Open Access

A logarithmically improved blow-up criterion for smooth solutions to the micropolar fluid equations in weak multiplier spaces

Boundary Value Problems20132013:122

DOI: 10.1186/1687-2770-2013-122

Received: 11 March 2013

Accepted: 26 April 2013

Published: 10 May 2013

Abstract

In this paper, we study the initial value problem for the three-dimensional micropolar fluid equations. A new logarithmically improved blow-up criterion for the three-dimensional micropolar fluid equations in a weak multiplier space is established.

MSC:35K15, 35K45.

Keywords

micropolar fluid equations smooth solution blow-up criterion

1 Introduction

In the paper, we consider the initial value problem for the micropolar fluid equations in  R 3
{ t v ( ν + κ ) Δ v + v v + p 2 κ × w = 0 , t w γ Δ w ( α + β ) w + 4 κ w + v w 2 κ × v = 0 , v = 0
(1.1)
with the initial value
t = 0 : v = v 0 ( x ) , w = w 0 ( x ) ,
(1.2)

where v ( t , x ) , w ( t , x ) and p ( t , x ) represent the divergence free velocity field, non-divergence free micro-rotation field and the scalar pressure, respectively. ν > 0 is the Newtonian kinetic viscosity and κ > 0 is the dynamics micro-rotation viscosity, α , β , γ > 0 are the angular viscosity (see [1]).

The micropolar fluid equations were first proposed by Eringen [2]. The micropolar fluid equations are a generalization of the Navier-Stokes model. It takes into account the microstructure of the fluid, by which we mean the geometry and microrotation of particles. It is a type of fluids which exhibit the micro-rotational effects and micro-rotational inertia, and can be viewed as a non-Newtonian fluid. Physically, it may represent adequately the fluids consisting of bar-like elements. Certain anisotropic fluids, e.g., liquid crystals that are made up of dumbbell molecules, are of the type. For more background, we refer to [1] and references therein.

Due to its importance in mathematics and physics, there is lots of literature devoted to the mathematical theory of the 3D micropolar fluid equations. Fundamental mathematical issues such as the global regularity of their solutions have generated extensive research and many interesting results have been established (see [312]). The regularity of weak solutions is examined by imposing some critical growth conditions only on the pressure field in the Lebesgue space, Morrey space, multiplier space, BMO space and Besov space, respectively (see [4]). A new logarithmically improved blow-up criterion for the 3D micropolar fluid equations in an appropriate homogeneous Besov space was obtained by Wang and Yuan [9]. A Serrin-type regularity criterion for the weak solutions to the micropolar fluid equations in R 3 in the critical Morrey-Campanato space was built [10]. Wang and Zhao [12] established logarithmically improved blow-up criteria of a smooth solution to (1.1), (1.2) in the Morrey-Campanto space.

If κ = 0 and w = 0 , then equations (1.1) reduce to be the Navier-Stokes equations. The Leray-Hopf weak solution was constructed by Leray [13] and Hopf [14], respectively. Later on, much effort has been devoted to establishing the global existence and uniqueness of smooth solutions to the Navier-Stokes equations. Different criteria for regularity of the weak solutions have been proposed and many interesting results were established (see [1524]).

Without loss of generality, we set ν = κ = 1 2 , γ = α + β = 1 in the rest of the paper. The purpose of this paper is to establish a new logarithmically improved blow-up criterion to (1.1), (1.2) in a weak multiplier space. Now we state our results as follows.

Theorem 1.1 Assume that v 0 , w 0 H m ( R 3 ) , m 3 with v 0 = 0 . Let ( v , w ) be a smooth solution to equations (1.1), (1.2) for 0 t < T . If v satisfies
0 T v M ( H ˙ r H ˙ r ) 1 r 1 + ln ( e + v L ) d t < , 0 r < 1 ,
(1.3)

then the solution ( v , w ) can be extended beyond t = T .

We have the following corollary immediately.

Corollary 1.1 Assume that v 0 , w 0 H m ( R 3 ) , m 3 with v 0 = 0 . Let ( v , w ) be a smooth solution to equations (1.1), (1.2) for 0 t < T . Suppose that T is the maximal existence time, then
0 T v M ( H ˙ r H ˙ r ) 1 r 1 + ln ( e + v L ) d t = , 0 r < 1 .
(1.4)

The paper is organized as follows. We first state some preliminaries on functional settings and some important inequalities in Section 2, which play an important role in the proof of our main result. Then we prove the main result in Section 3.

