Blow-up criterion for 3D compressible viscous magneto-micropolar fluids with initial vacuum

Boundary Value Problems20132013:160

DOI: 10.1186/1687-2770-2013-160

Received: 18 February 2013

Accepted: 16 June 2013

Published: 1 July 2013

Abstract

In this paper, the author establishes a blow-up criterion of strong solutions to 3D compressible viscous magneto-micropolar fluids. It is shown that if the density and the velocity satisfy ρ L ( 0 , T ; L ) + u L s ( 0 , T ; L r ) < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq1_HTML.gif, where 2 s + 3 r 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq2_HTML.gif and 3 < r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq3_HTML.gif, then the strong solutions to the Cauchy problem can exist globally over R 3 × [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq4_HTML.gif. The initial density may vanish on open sets, that is, the initial vacuum is allowed.

MSC:76N10, 35B44, 35B45.

Keywords

compressible magneto-micropolar fluids blow-up criterion strong solution vacuum

1 Introduction

In this paper, we consider the following 3D compressible viscous magneto-micropolar fluids:
{ ρ t + div ( ρ u ) = 0 , ( ρ u ) t + div ( ρ u u ) ( μ + ξ ) u ( μ + λ ξ ) div u + P = 2 ξ × w + ( × H ) × H , ( ρ w ) t + div ( ρ u w ) μ w ( μ + λ ) div w + 4 ξ w = 2 ξ × u , H t × ( u × H ) = × ( σ × H ) , div H = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ1_HTML.gif
(1.1)
where x = ( x 1 , x 2 , x 3 ) R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq5_HTML.gif is the spacial coordinate and t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq6_HTML.gif is the time. The unknown functions ρ = ρ ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq7_HTML.gif, u = u ( t , x ) = ( u 1 , u 2 , u 3 ) ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq8_HTML.gif, w = w ( t , x ) = ( w 1 , w 2 , w 3 ) ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq9_HTML.gif, H = H ( t , x ) = ( H 1 , H 2 , H 3 ) ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq10_HTML.gif and P ( ρ ) = A ρ γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq11_HTML.gif ( A > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq12_HTML.gif, γ > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq13_HTML.gif) are the fluid density, velocity, micro-rotational velocity, magnetic field and pressure, respectively. The constants μ, λ, ξ, μ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq14_HTML.gif, λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq15_HTML.gif and σ are the viscosity coefficients of the fluid satisfying
μ , μ , ξ , σ > 0 , 2 μ + 3 λ 4 ξ 0 , and 2 μ + 3 λ 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ2_HTML.gif
(1.2)

System (1.1)-(1.2) describing the motion of aggregates of small solid ferromagnetic particles relative to viscous magnetic fluids, such as water, hydrocarbon, ester, fluorocarbon, etc., in which they are immersed, covers a wide range of heat and mass transfer phenomena, under the action of magnetic fields, and is of great importance in practical and mathematics applications (see [1]). Indeed, (1.1) is composed of the balance laws of mass, momentum, moment of momentum and magnetohydrodynamic, respectively. Due to its importance in mathematics and physics, there is a lot of literature devoted to the mathematical theory of the compressible viscous magneto-micropolar system (see [24]).

For the incompressible magneto-micropolar fluid models where ρ = C o n s t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq16_HTML.gif, Rojas-Medar [5] established local existence and uniqueness of strong solutions by the Galerkin method. Ortega-Torres and Rojas-Medar [6] proved global existence of strong solutions for small initial data. A BKM type blow-up criterion for smooth solution that relies on the vorticity of velocity only was obtained by Yuan [7]. For regularity results, refer to Yuan [8] and Gala [9].

