Unique solvability for the non-Newtonian magneto-micropolar fluid

Boundary Value Problems20132013:182

DOI: 10.1186/1687-2770-2013-182

Received: 8 January 2013

Accepted: 24 July 2013

Published: 8 August 2013

Abstract

In this paper, a motion of an incompressible non-Newtonian magneto-micropolar fluid is considered. We assume that the stress tensor has a p-structure, and we establish the global in time existence and uniqueness of the weak solutions with p 5 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq1_HTML.gif in three dimensions.

1 Introduction and main results

This paper is concerned about the existence and uniqueness of the weak solutions to the non-Newtonian magneto-micropolar fluid equations in Ω × ( 0 , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq2_HTML.gif, which are described by
{ u t + ( u ) u div ( | e ( u ) | p 2 e ( u ) ) + ( π + 1 2 | b | 2 ) = χ rot ω + ( b ) b , ω t + ( u ) ω μ Δ ω + 2 χ ω = χ rot u , b t λ Δ b + ( u ) b ( b ) u = 0 , div u = div b = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ1_HTML.gif
(1.1)
here Ω R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq3_HTML.gif is an open-bounded domain with Lipschitz boundaries, and the unknowns u, ω, b, π denote the velocity of the fluid, the micro-rotational velocity, magnetic field and hydrostatic pressure, respectively. χ, μ, λ are positive numbers associated with properties of the material: χ is the vortex viscosity, μ is spin viscosity and 1 λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq4_HTML.gif is the magnetic Reynold. In (1.1), e ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq5_HTML.gif is the symmetric part of the velocity gradient, i.e.,
2 e ( u ) = u + ( u ) T . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equa_HTML.gif
To (1.1) we append the following initial and boundary conditions
u ( x , 0 ) = u 0 ( x ) , ω ( x , 0 ) = ω 0 ( x ) , b ( x , 0 ) = b 0 ( x ) , x Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ2_HTML.gif
(1.2)
u ( x , t ) = ω ( x , t ) = b ( x , t ) = 0 , ( x , t ) Σ T , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ3_HTML.gif
(1.3)

where Σ T = Ω × ( 0 , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq6_HTML.gif.

The theory of micropolar fluid was first proposed by Eringen [1] in 1966, which enabled us to consider some physical phenomena that cannot be treated by the classical Navier-Stokes equations for the viscous incompressible fluid, for example, the motion of animal blood, liquid crystals and dilute aqueous polymer solutions, etc.

If p = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq7_HTML.gif, system (1.1) reduces to the classical magneto-micropolar fluid equations, and there are many earlier results concerning the weak and strong solvability in a bounded domain Ω R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq3_HTML.gif. For strong solutions, Galdi and Rionero [2] stated, without proof, the results of existence and uniqueness of strong solutions. Rojas-Medar [3] studied it and established the local in time existence and uniqueness of strong solutions by the spectral Galerkin method. In 1999, Ortega-Torres and Rojas-Medar [4] proved global existence of strong solutions with the small initial values. For weak solutions, Rojas-Medar and Boldrini [5] proved the existence of weak solutions, and in the 2D case, also proved the uniqueness of the weak solutions.

On the other hand, there are few existence results about the non-Newtonian magneto-micropolar fluid, i.e., the p 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq8_HTML.gif case. In a recent work, Gunzburger et al. [6] studied the reduced problem (with both ω = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq9_HTML.gif and χ = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq10_HTML.gif), and gave the global unique solvability of the first initial-boundary value problem in a bounded two or three-dimensional domain. Improved results are proved for the periodic boundary condition case.

In this paper, we will prove the global existence and uniqueness of the weak solutions for the full system (1.1)-(1.3) under the condition that p 5 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq11_HTML.gif. These results are based on the Galerkin method and a series of uniform estimates, which do not depend on the parameters.

Throughout this work, we use a standard notation L p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq12_HTML.gif (normed p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq13_HTML.gif) for Lebesgue L p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq14_HTML.gif-spaces, as well as W k , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq15_HTML.gif (normed k , p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq16_HTML.gif) for the usual Sobolev spaces. As usual, C 0 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq17_HTML.gif denotes the set of all C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq18_HTML.gif-functions with the compact support in Ω. Given T > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq19_HTML.gif and a Banach space X, we denote by L q ( 0 , T ; X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq20_HTML.gif Bochner spaces, which are equipped with the norm
L q ( 0 , T ; X ) : = ( 0 T X q d s ) 1 q . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equb_HTML.gif
We also introduce the following functional vector spaces:
V { u C 0 ( Ω ) , div u = 0 } , H the closure of  V  in  L 2 ( Ω ) , V p the closure of  V  in  W 1 , p ( Ω ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equc_HTML.gif

We next introduce the definition of a weak solution for problems (1.1)-(1.3).