2 Preliminaries

Definition 2.1 [25]

For 0 < r < 3 2 , M ( H ˙ r H ˙ r ) is a Banach space of all distributions f on R 3 such that there exists a constant C such that for all u D , we have f u H ˙ r and
f u H ˙ r C u H ˙ r ,

where we denote by H ˙ r the completion of the space C 0 ( R 3 ) with respect to the norm u H ˙ r = ( Δ ) r 2 u L 2 .

The norm of M ( H ˙ r H ˙ r ) is given by the operator norm of pointwise multiplication
f M ( H ˙ r H ˙ r ) = sup { f u H ˙ r : u H ˙ r 1 , u D } .

The following lemma comes from [26].

Lemma 2.2 Assume that 1 < p < . For f , g W m , p , and 1 < q , 1 < r < , we have
α ( f g ) f α g L p C ( f L q 1 α 1 g L r 1 + g L q 2 α f L r 2 ) ,
(2.1)

where 1 α m and 1 p = 1 q 1 + 1 r 1 = 1 q 2 + 1 r 2 .

We also need the following interpolation inequalities in three space dimensions.

Lemma 2.3 In three space dimensions, the following inequalities hold:
{ f L 4 C f L 2 1 8 2 f L 2 7 8 , f L 4 C f L 2 5 8 2 f L 2 3 8 , 2 f L 4 C f L 2 1 12 3 f L 2 11 12 , 2 f L 2 C f L 2 1 3 3 f L 2 2 3 .
(2.2)