In particular, if the effect of angular velocity field of the particle’s rotation is omitted, i.e., w = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq17_HTML.gif, then (1.1) reduces to compressible magnetohydrodynamic equations (MHD). There are numerous important progress on compressible MHD (see [1012] and the references therein). The local strong solutions to the compressible MHD with large initial data were respectively obtained by Vol’pert-Khudiaev [10] and Fan-Yu [11] in cases that the initial density is strictly positive and the initial density may vanish. Xu-Zhang [12] proved a blow-up criterion that if T < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq18_HTML.gif is the maximal time of existence of a strong solution, then
sup T T ( ρ L ( 0 , T ; L ) + u L s ( 0 , T ; L w r ) ) = , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equa_HTML.gif
where L w r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq19_HTML.gif is the weak L r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq20_HTML.gif space and r, s satisfy
2 s + 3 r 1 , 3 < r . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ3_HTML.gif
(1.3)
If H = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq21_HTML.gif, (1.1) reduces to compressible micropolar fluid equations. Mujakovic [13, 14] considered the one-dimensional motion of compressible viscous micropolar fluids and studied the local/global existence. The global existence of strong solutions to the 1D model with initial vacuum was also obtained in [15]. For multi-dimensional compressible magneto-micropolar equations, Amirat and Hamdache [16] proved the global existence of weak solutions with finite energy and the adiabatic constant for γ > 3 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq22_HTML.gif, which generalized Lions’ pioneering work [17] and the work by Feireisl et al. [18]. Chen [19] established the local existence and uniqueness of strong solutions under the assumption that the initial density may vanish, and in [20] Chen et al. proved a blow-up criterion that
sup T T ( ρ L ( 0 , T ; L ) + ρ u L s ( 0 , T ; L r ) ) = , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equb_HTML.gif

where r, s satisfy (1.3).

If H = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq21_HTML.gif and w = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq17_HTML.gif, (1.1) reduces to isentropic compressible Navier-Stokes equations. In [21], the authors established a Serrin-type blow-up criterion that
sup T T ( div u L 1 ( 0 , T ; L ) + ρ u L s ( 0 , T ; L r ) ) = , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equc_HTML.gif
or
sup T T ( ρ L ( 0 , T ; L ) + ρ u L s ( 0 , T ; L r ) ) = , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equd_HTML.gif

where r, s satisfy (1.3).

In this paper, our main purpose is to establish a blow-up criterion of strong solutions for system (1.1) with the following conditions:
{ ( ρ , u , w , H ) ( x , 0 ) = ( ρ 0 , u 0 , w 0 , H 0 ) ( x ) in  R 3 , ( ρ , u , w , H ) ( x , t ) 0 as  | x | . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ4_HTML.gif
(1.4)
To proceed, we introduce the following notations. For 1 r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq23_HTML.gif, we denote the standard homogeneous and inhomogeneous Sobolev spaces as follows:
{ L r = L r ( R 3 ) , D k , r = { u L loc 1 ( R 3 ) | k u L r < } , u D k , r : = k u L r , W k , r = L r D k , r , H k = W k , 2 , D k = D k , 2 , D 1 = { u L 6 | u L 2 < } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Eque_HTML.gif

To present the main result, we first give the following local existence and uniqueness of strong solutions to the Cauchy problem (1.1), (1.2) and (1.4) with initial vacuum (without proof), which can be obtained by the same method developed by Choe-Kim in [22] (see also Fan-Yu [11] and Chen [19] for MHD and compressible micropolar fluids, respectively).

Theorem 1.1 Assume that for some q ( 3 , 6 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq24_HTML.gif, the initial data ( ρ 0 , u 0 , w 0 , H 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq25_HTML.gif satisfy
0 ρ 0 L 1 H 1 W 1 , q , u 0 D 1 D 2 , w 0 H 2 , H 0 H 2 , div H 0 = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ5_HTML.gif
(1.5)
and the compatibility conditions
( μ + ξ ) u 0 ( μ + λ ξ ) div u 0 + P 0 2 ξ × w 0 ( × H 0 ) × H 0 = ρ 0 1 / 2 g 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ6_HTML.gif
(1.6)
μ w 0 ( μ + λ ) div w 0 2 ξ × u 0 + 4 ξ w 0 = ρ 0 1 / 2 g 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ7_HTML.gif
(1.7)
with some ( g 1 , g 2 ) L 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq26_HTML.gif. Then there exists a positive time T ( 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq27_HTML.gif such that the problem (1.1), (1.2) and (1.4) has a unique strong solution ( ρ , u , w , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq28_HTML.gif in R 3 × [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq29_HTML.gif satisfying, for some q 0 ( 3 , 6 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq30_HTML.gif,
{ ρ C ( [ 0 , T ] ; L 1 H 1 W 1 , q 0 ) , ρ t L ( 0 , T ; L 2 L q 0 ) , ρ 0 , ( u , w , H ) C ( [ 0 , T ] ; D 1 D 2 ) L 2 ( 0 , T ; D 2 , q 0 ) , w C ( [ 0 , T ] ; L 2 ) , H C ( [ 0 , T ] ; H 2 ) , ( ρ u t , ρ w t , H t ) L ( 0 , T ; L 2 ) , ( u t , w t , H t ) L 2 ( 0 , T ; D 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ8_HTML.gif
(1.8)

