Skip to main content

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

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.

1 Introduction

Let Ω 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, ∞):

t u + u u + π - Δ u = θ e 2 ,
(1.1)
div u = 0 ,
(1.2)
t θ + u θ = ε Δ θ ,
(1.3)
u n = 0 , curl u = 0 , θ = 0 , on Ω × (0, ),
(1.4)
( u , θ ) ( x , 0 ) = ( u 0 , θ 0 ) ( x ) , x Ω ,
(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:= 2, 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 [27] when Ω is a bounded domain. However, the methods in [16] 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 θ 0 H 0 1 H 2 . Then there exists a positive constant C independent of ϵ such that

u ε L ( 0 , T ; H 3 ) L 2 ( 0 , T ; H 4 ) C , θ ε L ( 0 , T ; H 2 ) C , t u ε L 2 ( 0 , T ; L 2 ) C , t θ ε L 2 ( 0 , T ; L 2 ) C
(1.6)

for any T > 0, which implies

( u ε , q ε ) ( u , θ ) s t r o n g l y i n L 2 ( 0 , T ; H 1 ) w h e n ε 0 .
(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 [810].

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

u L ( Ω ) C ( 1 + curl u L ( Ω ) log ( e + u H 3 ( Ω ) ) )

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

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

u W s , p C u L p + curl u W s - 1 , p

for any s > 1 and p (1, ∞).

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

θ L ( 0 , T ; L ) θ 0 L C .
(2.1)

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

1 2 d d t θ 2 d x + ε θ 2 d x = 0 ,

which gives

ε θ L 2 ( 0 , T ; H 1 ) C .
(2.2)

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

1 2 d d t u 2 d x + C u 2 d x = θ e 2 u θ L 2 u L 2 C u L 2 ,

which gives

u L ( 0 , T ; L 2 ) + u L 2 ( 0 , T ; H 1 ) C .
(2.3)

Here we used the well-known inequality:

u H 1 C curl u L 2 .

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

t ω + u ω - Δ ω = curl( θ e 2 ) .
(2.4)

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

1 p d d t ω p d x + 1 2 ω p - 2 ω 2 d x + 4 p - 2 p 2 ω p / 2 2 d x = curl( θ e 2 ) ω p - 2 ω d x C θ L ω p - 2 ω d x 1 2 1 2 ω p - 2 ω 2 d x + 4 p - 2 p 2 ω p / 2 2 d x + C ω p d x + C ,

which gives

u L ( 0 , T ; W 1 , p ) C ω L ( 0 , T ; L p ) C .
(2.5)

(2.4) can be rewritten as

t ω - Δ ω = div f : = curl ( θ e 2 ) - div ( u ω ) , ω = 0 on Ω × ( 0 , ) ω ( x , 0 ) = ω 0 ( x ) in Ω

with f1: = θ - u1ω, f2:= -u2ω.

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

ω L ( 0 , T ; L ) C ω 0 L + f L ( 0 , T ; L p ) C .
(2.6)

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

u θ = ( ( u τ ) τ + ( u n ) n ) θ = ( u τ ) τ θ = ( u τ ) θ τ = 0
(2.7)

on ∂Ω × (0, ∞).

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

Δ θ = 0 on Ω × ( 0 , ) .
(2.8)

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

1 2 d d t Δ θ 2 d x + ε Δ θ 2 d x = - ( Δ ( u θ ) - u Δ θ ) Δ θ d x = - ( Δ u θ + 2 i i u i θ ) Δ θ d x C Δ u L 4 θ L 4 + u L Δ θ L 2 Δ θ L 2 .
(2.9)

Now using the Gagliardo-Nirenberg inequalities

θ L 4 2 C θ L Δ θ L 2 , Δ u L 4 2 C u L u H 3 ,
(2.10)

we have

1 2 d d t Δ θ 2 d x + ε Δ θ 2 d x C u L Δ θ L 2 2 + C Δ θ L 2 2 + C u L u H 3 2 C 1 + u L u H 3 2 + Δ θ L 2 2 C 1 + ω L log e + u H 3 1 + Δ ω L 2 2 + Δ θ L 2 2 C 1 + log e + Δ ω L 2 + Δ θ L 2 1 + Δ ω L 2 2 + Δ θ L 2 2 .
(2.11)

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

u ω = 0 on Ω × ( 0 , ) ,
(2.12)
Δ ω + curl ( θ e 2 ) = 0 on Ω × ( 0 , ) .
(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

1 2 d d t Δ ω 2 d x + Δ ω 2 d x = - ( Δ ( u ω ) - u Δ ω ) Δ ω d x - curl ( θ e 2 ) Δ ω d x C Δ u L 4 ω L 4 + u L Δ ω L 2 Δ ω L 2 + C Δ θ L 2 Δ ω L 2 C Δ u L 4 2 + u L Δ ω L 2 Δ ω L 2 + C Δ θ L 2 Δ ω L 2 C u L u H 3 Δ ω L 2 + C Δ θ L 2 Δ ω L 2 C u L 1 + Δ ω L 2 Δ ω L 2 + C Δ θ L 2 2 + 1 2 Δ ω L 2 2

which yields

d d t Δ ω 2 d x + Δ ω 2 d x C u L 1 + Δ ω L 2 Δ ω L 2 + C Δ θ L 2 2 C 1 + log e + Δ ω L 2 + Δ θ L 2 1 + Δ ω L 2 2 + Δ θ L 2 2 .
(2.14)

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

θ L ( 0 , T ; H 2 ) + ε θ L ( 0 , T ; H 3 ) C ,
(2.15)
u L ( 0 , T ; H 3 ) + u L 2 ( 0 , T ; H 4 ) C .
(2.16)

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

t u L 2 ( 0 , T : L 2 ) C , t θ L 2 ( 0 , T : L 2 ) C .

This completes the proof.

References

  1. Chae D: Global regularity for the 2D Boussinesq equations with partial viscosity terms. Adv Math 2006, 203: 497-513. 10.1016/j.aim.2005.05.001

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  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 2011, 13: 117-135. 10.1007/s00021-009-0012-3

    Article  MathSciNet  Google Scholar 

  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 1998, 11: 1625-1636. 10.1088/0951-7715/11/6/011

    Article  Google Scholar 

  5. Iftimie D, Planas G: Inviscid limits for the Navier-Stokes equations with Navier friction boundary conditions. Nonlinearity 2006, 19: 899-918. 10.1088/0951-7715/19/4/007

    Article  MathSciNet  Google Scholar 

  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 2007, 60: 1027-1055. 10.1002/cpa.20187

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  8. Ferrari AB: On the blow-up of solutions of 3-D Euler equations in a bounded domain. Commun Math Phys 1993, 155: 277-294. 10.1007/BF02097394

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. Bourguignon JP, Brezis H: Remarks on the Euler equation. J Funct Anal 1974, 15: 341-363. 10.1016/0022-1236(74)90027-5

    Article  MathSciNet  Google Scholar 

Download references

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).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yong Zhou.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All authors read and approved the final manuscript.

Rights and permissions

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

Reprints and permissions

About this article

Cite this article

Jin, L., Fan, J., Nakamura, G. et al. Partial vanishing viscosity limit for the 2D Boussinesq system with a slip boundary condition. Bound Value Probl 2012, 20 (2012). https://doi.org/10.1186/1687-2770-2012-20

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1687-2770-2012-20

Keywords