3 Proof of Theorem 1.1

Multiplying the first equation of (1.1) by v and integrating with x respect to on R 3 , using integration by parts, we obtain
1 2 d d t v ( t ) L 2 2 + v ( t ) L 2 2 = R 3 ( × w ) v d x .
(3.1)
Similarly, we get
1 2 d d t w ( t ) L 2 2 + w ( t ) L 2 2 + w L 2 2 + 2 w L 2 2 = R 3 ( × v ) w d x .
(3.2)
Summing up (3.1)-(3.2), we deduce that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-122/MediaObjects/13661_2013_Article_344_Equ9_HTML.gif
(3.3)
We apply integration by parts and the Cauchy inequality. This yields
R 3 ( × w ) v d x + R 3 ( × v ) w d x 1 2 v L 2 2 + 2 w L 2 2 .
(3.4)
Substituting (3.3) into (3.4) yields
1 2 d d t ( v ( t ) L 2 2 + w ( t ) L 2 2 ) + 1 2 v ( t ) L 2 2 + w ( t ) L 2 2 + w L 2 2 0 .
Integrating with respect to t, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-122/MediaObjects/13661_2013_Article_344_Equ11_HTML.gif
(3.5)
Multiplying the first equation of (1.1) by | v | 2 v , then integrating the resulting equation with respect to x over R 3 and using integrating by parts, we obtain
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-122/MediaObjects/13661_2013_Article_344_Equ12_HTML.gif
(3.6)
We multiply the second equation of (1.1) by | w | 2 w , then integrate the resulting equation with respect to x over R 3 and use integrating by parts. This yields
1 4 d d t w L 4 4 + | w | w L 2 2 + 1 2 | w | w L 2 2 + 2 w L 4 4 R 3 v × ( | w | 2 w ) d x ,
(3.7)
where we have used
R 3 ( w ) ( w | w | 2 ) d x = R 3 | w | 2 | w | 2 d x + R 3 ( w ) w | w | 2 d x 1 2 R 3 | w | 2 | w | 2 d x 1 2 R 3 | | w | 2 | 2 d x .
Equations (3.6) and (3.7) give
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-122/MediaObjects/13661_2013_Article_344_Equ14_HTML.gif
(3.8)
Making use of the Young inequality, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-122/MediaObjects/13661_2013_Article_344_Equ15_HTML.gif
(3.9)
Applying the divergence operator to the first equation of (1.1) produces the expression of the pressure
p = ( Δ ) 1 ( v v ) .
(3.10)
It follows from (3.10) that
p L 2 C v L 4 2 , p L 2 C | v | v L 2 .
(3.11)
By integration by parts and (3.11), we obtain
R 3 ( v p ) | v | 2 d x C | v | 2 v H ˙ r p H ˙ r C v M ( H ˙ r H ˙ r ) | v | 2 H ˙ r p L 2 1 r p L 2 r C v M ( H ˙ r H ˙ r ) | v | 2 L 2 1 r | v | 2 L 2 r p L 2 1 r p L 2 r C v M ( H ˙ r H ˙ r ) v L 4 4 ( 1 r ) | v | v L 2 2 r ϵ | v | v L 2 2 + C v M ( H ˙ r H ˙ r ) 1 r v L 4 4 .
(3.12)
Combining (3.8), (3.9), (3.12) and (3.5) yields
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-122/MediaObjects/13661_2013_Article_344_Equ19_HTML.gif
(3.13)
where we have used
H 2 L .
By (1.3), we know that for any small constant ε > 0 , there exists T < T such that
T T v M ( H ˙ r H ˙ r ) 1 r 1 + ln ( e + v L ) d t ε .
(3.14)
Let
A ( t ) = sup T τ t ( 3 v L 2 2 + 3 w L 2 2 ) , T t < T .
(3.15)
The Gronwall inequality and (3.13)-(3.15) give
v L 4 4 + w L 4 4 C exp { C ϵ ( 1 + ln ( e + A ( t ) ) ) } C exp { 2 C ϵ ln ( e + A ( t ) ) } C ( e + A ( t ) ) 2 C ε , T t < T .
(3.16)
Applying to the first equation, then multiplying the resulting equation by v and using integration by parts, the Hölder inequality, (2.2) and the Young inequality, we obtain
1 2 d d t v L 2 2 + 2 v L 2 2 = R 3 v v v d x + R 3 ( × w ) 2 v d x C v L 4 v L 4 2 v L 2 + C w L 2 2 v L 2 C v L 4 v L 2 1 8 2 v L 2 15 8 + 1 8 2 v L 2 2 + C w L 2 2 w L 2 1 4 2 v L 2 2 + 1 4 2 w L 2 2 + C v L 4 16 v L 2 2 + C w L 2 2 .
(3.17)
Similarly, we have
1 2 d d t w L 2 2 + 2 w L 2 2 + w L 2 2 + 2 w L 2 2 = R 3 v w w d x + R 3 ( × v ) 2 w d x C v L 4 w L 4 2 w L 2 + C v L 2 2 w L 2 C v L 4 w L 2 1 8 2 w L 2 15 8 + 1 8 2 w L 2 2 + C v L 2 2 v L 2 1 4 2 w L 2 2 + 1 4 2 v L 2 2 + C v L 4 16 w L 2 2 + C v L 2 2 .
(3.18)
Adding (3.17) and (3.18), we arrive at
d d t ( v L 2 2 + w L 2 2 ) + 2 v L 2 2 + 2 w L 2 2 + w L 2 2 + w L 2 2 C v L 4 16 ( v L 2 2 + w L 2 2 ) + C ( v L 2 2 + w L 2 2 ) .
(3.19)
Equations (3.5), (3.16), (3.19) and the Gronwall inequality give
v L 2 2 + w L 2 2 C ( e + A ( t ) ) 8 C ε , T t < T .
(3.20)
Applying m to the first equation in (1.1), then taking L 2 inner product of the resulting equation with m v and using integration by parts, we have
1 2 d d t m v L 2 2 + m + 1 v L 2 2 = R 3 m ( v v ) m v d x + R 3 m ( × w ) m v d x .
(3.21)
Similarly, we obtain
1 2 d d t m w L 2 2 + m + 1 w L 2 2 + m w L 2 2 + 2 m w L 2 2 = R 3 m ( v w ) m w d x + R 3 m ( × v ) m w d x .
(3.22)
Summing (3.21), (3.22) and using v = 0 , we get
1 2 d d t ( m v L 2 2 + m w L 2 2 ) + m + 1 v L 2 2 + m + 1 w L 2 2 + m w L 2 2 + 2 m w L 2 2 = R 3 [ m ( v v ) v m v ] m v d x R 3 [ m ( v w ) v m w ] m w d x + R 3 m ( × w ) m v d x + R 3 m ( × v ) m w d x I 1 + I 2 + I 3 + I 4 .
(3.23)

In what follows, for simplicity, we set m = 3 .

