Báo cáo sinh học: " Vanishing heat conductivity limit for the 2D Cahn-Hilliard-Boussinesq system" ppt

This Provisional PDF corresponds to the article as it appeared upon acceptance.. Vanishing heat conductivity limit for the 2D Cahn-Hilliard-Boussinesq system Boundary Value Problems 2011

This Provisional PDF corresponds to the article as it appeared upon acceptance Fully formatted

PDF and full text (HTML) versions will be made available soon

Vanishing heat conductivity limit for the 2D Cahn-Hilliard-Boussinesq system

Boundary Value Problems 2011, 2011:54 doi:10.1186/1687-2770-2011-54

Zaihong Jiang (jzhong@zjnu.cn) Jishan Fan (fanjishan@njfu.com.cn)

ISSN 1687-2770

This peer-reviewed article was published immediately upon acceptance It can be downloaded,

printed and distributed freely for any purposes (see copyright notice below)

For information about publishing your research in Boundary Value Problems go to

http://www.boundaryvalueproblems.com/authors/instructions/

For information about other SpringerOpen publications go to

http://www.springeropen.com

Boundary Value Problems

Vanishing heat conductivity limit for the 2D Cahn-Hilliard-Boussinesq system

Jinhua 321004, People’s Republic of China

Nanjing 210037, People’s Republic of China

Email address:

fanjishan@njfu.com.cn

Abstract This article studies the vanishing heat conductivity limit for the 2D Cahn-Hilliard-boussinesq system in a bounded domain with non-slip boundary condi-tion The result has been proved globally in time

2010 MSC: 35Q30; 76D03; 76D05; 76D07

Keywords: Cahn–Hilliard–Boussinesq; inviscid limit; non-slip boundary con-dition

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 following

Cahn-Hilliard-Boussinesq system in Ω × (0, ∞) [1]:

∂ t u + (u · ∇)u + ∇π − ∆u = µ∇φ + θe2, (1.1)

u = 0, θ = 0, ∂φ

∂n =

∂µ

∂n = 0 on ∂Ω × (0, ∞), (1.6)

(u, θ, φ)(x, 0) = (u0, θ0, φ0)(x), x ∈ Ω, (1.7)

where u, π, θ and φ denote unknown velocity field, pressure scalar, temperature of the fluid and the order parameter, respectively ² > 0 is the heat conductivity coefficient and e2 := (0, 1) t µ is a chemical potential and f (φ) := 1

4(φ2− 1)2 is the double well potential

When φ = 0, (1.1), (1.2) and (1.3) is the well-known Boussinesq system In [2] Zhou and Fan proved a regularity criterion ω = curlu ∈ L . 1(0, T ; ˙ B0

∞,∞) for the 3D Boussinesq system with partial viscosity Later, in [3] Zhou and Fan studied the

Cauchy problem of certain Boussinesq−α equations in n dimensions with n = 2 or

3 We establish regularity for the solution under ∇u ∈ L1(0, T ; ˙ B0

∞,∞) Here ˙B0

∞,∞

denotes the homogeneous Besov space Chae [4] studied the vanishing viscosity limit

² → 0 when Ω = R2 The aim of this article is to prove a similar result We will prove that

Theorem 1.1 Let (u0, θ0) ∈ H1

0 ∩ H2, φ0 ∈ H4, div u0 = 0 in Ω and ∂φ0

∂n = ∂µ0

∂n = 0

on ∂Ω Then, there exists a positive constant C independent of ² such that

ku ² k L ∞ (0,T ;H2 )≤ C, kθ ² k L ∞ (0,T ;H2 ) ≤ C,

kφ ² k L ∞ (0,T ;H4 ) ≤ C, k∂ t (u ² , θ ² , φ ² )k L2(0,T ;L2 )≤ C, (1.8) for any T > 0, which implies

(u ² , θ ² , φ ² ) → (u, θ, φ) strongly in L2(0, T ; H1) when ² → 0. (1.9)

Here, (u, θ, φ) is the solution of the problem (1.1)–(1.7) with ² = 0.