Definition 1.1 We say that ( u , ω , b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq21_HTML.gif is a weak solution to problems (1.1)-(1.3) if
u L ( 0 , T ; V p ) , u t L 2 ( 0 , T ; H ) , ω L ( 0 , T ; W 1 , 2 ( Ω ) ) L 2 ( 0 , T ; W 2 , 2 ( Ω ) ) , b L ( 0 , T ; V 2 ) L 2 ( 0 , T ; W 2 , 2 ( Ω ) ) , ω t L 2 ( 0 , T ; W 1 , 2 ( Ω ) ) , b t L 2 ( 0 , T ; H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ4_HTML.gif
(1.4)
satisfy
u t , ϕ + | e ( u ) | p 2 e ( u ) , e ( ϕ ) = χ rot ω , ϕ + ( b ) b , ϕ ( u ) u , ϕ for all  ϕ V p , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ5_HTML.gif
(1.5)
ω t , ψ + μ ω , ψ = χ rot u , ψ ( u ) ω , ψ 2 χ ω , ψ for all  ψ W 0 1 , 2 ( Ω ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ6_HTML.gif
(1.6)
b t , η + λ b , η = ( b ) u , η ( u ) b , η for all  η V 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ7_HTML.gif
(1.7)

where the symbol , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq22_HTML.gif denotes a generic duality pairing.

The following theorem gives the main results of this paper.

Theorem 1.1 Let Ω R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq3_HTML.gif be an open-bounded domain with a Lipschitz boundary Ω. Assume that p 5 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq23_HTML.gif, u 0 W 1 , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq24_HTML.gif, ω 0 , b 0 W 1 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq25_HTML.gif. Then, for T ( 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq26_HTML.gif, there exists a unique weak solution to problem (1.1)-(1.3) in the sense of Definition  1.1.

Remark 1.1 If (1.4)-(1.5) hold, it could be easy to introduce the pressure π L ( 0 , T ; L p ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq27_HTML.gif, p = p / ( p 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq28_HTML.gif. This will be done at the end of Section 3.

For latter use, let us state some useful inequalities.

Lemma 1.1 (See [7]) (Korn’s inequality)

Let 1 < p < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq29_HTML.gif. Then there exists a constant C p = C p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq30_HTML.gif such that
C p v 1 , p e ( v ) p for all v W 0 1 , p ( Ω ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ8_HTML.gif
(1.8)

where Ω R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq31_HTML.gif is open and bounded with a Lipschitz boundary.

Lemma 1.2 (See [8]) (On negative norm)

Let 1 < p < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq29_HTML.gif, and let v W 0 1 , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq32_HTML.gif. Then there exists a constant C such that
C v p v 1 , p + v 1 , p . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equd_HTML.gif

Lemma 1.3 (See [9])

Let v , w W 1 , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq33_HTML.gif. For each 2 < p < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq34_HTML.gif, there exists a constant C = C ( p ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq35_HTML.gif such that
e ( v ) e ( w ) p p C Ω ( | e ( v ) | p 2 e ( v ) | e ( w ) | p 2 e ( w ) ) ( e ( v ) e ( w ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Eque_HTML.gif
By using Hölder’s inequality and the imbedding inequality, we could arrive at
u m C ( m , r ) u r + C ˜ ( m , r ) u 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ9_HTML.gif
(1.9)
with
{ m 3 r / ( 3 r ) for  r [ 1 , 3 ) , for any  m < for  r = 3 , m = for  r > 3 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equf_HTML.gif
Here, C ˜ ( m , r ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq36_HTML.gif if u | Ω = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq37_HTML.gif or Ω u d x = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq38_HTML.gif. We will also apply the so-called multiplicative inequalities
u q C ( q ) u 2 α u 2 1 α + C ˆ ( q ) u 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ10_HTML.gif
(1.10)
with
α = 3 ( 1 2 1 q ) [ 0 , 1 ] , q [ 2 , 6 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equg_HTML.gif

If u | Ω = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq37_HTML.gif or Ω u d x = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq39_HTML.gif, then C ˆ ( q ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq40_HTML.gif.