Motivated by [20, 21] and [12], we have the main purpose in this paper to prove a blow-up criterion for the problem (1.1), (1.2) and (1.4). More precisely, the main result in this paper reads as follows.

Theorem 1.2 Assume that the initial data ( ρ 0 , u 0 , w 0 , H 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq31_HTML.gif satisfies (1.5)-(1.7). Let ( ρ , u , w , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq32_HTML.gif be a strong solution of the Cauchy problem (1.1), (1.2) and (1.4) with the regularities (1.8). If T ( 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq33_HTML.gif is the maximal time of existence, then
lim T T ( ρ L ( 0 , T ; L ) + u L s ( 0 , T ; L r ) ) = , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ9_HTML.gif
(1.9)

for any r and s satisfying (1.3).

Remark 1.3 Theorem 1.1 proves that the strong solutions of (1.1), (1.2) and (1.4) can exist only in a small time T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq34_HTML.gif, which means that if T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq34_HTML.gif is the maximal time of existence, then there must be some component of the fluid mechanics blow-ups. Theorem 1.2 points out one kind of blow-up mechanics.

Remark 1.4 There is no any additional growth condition on the micro-rotational velocity w and magnetic field H. This reveals that the density and the linear velocity play a more important role compared to the angular velocity of rotation of particles and the magnetic field in the regularity theory of solutions to 3D compressible magneto-micropolar fluid flows.

The rest of the paper is devoted to completing the proof of Theorem 1.2.

2 Proof of Theorem 1.2

First, we give the following well-known Gagliardo-Nirenberg inequality that will be used frequently.

Lemma 2.1 For p [ 2 , 6 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq35_HTML.gif, q ( 1 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq36_HTML.gif and r ( 3 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq37_HTML.gif, there exists some generic constant C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq38_HTML.gif, which may depend on p, q and r, such that for any f H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq39_HTML.gif and g L q D 1 , r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq40_HTML.gif, we have
f L p C f L 2 ( 6 p ) / 2 p f L 2 ( 3 p 6 ) / 2 p , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ10_HTML.gif
(2.1)
g L C g L q q ( r 3 ) / ( 3 r + q ( r 3 ) ) g L r 3 r / ( 3 r + q ( r 3 ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ11_HTML.gif
(2.2)

The following BKM’s type inequality which will be used to estimate u L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq41_HTML.gif and ρ L q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq42_HTML.gif with q ( 3 , 6 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq43_HTML.gif can be found in [12].

Lemma 2.2 For 3 < q < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq44_HTML.gif, there is a constant C = C ( q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq45_HTML.gif, depending only on q, such that the following estimate holds for all u L 2 D 1 , q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq46_HTML.gif:
u L C ( div u L + × u L ) log ( e + 2 u L q ) + C u L 2 + C . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ12_HTML.gif
(2.3)
The proof of Theorem 1.2 is based on the contradiction arguments. Let ( ρ , u , w , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq28_HTML.gif be a strong solution of the problem (1.1), (1.2) and (1.4) as described in Theorem 1.1. Suppose that (1.9) is false, that is,
lim T T ( ρ L ( 0 , T ; L ) + u L s ( 0 , T ; L r ) ) M 0 < , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ13_HTML.gif
(2.4)

where r, s satisfy (1.3) and M 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq47_HTML.gif is a constant.

One can easily deduce from the following energy estimate (1.1), (1.2) and (1.4).