Testing (1.3) by θ, using (1.2) and (1.6), we see that

1 2

d

dt

Z

θ2dx + ²

Z

|∇θ|2dx = 0,

whence

√

Testing (1.1) and (1.4) by u and µ, respectively, using (1.2), (1.6), (2.1), and

summing up the result, we find that

d

dt

Z 1

2u

2+1

2|∇φ|

2+ f (φ)dx +

Z

|∇u|2+ |∇µ|2dx

=

Z

θe2udx ≤ kθk L2kuk L2 ≤ Ckuk L2,

which gives

Testing (1.4) by φ, using (1.2), (1.5) and (1.6), we infer that

1 2

d

dt

Z

φ2dx +

Z

|∆φ|2dx =

Z

(φ3− φ)∆φdx

= −3

Z

φ2|∇φ|2dx −

Z

φ∆φdx ≤ −

Z

φ∆φdx

≤ 1

2

Z

|∆φ|2dx + 1

2

Z

φ2dx,

which leads to

We will use the following Gagliardo-Nirenberg inequality:

kφk2

L ∞ ≤ Ckφk L6kφk H2. (2.7)

It follows from (2.6), (2.7), (2.5), (2.3) and (1.5) that

Z T

0

Z

|∇∆φ|2dxdt

=

Z T

0

Z

|∇(f 0 (φ) − µ)|2dxdt

≤ C

Z T

0

Z

|∇µ|2dxdt + C

Z T

0

Z

|∇(φ3− φ)|2dxdt

≤ C + C

Z T

0

Z

φ4|∇φ|2dxdt

≤ C + Ck∇φk2

L ∞ (0,T ;L2 )

Z T

0

kφk4

L ∞ dt

≤ C + C

Z T

0

kφk2

L6kφk2

H2dt

≤ C + Ckφk2L ∞ (0,T ;H1 )

Z T

0

kφk2H2dt ≤ C, (2.8) which yields

Testing (1.4) by ∆2φ, using (1.5), (2.4), (2.3), (2.10) and (2.11), we derive

1

2

d

dt

Z

|∆φ|2dx +

Z

|∆2φ|2dx

= −

Z

u · ∇φ · ∆2φdx +

Z

∆(φ3− φ) · ∆2φdx

≤ kuk L2k∇φk L ∞ k∆2φk L2 + k∆(φ3− φ)k L2k∆2φk L2

≤ Ck∇φk L ∞ k∆2φk L2

+C(kφk2

L ∞ k∆φk L2 + kφk L ∞ k∇φk L ∞ k∇φk L2 + k∆φk L2)k∆2φk L2

≤ Ck∇φk L ∞ k∆2φk L2

+C(kφk2 k∆φk + kφk k∇φk + k∆φk )k∆2φk

Testing (1.1) by −∆u + ∇π, using (1.2), (1.6), (2.12), (2.1) and (2.4), we reach

1

2

d

dt

Z

|∇u|2dx +

Z

(−∆u + ∇π)2dx

=

Z

(µ∇φ + θe2− u · ∇u)(−∆u + ∇π)dx

≤ (kµk L2k∇φk L ∞ + kθk L2 + kuk L4k∇uk L4)k − ∆u + ∇πk L2

≤ C(k∇φk L ∞ + 1 + kuk 1/2 L2 k∇uk 1/2 L2 · k∇uk 1/2 L2 k∆uk 1/2 L2 )k − ∆u + ∇πk L2

≤ Ck∇φk2

L ∞ + C + Ck∇uk4

L2 +1

2k − ∆u + ∇πk

2

L2,

which yields

Here, we have used the Gagliardo-Nirenberg inequalities:

kuk2

L4 ≤ Ckuk L2k∇uk L2, k∇uk2

L4 ≤ Ck∇uk L2kuk H2,

and the H2-theory of the Stokes system:

kuk H2 + kπk H1 ≤ Ck − ∆u + ∇πk L2. (2.14) Similarly to (2.13), we have

(1.1), (1.2), (1.6) and (1.7) can be rewritten as







∂ t u − ∆u + ∇π = g := µ∇φ + θe2 − u · ∇u, in Ω × (0, ∞),

u = 0, on ∂Ω × (0, ∞), u(x, 0) = u0(x).

Using (2.12), (2.1), (2.13), and the regularity theory of Stokes system, we have

k∂ t uk L2(0,T ;L p)+ kuk L2(0,T ;W 2,p) ≤ Ckgk L2(0,T ;L p)

≤ Ckµk L2(0,T ;L ∞)k∇φk L ∞ (0,T ;L p)+ Ckθk L ∞ (0,T ;L ∞)

+Ckuk L ∞ (0,T ;L 2p)k∇uk L2(0,T ;L 2p) ≤ C, (2.16)

for any 2 < p < ∞.