Finally, the paper is organized as follows. In Section 2, we focus on the derivation of the priori estimates for the smooth solutions. On the bases of these estimates, in Section 3, we get the existence result with the help of the Galerkin method. The aim of Section 4 is to give the uniqueness criterion.

2 The priori estimates

Let ( u , ω , b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq21_HTML.gif be a smooth solution to system (1.1)-(1.3). The goal of this section is to derive some priori estimates about it. In all the following sections, we always assume that p 5 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq23_HTML.gif holds.

Setting ϕ = u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq41_HTML.gif in (1.5), ψ = ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq42_HTML.gif in (1.6), η = b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq43_HTML.gif in (1.7), and observing that ( u ) u , u = ( u ) ω , ω = ( u ) b , b = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq44_HTML.gif, we obtain
1 2 d d t u 2 2 + e ( u ) p p = χ rot ω , u + ( b ) b , u , 1 2 d d t ω 2 2 + μ ω 2 2 + 2 χ ω 2 2 = χ rot u , ω , 1 2 d d t b 2 2 + λ b 2 2 = ( b ) u , b . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equh_HTML.gif
Adding the identities above, noting that ( b ) b , u + ( b ) u , b = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq45_HTML.gif, and Korn’s inequality (1.8), we get
1 2 d d t ( u 2 2 + ω 2 2 + b 2 2 ) + ( C p u p p + μ ω 2 2 + 2 χ ω 2 2 + λ b 2 2 ) = χ rot ω , u + χ rot u , ω C χ ( ω 2 u 2 + u 2 ω 2 ) ε ( ω 2 2 + u p p ) + C ε ( ω 2 2 + u 2 2 ) ( for  p 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equi_HTML.gif
After choosing ε properly small, integrating over ( 0 , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq46_HTML.gif, t ( 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq47_HTML.gif, the Gronwall’s inequality yields that
sup t ( 0 , T ) ( u 2 2 + ω 2 2 + b 2 2 ) + 0 T ( u p p + ω 2 2 + b 2 2 ) C 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ11_HTML.gif
(2.1)

where C 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq48_HTML.gif is a constant depending on the time T and u 0 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq49_HTML.gif, w 0 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq50_HTML.gif, b 0 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq51_HTML.gif.

Next, we derive the higher order estimates for ω and b. Setting ψ = Δ ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq52_HTML.gif in (1.6), we find
1 2 d d t ω 2 2 + μ Δ ω 2 2 = ( u ) ω , Δ ω + 2 χ ω , Δ ω χ rot u , Δ ω | ( u ) ω , Δ ω | + ε Δ ω 2 2 + C ε ( ω 2 2 + u 2 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ12_HTML.gif
(2.2)
for the first term on the right hand side, we compute by the divergence free conditions
| ( u ) ω , Δ ω | = | i j k k u i i ω j , k ω j | C Ω | u | | ω | 2 C u p ω 2 p p 1 2 C u p ( ω 2 2 p 3 p Δ ω 2 3 p + ω 2 2 ) ε Δ ω 2 2 + C ε u p 2 p 2 p 3 ω 2 2 + C u p ω 2 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ13_HTML.gif
(2.3)

where Hölder’s, Young’s inequality and (1.9), (1.10) have been used.

Inserting (2.3) into (2.2), choosing ε = μ 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq53_HTML.gif and integrating over ( 0 , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq54_HTML.gif, t ( 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq55_HTML.gif, we have
1 2 ω 2 2 + μ 2 0 t Δ ω 2 2 C 0 t ( u p 2 p 2 p 3 + u p ) ω 2 2 + C 0 t ( ω 2 2 + u 2 2 ) + 1 2 ω 0 2 2 (2.1) C 0 t ( u p 2 p 2 p 3 + u p ) ω 2 2 + 1 2 ω 0 2 2 + C 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ14_HTML.gif
(2.4)
since p 5 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq23_HTML.gif, we have
2 p 2 p 3 p . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ15_HTML.gif
(2.5)
Gronwall’s inequality and estimate (2.1) now provide the bound
sup t ( 0 , T ) ω 2 2 + 0 T Δ ω 2 2 C 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ16_HTML.gif
(2.6)

where C 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq56_HTML.gif is a constant depending on the time T, C 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq48_HTML.gif and ω 0 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq57_HTML.gif.

