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A regularity criterion for the Cahn-Hilliard-Boussinesq system with zero viscosity

  • 1,
  • 2,
  • 2, 3 and
  • 2, 4Email author
Journal of Inequalities and Applications20142014:287

https://doi.org/10.1186/1029-242X-2014-287

  • Received: 20 May 2014
  • Accepted: 15 July 2014
  • Published:

Abstract

This paper studies a coupled Cahn-Hilliard-Boussinesq system with zero viscosity. We prove a regularity criterion in terms of vorticity in the homogeneous Besov space B ˙ , 0 .

MSC:35Q30, 76D03, 76D05, 76D07.

Keywords

  • regularity criterion
  • Cahn-Hilliard-Boussinesq
  • zero viscosity

1 Introduction

In this paper, we study the following Cahn-Hilliard-Boussinesq system with zero viscosity [1]:
t u + ( u ) u + π = μ ϕ + θ e 3 ,
(1.1)
div u = 0 ,
(1.2)
t θ + u θ = Δ θ ,
(1.3)
t ϕ + u ϕ = Δ μ ,
(1.4)
Δ ϕ + f ( ϕ ) = μ ,
(1.5)
( u , θ , ϕ ) ( x , 0 ) = ( u 0 , θ 0 , ϕ 0 ) ( x ) , x R 3 ,
(1.6)

with u the fluid velocity field, θ the temperature, ϕ the order parameter, and π the pressure, are the unknowns. e 3 : = ( 0 , 0 , 1 ) t . μ is the chemical potential. f ( ϕ ) : = 1 4 ( ϕ 2 1 ) 2 is the double well potential.

When θ = ϕ 0 , (1.1) and (1.2) are the well-known Euler system; Kozono, Ogawa and Taniuchi [2] proved the following regularity criterion:
ω : = curl u L 1 ( 0 , T ; B ˙ , 0 ) .
(1.7)

Here B ˙ , 0 denotes the homogeneous Besov space.

When ϕ = 0 , (1.1), (1.2), and (1.3) are the well-known Boussinesq system with zero viscosity; Fan and Zhou [3] also showed the regularity criterion (1.7).

When u = 0 , (1.4) and (1.5) are the well-known Cahn-Hilliard system.

It is easy to show that the problem (1.1)-(1.6) has a unique local smooth solution, thus we omit the details here. However, the global regularity is still open. The aim of this paper is to study the blow up criterion. We will prove the following.

Theorem 1.1 Let T > 0 and u 0 H 3 , θ 0 H 2 , ϕ 0 H 3 with div u 0 = 0 in R 3 . Suppose that ( u , θ , ϕ ) is a local smooth solution to the problem (1.1)-(1.6). Then ( u , θ , ϕ ) is smooth up to time T provided that (1.7) is satisfied.

We will use the following logarithmic Sobolev inequality [2]:
u L C ( 1 + curl u B ˙ , 0 log ( e + Λ 3 u L 2 ) ) ,
(1.8)
and the bilinear product and commutator estimates due to Kato and Ponce [4]:
Λ s ( f g ) L p C ( Λ s f L p 1 g L q 1 + f L p 2 Λ s g L q 2 ) ,
(1.9)
Λ s ( f g ) f Λ s g L p C ( f L p 1 Λ s 1 g L q 1 + Λ s f L p 2 g L q 2 ) ,
(1.10)

with s > 0 , Λ : = ( Δ ) 1 / 2 and 1 p = 1 p 1 + 1 q 1 = 1 p 2 + 1 q 2 .

2 Proof of Theorem 1.1

First, it follows from (1.2), (1.3), and the maximum principle that
θ L ( 0 , T ; L ) C .
(2.1)
Testing (1.3) by θ, using (1.2), we see that
1 2 d d t θ 2 d x + | θ | 2 d x = 0 ,
whence
θ L ( 0 , T ; L 2 ) + θ L 2 ( 0 , T ; H 1 ) C .
(2.2)
Testing (1.4) by ϕ, using (1.2) and (1.5), we find that
1 2 d d t ϕ 2 d x + | Δ ϕ | 2 d x = f ( ϕ ) Δ ϕ d x = ( ϕ 3 ϕ ) Δ ϕ d x = 3 ϕ 2 | ϕ | 2 d x ϕ Δ ϕ d x ϕ Δ ϕ d x ϕ L 2 Δ ϕ L 2 1 2 Δ ϕ L 2 2 + 1 2 ϕ L 2 2 ,
which gives
ϕ L ( 0 , T ; L 2 ) + ϕ L 2 ( 0 , T ; H 2 ) C .
(2.3)
Testing (1.1) and (1.4) by u and μ, respectively, using (1.2), (1.5), and (2.2), summing up the result, we deduce that
d d t 1 2 | ϕ | 2 + f ( ϕ ) + 1 2 u 2 d x + | μ | 2 d x = θ e 3 u d x θ L 2 u L 2 C u L 2 ,
which gives
ϕ L ( 0 , T ; H 1 ) C ,
(2.4)
u L ( 0 , T ; L 2 ) C ,
(2.5)
μ L 2 ( 0 , T ; L 2 ) C .
(2.6)
In the following calculations, we will use the following Gagliardo-Nirenberg inequality:
ϕ L 2 C ϕ L 2 Δ ϕ L 2 .
(2.7)
It follows from (2.6), (1.5), (2.3), (2.4), and (2.7) that
0 T | Δ ϕ | 2 d x d t = 0 T | ( f ( ϕ ) μ ) | 2 d x d t C 0 T | μ | 2 d x d t + C 0 T | f ( ϕ ) | 2 d x d t C + C 0 T | ( ϕ 3 ϕ ) | 2 d x d t C + C 0 T ϕ 4 | ϕ | 2 d x d t C + C ϕ L ( 0 , T ; L 2 ) 2 0 T ϕ L 4 d t C + C 0 T ϕ L 4 d t C + C 0 T ϕ L 2 2 Δ ϕ L 2 2 d t C + C sup t ϕ ( t ) L 2 2 0 T Δ ϕ L 2 2 d t C ,
(2.8)
which yields
ϕ L 2 ( 0 , T ; L ) C .
(2.9)
Applying Λ 3 to (1.1), testing by Λ 3 u , using (1.2), (1.9), (1.10), and noting that
Δ ϕ ϕ = j j ( j ϕ ϕ ) 1 2 | ϕ | 2 ,
we derive
1 2 d d t | Λ 3 u | 2 d x = ( Λ 3 ( u u ) u Λ 3 u ) Λ 3 u d x j Λ 3 j ( j ϕ ϕ ) Λ 3 u d x + Λ 3 θ e 3 Λ 3 u d x C u L Λ 3 u L 2 2 + C ϕ L Λ 5 ϕ L 2 Λ 3 u L 2 + C Λ 3 θ L 2 Λ 3 u L 2 C u L Λ 3 u L 2 2 + C ϕ L 2 Λ 3 u L 2 2 + C Λ 3 u L 2 2 + ϵ Λ 5 ϕ L 2 2 + ϵ Λ 3 θ L 2 2
(2.10)