(2.16) gives

It follows from (1.3) and (1.6) that

Applying ∆ to (1.3), testing by ∆θ, using (1.2), (1.6), (2.16), (2.17) and (2.18), we

obtain

1 2

d

dt

Z

|∆θ|2dx + ²

Z

|∇∆θ|2dx

= −

Z

(∆(u · ∇θ) − u∇∆θ)∆θdx

≤ C(k∆uk L4k∇θk L4 + k∇uk L ∞ k∆θk L2)k∆θk L2

≤ C(k∆uk L4 + k∇uk L ∞ )k∆θk2L2,

which implies

kθk L ∞ (0,T ;H2 )+√ ²kθk L2(0,T ;H3 ) ≤ C. (2.19)

It follows from (1.3), (1.6), (2.19) and (2.13) that

Taking ∂ t to (1.4) and (1.5), testing by ∂ t φ, using (1.2), (1.6), (2.12), and (2.15),

we have

1 2

d

dt

Z

|∂ t φ|2dx +

Z

|∆∂ t φ|2dx

= −

Z

∂ t u · ∇φ · ∂ t φdx +

Z

∆(3φ2∂ t φ − ∂ t φ) · ∂ t φdx

= −

Z

∂ t u · ∇φ · ∂ t φdx +

Z

(3φ2∂ t φ − ∂ t φ)∆∂ t φdx

≤ k∂ t uk L2k∇φk L ∞ k∂ t φk L2 + (k3φk2

L ∞ + 1)k∂ t φk L2k∆∂ t φk L2

≤ k∂ t uk L2k∇φk L ∞ k∂ t φk L2 +1

2k∆∂ t φk

2

L2 + Ck∂ t φk2

L2,

which gives

k∂ t φk L ∞ (0,T ;L2 )+ k∂ t φk L2(0,T ;H2 ) ≤ C. (2.21)

By the regularity theory of elliptic equation, it follows from (1.4), (1.5), (1.6), (2.21), (2.13) and (2.12) that

kφk L ∞ (0,T ;H4 )≤ Ck∆φk L ∞ (0,T ;H2 )≤ Ckµ − f 0 (φ)k L ∞ (0,T ;H2 )

≤ Ckµk + Ckf 0 (φ)k

Taking ∂ t to (1.1), testing by ∂ t u, using (1.2), (1.6), (2.17), (2.22), (2.21) and (1.5),

we conclude that

1

2

d

dt

Z

|∂ t u|2dx +

Z

|∇∂ t u|2dx

= −

Z

∂ t u · ∇u · ∂ t udx +

Z

(∂ t µ · ∇φ + µ · ∇∂ t φ + ∂ t θe2)∂ t udx

≤ k∇uk L ∞ k∂ t uk2

L2 + (k∂ t uk L2k∇φk L ∞ + kµk L ∞ k∇∂ t φk L2 + k∂ t θk L2)k∂ t uk L2

≤ k∇uk L ∞ k∂ t uk2L2 + C(k∆∂ t φk L2 + k∂ t (φ3− φ)k L2 + k∇∂ t φk L2 + 1)k∂ t uk L2,

which implies

k∂ t uk L ∞ (0,T ;L2 )+ k∂ t uk L2(0,T ;H1 )≤ C. (2.23)

Using (2.23), (2.22), (2.1), (2.13), (1.1), (1.2), (1.6) and the H2-theory of the Stokes system, we arrive at

kuk L ∞ (0,T ;H2 )≤ C.

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

Acknowledgment

This study was supported by the NSFC (No 11171154) and NSFC (Grant No 11101376)

References

[1] Boyer, Franck: Mathematical study of multi-phase flow under shear through order parameter formulation Asymptot Anal 20, 175–212 (1999)

[2] Fan, Jishan; Zhou, Yong: A note on regularity criterion for the 3D Boussinesq system with partial viscosity Appl Math Lett 22, 802–C805 (2009)

[3] Zhou, Yong; Fan, Jishan: On the Cauchy problems for certain Boussinesq-α

equations Proc R Soc Edinburgh Sect A 140, 319–C327 (2010)

[4] Chae, Dongho: Global regularity for the 2D Boussinesq equations with partial viscosity terms Adv Math 203, 497–513 (2006)