By the Hölder inequality, (2.1), (2.2) and the Young inequality, we obtain
R 3 [ 3 ( v v ) v 3 v ] 3 v d x 3 ( v v ) v 3 v L 2 3 v L 2 C v L 4 3 v L 4 3 v L 2 C v L 2 5 8 3 v L 2 3 8 v L 2 1 12 4 v L 2 11 12 v L 2 1 3 4 v L 2 2 3 C v L 2 25 24 3 v L 2 3 8 4 v L 2 19 12 1 8 4 v L 2 2 + C v L 2 5 3 v L 2 9 5 1 8 4 v L 2 2 + C ( e + A ( t ) ) 20 C ε + 9 10
(3.24)
and
R 3 [ 3 ( v w ) v 3 w ] 3 w d x 3 ( v w ) v 3 w L 2 3 w L 2 C v L 4 3 w L 4 3 w L 2 + w L 4 3 v L 4 3 w L 2 C v L 2 5 8 3 v L 2 3 8 w L 2 1 12 4 w L 2 11 12 w L 2 1 3 4 w L 2 2 3 + C w L 2 5 8 3 w L 2 3 8 v L 2 1 12 4 v L 2 11 12 w L 2 1 3 4 w L 2 2 3 C v L 2 5 8 3 v L 2 3 8 w L 2 5 12 4 w L 2 19 12 + C w L 2 23 24 3 w L 2 3 8 v L 2 1 12 4 v L 2 11 12 4 w L 2 2 3 1 8 4 v L 2 2 + 1 8 4 w L 2 2 + C v L 2 3 w L 2 2 3 v L 2 9 5 + C v L 2 2 5 w L 2 23 5 3 w L 2 9 5 1 8 4 v L 2 2 + 1 8 4 w L 2 2 + C ( e + A ( t ) ) 20 C ε + 9 10 .
(3.25)
From the Young inequality and (2.2), we deduce that
I 3 C 4 v L 2 3 w L 2 1 8 4 v L 2 2 + C 3 w L 2 2 1 8 4 v L 2 2 + C w L 2 2 3 4 w L 2 4 3 1 8 4 v L 2 2 + 1 8 4 w L 2 2 + C w L 2 2 1 8 4 v L 2 2 + 1 8 4 w L 2 2 + C ( e + A ( t ) ) 8 C ε , T t < T .
(3.26)
Similarly, we have
I 4 1 8 4 v L 2 2 + 1 8 4 w L 2 2 + C ( e + A ( t ) ) 8 C ε , T t < T .
(3.27)
Inserting (3.24)-(3.27) into (3.23) and taking ε small enough such that 20 C ε < 1 10 , we obtain
d d t ( 3 v L 2 2 + 3 w L 2 2 ) C ( e + A ( t ) ) , T t < T ,
(3.28)

for all T t < T .

Integrating (3.28) with respect to time from T to τ, we have
e + 3 v ( τ ) L 2 2 + 3 w ( τ ) L 2 2 e + 3 v ( T ) L 2 2 + 3 w ( T ) L 2 2 + C 2 T τ ( e + A ( s ) ) d s .
(3.29)
We get from (3.29)
e + A ( t ) e + 3 v ( T ) L 2 2 + 3 w ( T ) L 2 2 + C 2 T t ( e + A ( τ ) ) d τ .
(3.30)
For all T t < T , with the help of Gronwall inequality and (3.30), we have
e + 3 v ( t ) L 2 2 + 3 w ( t ) L 2 2 C ,
(3.31)

where C depends on v ( T ) L 2 2 + w ( T ) L 2 2 . (3.31) and (3.5) imply ( v , w ) L ( 0 , T ; H 3 ( R 3 ) ) . Thus, ( v , w ) can be extended smoothly beyond t = T . We have completed the proof of Theorem 1.1.

Declarations

Authors’ Affiliations

(1)
School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power

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