for any 0 < ϵ < 1 .

Taking Δ to (1.3), testing by Δθ, using (1.2), (1.10), and (2.1), we obtain
1 2 d d t | Δ θ | 2 d x + | Δ θ | 2 d x = ( Δ ( u θ ) u Δ θ ) Δ θ d x C u L Δ θ L 2 2 + C θ L 4 Δ u L 4 Δ θ L 2 C u L Δ θ L 2 2 + C θ L 1 / 2 Δ θ L 2 1 / 2 u L 1 / 2 Δ u L 2 1 / 2 Δ θ L 2 C u L Δ θ L 2 2 + C ( u L Δ θ L 2 + Δ u L 2 ) Δ θ L 2 C u L Δ θ L 2 2 + C Δ θ L 2 2 + C Δ u L 2 2 .
(2.11)
Here we have used the Gagliardo-Nirenberg inequalities:
θ L 4 2 C θ L Δ θ L 2 , Δ u L 4 2 C u L Δ u L 2 .
Taking Δ to (1.4), testing by Δ ϕ , using (1.5), (1.2), and (1.10), we have
1 2 d d t | Δ ϕ | 2 d x + | Λ 5 ϕ | 2 d x = i ( Δ ( u i i ϕ ) u i Δ i ϕ ) Δ ϕ d x + Δ Δ f ( ϕ ) Δ ϕ d x C u L Δ ϕ L 2 2 + C ϕ L Λ 3 u L 2 Δ ϕ L 2 + Δ f ( ϕ ) Δ 2 ϕ d x .
(2.12)
Using (2.4), we get
Δ f ( ϕ ) Δ 2 ϕ d x C ( | Δ ϕ | + | ϕ 2 | | Δ ϕ | + | ϕ | | ϕ | | 2 ϕ | + | ϕ | 3 ) | Δ 2 ϕ | d x C ( Δ ϕ L 2 + ϕ L 2 Δ ϕ L 2 + ϕ L ϕ L 2 2 ϕ L + ϕ L 2 ϕ L 2 ) Λ 5 ϕ L 2 C Δ ϕ L 2 2 + C ϕ L 4 Δ ϕ L 2 2 + C ϕ L 2 ϕ L Λ 5 ϕ L 2 + C ϕ L 2 Λ 5 ϕ L 2 + 1 4 Λ 5 ϕ L 2 2 .
(2.13)
Now we use the following Gagliardo-Nirenberg inequalities:
2 ϕ L C Δ ϕ L 2 3 / 4 Λ 5 ϕ L 2 1 / 4 , ϕ L 2 C Δ ϕ L 2 Λ 3 ϕ L 2 .
We obtain
Δ f ( ϕ ) Δ 2 ϕ d x C Δ ϕ L 2 2 + C ϕ L 4 Δ ϕ L 2 2 + C ϕ L 8 / 3 Δ ϕ L 2 2 + C Δ ϕ L 2 2 Λ 3 ϕ L 2 2 + 1 2 Λ 5 ϕ L 2 .
(2.14)
Combining (2.10), (2.11), (2.12), and (2.14), taking ϵ small enough, using (1.8), (2.5), Gronwall’s inequality and
ϕ L 4 ( 0 , T ; L ) C ,
we conclude that
u L ( 0 , T ; H 3 ) C , θ L ( 0 , T ; H 2 ) + θ L 2 ( 0 , T ; H 3 ) C , ϕ L ( 0 , T ; H 3 ) + ϕ L 2 ( 0 , T ; H 5 ) C .

This completes the proof.

Declarations

Acknowledgements

This project is funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under Grant No. 27-130-35-HiCi. The authors, therefore, acknowledge technical and financial support of KAU.

Authors’ Affiliations

(1)
Department of Mathematics, Tianshui Normal University, Tianshui, Gansu, 741001, P.R. China
(2)
Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia
(3)
Department of Mathematics, Quaid-i-Azam University, 45320 Islamabad, 44000, Pakistan
(4)
School of Mathematics, Shanghai University of Finance and Economics, Shanghai, 200433, P.R. China

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