Next, set η = Δ b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq58_HTML.gif in (1.7) to discover
1 2 d d t b 2 2 + λ Δ b 2 2 = ( u ) b , Δ b ( b ) u , Δ b | ( u ) b , Δ b | + | ( b ) u , Δ b | . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ17_HTML.gif
(2.7)
Reasoning similar to (2.3), we could find
| ( u ) b , Δ b | ε Δ b 2 2 + C ε u p 2 p 2 p 3 b 2 2 + C u p b 2 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ18_HTML.gif
(2.8)
For the second term on the right hand side of (2.7), we compute
| ( b ) u , Δ b | ε Δ b 2 2 + C ε ( b ) u 2 2 ε Δ b 2 2 + C ε u p 2 b 2 p p 2 2 ε Δ b 2 2 + C ε u p 2 b 6 p 5 p 6 2 ε Δ b 2 2 + C ε u p 2 ( b 2 2 ( 2 p 3 ) p Δ b 2 2 ( 3 p ) p + b 2 2 ) ε Δ b 2 2 + δ Δ b 2 2 + C ε δ u p 2 p 2 p 3 b 2 2 + C ε u p 2 b 2 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ19_HTML.gif
(2.9)
where we have used Hölder’s, Young’s inequality and (1.9), (1.10). Choosing ε and δ properly small, inserting (2.8)-(2.9) into (2.7) and integrating it over ( 0 , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq59_HTML.gif, t ( 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq55_HTML.gif, we find
1 2 b 2 2 + λ 2 0 t Δ b 2 2 C 0 t ( u p 2 p 2 p 3 + u p 2 ) b 2 2 + 1 2 b 0 2 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ20_HTML.gif
(2.10)
Observing (2.5) and estimate (2.1), then Gronwall’s inequality yields
sup t ( 0 , T ) b 2 2 + 0 T Δ b 2 2 C 3 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ21_HTML.gif
(2.11)

where C 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq60_HTML.gif is a constant depending on the time T, C 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq48_HTML.gif and b 0 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq61_HTML.gif.

Reasoning analogously to (2.6) and (2.11), it is easy to see that identity (1.6) with ψ = ω t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq62_HTML.gif, (1.7) with η = b t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq63_HTML.gif, with the help of (2.6) and (2.11), guarantee the estimate
0 T ω t 2 2 C 4 , 0 T b t 2 2 C 5 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ22_HTML.gif
(2.12)

where C 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq64_HTML.gif, C 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq65_HTML.gif are both constants depending only on the time T and some norm of the initial values.

In fact (we here only take b t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq66_HTML.gif as an example), set η = b t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq63_HTML.gif in (1.7), we deduce that
b t 2 2 + λ 2 d d t b 2 2 | ( b ) u , b t | + | ( u ) b , b t | ε b t 2 2 + C ε ( ( b ) u 2 2 + ( u ) b 2 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ23_HTML.gif
(2.13)
Now, we compute, by using Hölder’s, Young’s inequality and (1.9), (1.10)
( b ) u 2 2 u p 2 b 2 p p 2 2 C ( u p p + b 2 p p 2 2 p p 2 ) C ( u p p + b 6 p 5 p 6 2 p p 2 ) C u p p + C ( b 2 2 ( 2 p 3 ) p 2 Δ b 2 2 ( 3 p ) p 2 + b 2 2 p p 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ24_HTML.gif
(2.14)
and
( u ) b 2 2 u 3 p 3 p 2 b 6 p 5 p 6 2 C u p 2 b 6 p 5 p 6 2 C ( u p p + b 6 p 5 p 6 2 p p 2 ) C u p p + C ( b 2 2 ( 2 p 3 ) p 2 Δ b 2 2 ( 3 p ) p 2 + b 2 2 p p 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ25_HTML.gif
(2.15)
Combining (2.13)-(2.15), by choosing ε = 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq67_HTML.gif, we arrive at
1 2 b t 2 2 + λ 2 d d t b 2 2 C ( u p p + b 2 2 ( 2 p 3 ) p 2 Δ b 2 2 ( 3 p ) p 2 + b 2 2 p p 2 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equj_HTML.gif

noting that p 5 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq23_HTML.gif, so 2 ( 3 p ) / ( p 2 ) 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq68_HTML.gif, and now estimate (2.1), (2.11) and Gronwall’s inequality imply the estimate of b t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq66_HTML.gif in (2.12).