Lemma 2.3 It holds that
sup 0 t T ( ρ L 1 L γ + ρ u L 2 2 + ρ w L 2 2 + H L 2 2 ) + 0 T ( u L 2 2 + w H 1 2 + H L 2 2 ) d t C . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ14_HTML.gif
(2.5)

Here and hereafter, C denotes a generic positive constant which may depend on μ, μ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq14_HTML.gif, λ, λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq15_HTML.gif, ξ, σ, A, γ, ρ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq48_HTML.gif, u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq49_HTML.gif, w 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq50_HTML.gif, H 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq51_HTML.gif, g 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq52_HTML.gif, g 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq53_HTML.gif, T and M 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq47_HTML.gif.

We denote the material derivative of f by f ˙ = f t + u f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq54_HTML.gif and set
G 1 : = ( 2 μ + λ ) div u P ( ρ ) 1 2 | H | 2 , G 2 : = ( 2 μ + λ ) div w , V 1 : = × u , V 2 : = × w . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ15_HTML.gif
(2.6)
Since ( × H ) × H = H H 1 2 | H | 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq55_HTML.gif due to (1.1)5, we have from (1.1)2 and (1.1)3 that
{ G 1 = div ( ρ u ˙ H H ) , G 2 4 ξ 2 μ + λ G 2 = div ( ρ w ˙ ) , ( μ + ξ ) V 1 = × ( ρ u ˙ H H ) 2 ξ × V 2 , μ V 2 4 ξ V 2 = × ( ρ w ˙ ) 2 ξ × V 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ16_HTML.gif
(2.7)

Thus, from the standard L p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq56_HTML.gif-estimate of an elliptic system, we have the following lemma.

Lemma 2.4 Under the condition (2.4), it holds that
G 1 L 2 + V 1 L 2 + G 2 H 1 + V 2 H 1 C ( ρ u ˙ L 2 + ρ w ˙ L 2 + u L 2 + w L 2 + H H L 2 ) C ( ρ u t L 2 + ρ w t L 2 + u u L 2 + w w L 2 + u L 2 + w L 2 + H H L 2 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ17_HTML.gif
(2.8)
G 1 L 6 + G 2 L 6 + V 1 L 6 + V 2 L 6 C ( u ˙ L 2 + w ˙ L 2 + ρ u ˙ L 2 + ρ w ˙ L 2 + u L 2 + w L 2 + H H L 2 + H H L 6 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ18_HTML.gif
(2.9)
Proof In view of standard L 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq57_HTML.gif-estimates of elliptic system (2.7), one immediately obtains (2.8). By (2.1) and (2.4), we get that
G 1 L 6 + G 2 L 6 + V 1 L 6 + V 2 L 6 C ( ρ u ˙ L 6 + ρ w ˙ L 6 + G 2 L 2 + V 1 L 6 + V 2 L 2 + V 2 L 6 + H H L 6 ) C ( u ˙ L 2 + w ˙ L 2 + G 2 L 2 + V 1 L 2 + V 2 H 1 + H H L 6 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equf_HTML.gif

which, combined with (2.8), yields (2.9) immediately. □

The next lemma is concerned with the higher integrability of H under the assumption (2.4).

Lemma 2.5 Under the condition (2.4), it holds for any 0 T T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq58_HTML.gif that
H L ( 0 , T ; L q ) C ( q ) for any q [ 2 , ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ19_HTML.gif
(2.10)

where C ( q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq59_HTML.gif is a positive constant depending on q.

The proof is similar to Lemma 3.3 in [12] and is omitted here.

With the help of (2.4) and Lemmas 2.3-2.5, we can prove the following key lemma.

