3), testing by Δθ, using (1.2), (1.4), and (2.8), we derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M24">View MathML</a>

(2.9)

Now using the Gagliardo-Nirenberg inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M25">View MathML</a>

(2.10)

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M26">View MathML</a>

(2.11)

Similarly to (2.7) and (2.8), if follows from (2.4) and (1.4) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M27">View MathML</a>

(2.12)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M28">View MathML</a>

(2.13)

Applying Δ to (2.4), testing by Δω, using (1.2), (1.4), (2.13), (2.10), and Lemma 2.2, we reach

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M29">View MathML</a>

which yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M30">View MathML</a>

(2.14)

Combining (2.11) and (2.14), using the Gronwall inequality, we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M31">View MathML</a>

(2.15)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M32">View MathML</a>

(2.16)

It follows from (1.1), (1.3), (2.15), and (2.16) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/20/mathml/M33">View MathML</a>

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).

References

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

  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 Wk,p-inviscid limit for 3-D flows under a slip boundary condition. J Math Fluid Mech. 13, 117–135 (2011). Publisher Full Text OpenURL

  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 OpenURL

  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 OpenURL

  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 OpenURL

  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 OpenURL

  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 OpenURL