In the following, we will derive the bound for u t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq69_HTML.gif. Setting η = u t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq70_HTML.gif in (1.5), we deduce that
u t 2 2 + 1 p d d t e ( u ) p p | ( b ) b , u t | + | ( u ) u , u t | + χ | rot ω , u t | ε u t 2 2 + C ε ( ( b ) b 2 2 + ( u ) u 2 2 + ω 2 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equk_HTML.gif
Integrating it over ( 0 , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq54_HTML.gif, t ( 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq55_HTML.gif, by choosing ε = 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq67_HTML.gif and Korn’s inequality, we have
1 2 0 t u t 2 2 + C p u p p C 0 t ( ( b ) b 2 2 + ( u ) u 2 2 ) + C 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ26_HTML.gif
(2.16)
Now, we compute, by (2.11)
0 t ( b ) b 2 2 0 t b 6 2 b 3 2 C 0 t b 2 2 ( b 2 Δ b 2 + b 2 2 ) C sup τ ( 0 , t ) b 2 3 0 t Δ b 2 + C sup τ ( 0 , t ) b 2 4 t C 3 4 ( 1 + T ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ27_HTML.gif
(2.17)
0 t ( u ) u 2 2 0 t u 2 p p 2 2 u p 2 C 0 t u p 4 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ28_HTML.gif
(2.18)
Inserting (2.17)-(2.18) into (2.16), by appealing to Korn’s inequality, it follows that
u p p + 0 t u t 2 2 C 0 t u p p u p 4 p + C 6 + C u 0 p p , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ29_HTML.gif
(2.19)
where C 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq71_HTML.gif depends on T and C 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq60_HTML.gif. Now, Gronwall’s inequality and (2.1) yield that
sup t ( 0 , T ) u p + 0 T u t 2 2 C 7 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ30_HTML.gif
(2.20)

where C 7 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq72_HTML.gif depends on the time T, C 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq48_HTML.gif, C 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq60_HTML.gif and u 0 p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq73_HTML.gif.

3 Approximate solutions and existence result

In this section, we show the existence of a weak solution to the system (1.1)-(1.3) via the Galerkin approximations. For this purpose, we take the set { ϕ i } i = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq74_HTML.gif formed by the eigenvectors ϕ i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq75_HTML.gif, i = 1 , 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq76_HTML.gif , of the Stokes operator and the set { ψ i } i = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq77_HTML.gif formed by the eigenvectors ψ i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq78_HTML.gif, i = 1 , 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq76_HTML.gif , of the Laplace operator. According to the Appendix of [7], the functions { ϕ i } i = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq74_HTML.gif form a basis in the space V p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq79_HTML.gif, and V 2 W 2 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq80_HTML.gif. Setting R k = span { ϕ 1 , ϕ 2 , , ϕ k } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq81_HTML.gif and S k = span { ψ 1 , ψ 2 , , ψ k } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq82_HTML.gif, we construct the Galerkin approximations { u k , ω k , b k } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq83_HTML.gif being of the form
u k ( x , t ) = i = 1 k a i k ( t ) ϕ i ( x ) ; ω k ( x , t ) = i = 1 k c i k ( t ) ψ i ( x ) ; b k ( x , t ) = i = 1 k d i k ( t ) ϕ i ( x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equl_HTML.gif
where a k : = ( a 1 k , , a k k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq84_HTML.gif, c k : = ( c 1 k , , c k k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq85_HTML.gif, d k : = ( d 1 k , , d k k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq86_HTML.gif solve the system of ordinary equations
u t k , ϕ + | e ( u k ) | p 2 e ( u k ) , e ( ϕ ) = χ rot ω k , ϕ + ( b k ) b k , ϕ ( u k ) u k , ϕ for all  ϕ R k , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ31_HTML.gif
(3.1)
ω t k , ψ + μ ω k , ψ = χ rot u k , ψ ( u k ) ω k , ψ 2 χ ω k , ψ for all  ψ S k , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ32_HTML.gif
(3.2)
b t k , ϕ + λ b k , ϕ = ( b k ) u k , ϕ ( u k ) b k , ϕ for all  ϕ R k . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ33_HTML.gif
(3.3)
Moreover, we require that u k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq87_HTML.gif, ω k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq88_HTML.gif, b k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq89_HTML.gif satisfy the following initial conditions
u k | t = 0 = i = 1 k ( u 0 , ϕ i ) ϕ i , ω k | t = 0 = i = 1 k ( ω 0 , ψ i ) ψ i , b k | t = 0 = i = 1 k ( b 0 , ϕ i ) ϕ i . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ34_HTML.gif
(3.4)
The local solvability is guaranteed by the Carathéodory theorem, and the global unique solvability follows from the fact that
sup t ( 0 , T ) u k ( t ) 2 2 = sup t ( 0 , T ) i = 1 k ( a i k ( t ) ) 2 C , sup t ( 0 , T ) ω k ( t ) 2 2 = sup t ( 0 , T ) i = 1 k ( c i k ( t ) ) 2 C , sup t ( 0 , T ) b k ( t ) 2 2 = sup t ( 0 , T ) i = 1 k ( d i k ( t ) ) 2 C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equm_HTML.gif
with upper bounds C that do not depend on k. Moreover, we have for u k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq87_HTML.gif, ω k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq88_HTML.gif and b k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq89_HTML.gif the same estimates for all norms we have obtained in Section 2. More precisely, we have
sup t ( 0 , T ) u k 2 , sup t ( 0 , T ) u k p , 0 T u t 2 2 C , sup t ( 0 , T ) ω k 1 , 2 , 0 t ω k 2 , 2 2 , 0 T ω t 2 2 C , sup t ( 0 , T ) b k 1 , 2 , 0 t b k 2 , 2 2 , 0 T b t 2 2 C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ35_HTML.gif
(3.5)