Lemma 2.6 Under the condition (2.4), it holds that for any 0 T < T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq60_HTML.gif,
sup 0 t T ( u L 2 2 + w H 1 2 + H L 2 2 ) + 0 T ( ρ u t L 2 2 + ρ w t L 2 2 + H t L 2 2 + H H 1 2 ) d t C . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ20_HTML.gif
(2.11)
Proof Multiplying (1.1)2, (1.1)3 and (1.1)4 by u t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq61_HTML.gif, w t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq62_HTML.gif and H t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq63_HTML.gif, respectively, and integrating the resulting equations by parts, one obtains after summing up that
1 2 d d t ( μ | u | 2 + ( μ + λ ) ( div u ) 2 + μ | w | 2 + ( μ + λ ) ( div w ) 2 + σ | H | 2 ) d x + 1 2 d d t ξ ( | × u | 2 + 4 | w | 2 ) d x + ( ρ | u t | 2 + ρ | w t | 2 + | H t | 2 ) d x = P div u t d x + 2 ξ ( × w u t + × u w t ) d x ( ρ u u u t + ρ u w w t ) d x + ( H H 1 2 | H | 2 ) u t d x + H u H t d x u H H t d x H H t div u d x : = i = 1 7 I i . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ21_HTML.gif
(2.12)
To estimate the first term on the right-hand side of (2.12), we observe that P satisfies
P t + div ( P u ) + ( γ 1 ) P div u = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equg_HTML.gif
Hence, using (2.4), (2.5) and (2.10) yields that
I 1 = d d t P div u d x P u div u d x + ( γ 1 ) P ( div u ) 2 d x = d d t P div u d x 1 2 μ + λ P u G 1 d x + 1 2 ( 2 μ + λ ) P 2 div u d x 1 2 ( 2 μ + λ ) P u | H | 2 d x + ( γ 1 ) P ( div u ) 2 d x d d t P div u d x + C ( P L 3 u L 6 G 1 L 2 + P L P L 2 u L 2 + P L u L 6 H L 3 H L 2 + P L u L 2 2 ) d d t P div u d x + ε G 1 L 2 2 + C ( ε ) ( u L 2 2 + H L 2 2 + 1 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ22_HTML.gif
(2.13)

where we have used Young’s inequality and (2.1).

For the second term, we have, after integration by parts, that
I 2 = 2 ξ ( w ( × u t ) + ( × u ) w t ) d x = 2 ξ d d t w ( × u ) d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ23_HTML.gif
(2.14)
and by the Cauchy-Schwarz inequality, we have
I 3 1 4 ( ρ u t L 2 2 + ρ w t L 2 2 ) + C ( u u L 2 2 + u w L 2 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ24_HTML.gif
(2.15)
Similarly, integrating by parts and using the fact div H = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq64_HTML.gif, one has
I 4 = ( H u t H 1 2 | H | 2 div u t ) d x = d d t ( H u H 1 2 | H | 2 div u ) d x + ( H t u H + H u H t H H t div u ) d x d d t ( H u H 1 2 | H | 2 div u ) d x + 1 4 H t L 2 2 + C H u L 2 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ25_HTML.gif
(2.16)
For the last three terms on the right-hand side of (2.12), one has from (2.4) that
| i = 5 7 I i | 1 4 H t L 2 2 + C ( u H L 2 2 + H u L 2 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ26_HTML.gif
(2.17)
Thus, putting (2.13)-(2.17) into (2.12) and choosing ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq65_HTML.gif suitably small, we infer from (2.8) that
d d t ( μ | u | 2 + ( μ + λ ) ( div u ) 2 + μ | w | 2 + ( μ + λ ) ( div w ) 2 + σ | H | 2 ) d x + d d t ξ | × u 2 w | 2 d x + ( ρ | u t | 2 + ρ | w t | 2 + | H t | 2 ) d x d d t ( 2 P div u + | H | 2 div u 2 H u H ) d x + C ( u L 2 2 + w L 2 2 + H L 2 2 ) + C ( u u L 2 2 + u w L 2 2 + u H L 2 2 + H u L 2 2 + H H L 2 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ27_HTML.gif
(2.18)
For any r, s satisfying (1.3), we have by the Hölder and Sobolev inequalities that
f g L 2 C f L r g L 2 r r 2 C f L r g L 2 r 3 r g L 6 3 r δ g L 6 + C ( δ ) ( f L r s 2 + 1 ) g L 2 , r > 3 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ28_HTML.gif
(2.19)

for some δ ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq66_HTML.gif.