with a constant C that does not depend on k.

Uniform estimates (3.5) imply that there exists a subsequence of { u k } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq90_HTML.gif, { ω k } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq91_HTML.gif and { b k } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq92_HTML.gif (not relabeled) such that
u k u , weak-* in  L ( 0 , T ; H ) L ( 0 , T ; V p ) , ω k ω , weakly in  L 2 ( 0 , T ; W 2 , 2 ( Ω ) )  and  weak-* in  L ( 0 , T ; W 1 , 2 ( Ω ) ) , b k b , weakly in  L 2 ( 0 , T ; V 2 W 2 , 2 ( Ω ) )  and  weak-* in  L ( 0 , T ; V 2 ) , u t k u t , b t k b t , weakly in  L 2 ( 0 , T ; H ) , ω t k ω t , weakly in  L 2 ( 0 , T ; L 2 ( Ω ) ) , | e ( u k ) | p 2 e ( u k ) Λ , weakly in  L p ( 0 , T ; L p ( Ω ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equn_HTML.gif
where p = p / ( p 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq93_HTML.gif. Therefore, by making use of the Aubin-Lions lemma (see Lions [10], Theorem 1.5.1), we have
u k u , strongly in  L 2 ( 0 , T ; H ) , ω k ω , strongly in  L 2 ( 0 , T ; W 1 , 2 ( Ω ) ) , b k b , strongly in  L 2 ( 0 , T ; V 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equo_HTML.gif
With the convergence above, it is easy to pass to the limit as k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq94_HTML.gif in (3.1)-(3.3) to find
u t , ϕ + Λ , e ( ϕ ) = χ rot ω , ϕ + ( b ) b , ϕ ( u ) u , ϕ for all  ϕ V p , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ36_HTML.gif
(3.6)
ω t , ψ + μ ω , ψ = χ rot u , ψ ( u ) ω , ψ 2 χ ω , ψ for all  ψ W 2 , 2 ( Ω ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ37_HTML.gif
(3.7)
b t , ϕ + λ b , ϕ = ( b ) u , ϕ ( u ) b , ϕ for all  ϕ V 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ38_HTML.gif
(3.8)
Next, to complete the existence proof, we need to verify that
Λ = | e ( u ) | p 2 e ( u ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ39_HTML.gif
(3.9)
By Lemma 1.3, we have
e ( u k ) e ( u ) p p C Ω ( | e ( u k ) | p 2 e ( u k ) | e ( u ) | p 2 e ( u ) ) ( e ( u k ) e ( u ) ) = C Ω | e ( u k ) | p 2 e ( u k ) e ( u k ) Ω | e ( u ) | p 2 e ( u ) ( e ( u k ) e ( u ) ) Ω | e ( u k ) | p 2 e ( u k ) e ( u ) = (3.1) χ rot ω k , u k + ( b k ) b k , u k u t k , u k Ω | e ( u ) | p 2 e ( u ) ( e ( u k ) e ( u ) ) Ω | e ( u k ) | p 2 e ( u k ) e ( u ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equp_HTML.gif
Considering lim k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq95_HTML.gif of this identity together with (3.6) implies that
lim k e ( u k ) e ( u ) p p 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equq_HTML.gif

and thus (3.9) follows.

Having the estimates
u L ( 0 , T ; W 0 1 , p ( Ω ) ) , ω L ( 0 , T ; W 0 1 , 2 ( Ω ) ) , b L ( 0 , T ; W 0 1 , 2 ( Ω ) ) C , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ40_HTML.gif
(3.10)
we can now introduce the pressure from (1.5). For t ( 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq96_HTML.gif, define the functional F W 1 , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq97_HTML.gif as
F , ξ : = div ( | e ( u ) | p 2 e ( u ) ) , ξ + χ rot ω , ξ b × rot b , ξ ( u ) u , ξ u t , ξ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equr_HTML.gif
We have
F , ξ = 0 , ξ V p ,  a.e.  t ( 0 , T ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equs_HTML.gif
By using De Rahm’s theorem (see [11], Lemma 2.7), we obtain a function π L p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq98_HTML.gif such that
F = π , a.e.  t ( 0 , T ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equt_HTML.gif
Moreover, due to estimates (3.10),
π W 1 , p ( Ω ) C , a.e.  t ( 0 , T ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equu_HTML.gif
Then, by Lemma 1.2, there is a generic constant C, depending only on the data such that
π p C , a.e.  t ( 0 , T ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equv_HTML.gif

Now, we complete the proof of the existence part of Theorem 1.1.