Taking f = u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq67_HTML.gif, H, g = u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq68_HTML.gif, ∇w, ∇H into (2.19) and using (2.10), we obtain
u u L 2 2 + u w L 2 2 + u H L 2 2 + H u L 2 2 + H H L 2 2 δ ( u L 6 2 + w L 6 2 + H L 6 2 ) + C ( δ ) ( u L r s + 1 ) ( u L 2 2 + w L 2 2 + H L 2 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ29_HTML.gif
(2.20)
By the standard L p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq56_HTML.gif-estimate, one can deduce from (2.1), (2.4), (2.8) and (2.10) that
u L 6 + w L 6 C ( div u L 6 + × u L 6 + div w L 6 + × w L 6 ) C ( G 1 L 6 + G 2 L 6 + V 1 L 6 + V 2 L 6 + P L 6 + | H | 2 L 6 ) C ( 1 + G 1 L 2 + G 2 L 2 + V 1 L 2 + V 2 L 2 ) C ( 1 + ρ u t L 2 + ρ w t L 2 ) + C ( u u L 2 + u u L 2 + H H L 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ30_HTML.gif
(2.21)
Furthermore, it follows from (1.1)4 and Sobolev’s embedding inequality that
H L 6 C H H 1 C ( H t L 2 + u H L 2 + H u L 2 + H L 2 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ31_HTML.gif
(2.22)
putting (2.21) and (2.22) into (2.20), such that
u u L 2 2 + u w L 2 2 + u H L 2 2 + H u L 2 2 + H H L 2 2 C δ ( ρ u t L 2 2 + ρ w t L 2 2 + H t L 2 2 ) + C ( δ ) ( u L r s + 1 ) ( u L 2 2 + w L 2 2 + H L 2 2 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ32_HTML.gif
(2.23)
which, together (2.21) and (2.18), choosing δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq69_HTML.gif suitably small, gives
d d t ( μ | u | 2 + ( μ + λ ) ( div u ) 2 + μ | w | 2 + ( μ + λ ) ( div w ) 2 + σ | H | 2 ) d x + d d t ξ | × u 2 w | 2 d x + ( ρ | u t | 2 + ρ | w t | 2 + | H t | 2 ) d x d d t ( 2 P div u + | H | 2 div u 2 H u H ) d x + C ( u L r s + 1 ) ( u L 2 2 + w L 2 2 + H L 2 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ33_HTML.gif
(2.24)
It is easily seen that
| ( 2 P div u + | H | 2 div u 2 H u H ) d x | μ 4 u L 2 2 + C . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equh_HTML.gif

Taking this into account, we conclude from (2.4), (2.24) and Gronwall’s inequality that part of (2.11) holds for any 0 T < T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq60_HTML.gif. Note that the estimate of H L 2 ( 0 , T ; H 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq70_HTML.gif is a consequence of (2.4), (2.22) and (2.23). The proof of this lemma is completed. □

Next we prove the boundedness of ρ u ˙ L 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq71_HTML.gif, ρ w ˙ L 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq72_HTML.gif, H t L 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq73_HTML.gif and H H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq74_HTML.gif by the compatibility conditions (1.6) and (1.7).