4 Uniqueness criterion

Let ( u 1 , ω 1 , b 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq99_HTML.gif and ( u 2 , ω 2 , b 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq100_HTML.gif be both solutions of the problem. Then, for their difference u ¯ = u 1 u 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq101_HTML.gif, ω ¯ = ω 1 ω 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq102_HTML.gif, b ¯ = b 1 b 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq103_HTML.gif, we have
u ¯ t , ϕ + | e ( u 1 ) | p 2 e ( u 1 ) | e ( u 2 ) | p 2 e ( u 2 ) , e ( ϕ ) = χ rot ω ¯ , ϕ + ( b 1 ) b ¯ , ϕ + ( b ¯ ) b 2 , ϕ ( u 1 ) u ¯ , ϕ ( u ¯ ) u 2 , ϕ for all  ϕ V p , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ41_HTML.gif
(4.1)
ω ¯ t , ψ + μ ω ¯ , ψ + 2 χ ω ¯ , ψ = χ rot u ¯ , ψ ( u 1 ) ω ¯ , ψ ( u ¯ ) ω 2 , ψ for all  ψ W 0 1 , 2 ( Ω ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ42_HTML.gif
(4.2)
b ¯ t , η + λ b ¯ , η = ( b 1 ) u ¯ , η + ( b ¯ ) u 2 , η ( u 1 ) b ¯ , η ( u ¯ ) b 2 , η for all  η V 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ43_HTML.gif
(4.3)
Taking ϕ = u ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq104_HTML.gif in (4.1), by Lemma 1.3 and the fact that ( u 1 ) u ¯ , u ¯ = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq105_HTML.gif and p > 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq106_HTML.gif, we obtain
1 2 d d t u ¯ 2 2 + C p u ¯ p p χ rot ω ¯ , u ¯ + ( b ¯ ) b 2 , u ¯ ( u ¯ ) u 2 , u ¯ + ( b 1 ) b ¯ , u ¯ i = 1 3 J i + ( b 1 ) b ¯ , u ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equw_HTML.gif
for each J i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq107_HTML.gif, i = 1 , 2 , 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq108_HTML.gif, it follows from Hölder’s, Young’s inequality and (1.9), (1.10) that
J 1 χ | rot ω ¯ , u ¯ | χ ω ¯ 2 u ¯ 2 ε ω ¯ 2 2 + C ε u ¯ 2 2 , J 2 | ( b ¯ ) b 2 , u ¯ | b 2 6 b ¯ 3 u ¯ 2 C ( Δ b 2 2 + b 2 2 ) b ¯ 2 u ¯ 2 ε b ¯ 2 2 + C ε ( Δ b 2 2 2 + b 2 2 2 ) u ¯ 2 2 , J 3 | ( u ¯ ) u 2 , u ¯ | u 2 p u ¯ 2 p p 1 2 u 2 p u ¯ 2 2 p 3 p u ¯ 2 3 p ε u ¯ 2 2 + C ε u 2 p 2 p 2 p 3 u ¯ 2 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equx_HTML.gif
so, we have
1 2 d d t u ¯ 2 2 + C p u ¯ p p ε ( u ¯ 2 2 + ω ¯ 2 2 + b ¯ 2 2 ) + C ε ( 1 + b 2 2 2 + Δ b 2 2 2 + u 2 p 2 p 2 p 3 ) u ¯ 2 2 + ( b 1 ) b ¯ , u ¯ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ44_HTML.gif
(4.4)
Taking ψ = ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq109_HTML.gif in (4.2) and noting that ( u 1 ) ω ¯ , ω ¯ = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq110_HTML.gif, it follows
1 2 d d t ω ¯ 2 2 + μ ω ¯ 2 2 + 2 χ ω ¯ 2 2 = χ rot u ¯ , ω ¯ ( u ¯ ) ω 2 , ω ¯ i = 1 2 I i , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equy_HTML.gif
and for I i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq111_HTML.gif, i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq112_HTML.gif,
I 1 χ u ¯ 2 ω ¯ 2 ε u ¯ 2 2 + C ε ω ¯ 2 2 , I 2 ω 2 6 ω ¯ 3 u ¯ 2 C ( Δ ω 2 2 + ω 2 2 ) ω ¯ 2 u ¯ 2 ε ω ¯ 2 2 + C ε ( Δ ω 2 2 2 + ω 2 2 2 ) u ¯ 2 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equz_HTML.gif
thus, we obtain
1 2 d d t ω ¯ 2 2 + μ ω ¯ 2 2 ε ( u ¯ 2 2 + ω ¯ 2 2 ) + C ε ω ¯ 2 2 1 2 d d t ω ¯ 2 2 + μ ω ¯ 2 2 + C ε ( Δ ω 2 2 2 + ω 2 2 2 ) u ¯ 2 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ45_HTML.gif
(4.5)
Similarly, by taking η = b ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq113_HTML.gif in (4.3), reasoning analogous as above, we could get
1 2 d d t b ¯ 2 2 + λ b ¯ 2 2 ε b ¯ 2 2 + C ε ( Δ b 2 2 2 + b 2 2 2 ) u ¯ 2 2 + C ε u 2 p 2 p 2 p 3 b ¯ 2 2 + ( b 1 ) u ¯ , b ¯ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equ46_HTML.gif
(4.6)
Adding (4.4)-(4.6) and observing that ( b 1 ) b ¯ , u ¯ + ( b 1 ) u ¯ , b ¯ = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq114_HTML.gif, after choosing ε properly small, we finally get
1 2 d d t ( u ¯ 2 2 + ω ¯ 2 2 + b ¯ 2 2 ) + C ( u ¯ 2 2 + ω ¯ 2 2 + b ¯ 2 2 ) C F ( t ) ( u ¯ 2 2 + ω ¯ 2 2 + b ¯ 2 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equaa_HTML.gif
with
F ( t ) = 1 + b 2 2 2 + Δ b 2 2 2 + u 2 p 2 p 2 p 3 + ω 2 2 2 + Δ ω 2 2 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equab_HTML.gif
Since 2 p / ( 2 p 3 ) p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq115_HTML.gif for p 5 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_IEq23_HTML.gif, then Gronwall’s inequality and the estimates obtained in Section 2 yield that
u ¯ = ω ¯ = b ¯ = 0 for  t [ 0 , T ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-182/MediaObjects/13661_2013_Article_445_Equac_HTML.gif

This completes the proof of the theorem.

Declarations

Acknowledgements

The author would like to thank the referees for valuable comments and suggestions for improving this paper. This work was partially supported by the National NSF (Grant No. 10971080) of China.

Authors’ Affiliations

(1)
School of Science, Changchun University of Science and Technology

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