Lemma 2.7 Under the condition (2.4), it holds that for any 0 T < T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq60_HTML.gif,
sup 0 t T ( ρ u ˙ L 2 2 + ρ w ˙ L 2 2 + H t L 2 2 + H H 1 2 ) + 0 T ( u ˙ L 2 2 + w ˙ L 2 2 + H t L 2 2 ) d t C . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ34_HTML.gif
(2.25)
Proof Applying the operator u ˙ j [ t + div ( u ) ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq75_HTML.gif and u ˙ j [ t + div ( u ) ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq75_HTML.gif to both sides of (1.1)2 and (1.1)3, respectively, and using (1.1)1, one can obtain, after a straightforward calculation, that
1 2 d d t ( ρ | u ˙ | 2 + ρ | w ˙ | 2 ) d x = ( μ + ξ ) u ˙ j [ u t j + div ( u u j ) ] d x + ( μ + λ ξ ) u ˙ j [ t j div u + div ( u j div u ) ] d x + μ w ˙ j [ w t j + div ( u w j ) ] d x + ( μ + λ ) w ˙ j [ t j div w + div ( u j div w ) ] d x u ˙ j [ j P t + div ( u j P ) ] d x + 2 ξ u ˙ [ × w t + i ( u i × w ) ] d x + 2 ξ w ˙ [ × u t + i ( u i × u ) ] d x 4 ξ w ˙ j [ w t j + div ( u w j ) ] d x 1 2 u ˙ j [ t j | H | 2 + div ( u j | H | 2 ) ] d x + u ˙ j [ t ( H H j ) + div ( u ( H H j ) ) ] d x : = i = 1 10 J i . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ35_HTML.gif
(2.26)
We get after integration by parts that
J 1 = ( μ + ξ ) ( | u ˙ j | 2 i u ˙ j i u k k u j i u ˙ j u k k i u j + u u ˙ j u j ) d x = ( μ + ξ ) ( | u ˙ j | 2 i u ˙ j i u k k u j + i u ˙ j k u k i u j i u k k u ˙ j i u j ) d x ( μ + ξ ) u ˙ L 2 2 + C u ˙ L 2 u L 4 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ36_HTML.gif
(2.27)
Similarly, we also have
j = 2 4 J j ( μ + λ ξ ) div u ˙ L 2 2 μ w ˙ L 2 2 ( μ + λ ) div w ˙ L 2 2 + C u ˙ L 2 u L 4 2 + C w ˙ L 2 u L 4 w L 4 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ37_HTML.gif
(2.28)
After integration by parts, using (1.1)1 and (2.11), we obtain
J 5 = ( ρ P ( ρ ) div u div u ˙ P ( ρ ) k ( u k j u ˙ j ) P ( ρ ) j ( u k k u ˙ j ) ) d x C u L 2 u ˙ L 2 C u ˙ L 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ38_HTML.gif
(2.29)
Using the definition of the material derivation and integrating by parts, we deduce from (2.1), (2.5) and (2.11) that
J 6 = 2 ξ [ w t ( × u ˙ ) u u ˙ ( × w ) ] d x = 2 ξ w ˙ ( × u ˙ ) d x 2 ξ [ u w ( × u ˙ ) + u u ˙ ( × w ) ] d x 2 ξ w ˙ ( × u ˙ ) d x + C u L 6 w L 3 u ˙ L 2 2 ξ w ˙ ( × u ˙ ) d x + C w L 3 u ˙ L 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ39_HTML.gif
(2.30)
and, similarly,
J 7 = 2 ξ w ˙ ( × u ˙ ) d x 2 ξ [ u u ( × w ˙ ) + u w ˙ ( × u ) ] d x 2 ξ w ˙ ( × u ˙ ) d x + C u L 3 w ˙ L 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ40_HTML.gif
(2.31)
and
J 8 = 4 ξ | w ˙ | 2 d x + 4 ξ [ u w w ˙ + u w ˙ w ] d x = 4 ξ | w ˙ | 2 d x 4 ξ w w ˙ div u d x 4 ξ | w ˙ | 2 d x + C u L 2 w L 3 w ˙ L 6 4 ξ | w ˙ | 2 d x + C w ˙ L 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ41_HTML.gif
(2.32)
The ninth term on the right-hand side of (2.26) can be estimated as follows, integrating by parts, using (2.1), (2.5), (2.10), (2.11) and Hölder’s inequality:
J 9 = ( j u ˙ j H H t + k u ˙ j u k j H H ) d x C u ˙ L 2 ( H L 6 H t L 3 + u L 6 H L 6 H L 6 ) C u ˙ L 2 ( H t L 2 1 2 H t L 6 1 2 + u L 2 H H 1 ) δ u ˙ L 2 2 + C ( δ ) ( H t L 2 2 + H t L 2 2 + H H 1 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ42_HTML.gif
(2.33)
In a similar manner, one also has
J 10 = u ˙ j ( H t H j + H H t j + div ( u ( H H j ) ) ) d x = ( ( H t u ˙ j ) H j + ( H u ˙ j ) H t j + u k k u ˙ j ( H H j ) ) d x δ u ˙ L 2 2 + C ( δ ) ( H t L 2 2 + H t L 2 2 + H H 1 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ43_HTML.gif
(2.34)
Putting (2.27)-(2.34) into (2.26), using the Cauchy-Schwarz inequality and choosing δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq69_HTML.gif suitably small, we get
d d t ( ρ u ˙ L 2 2 + ρ w ˙ L 2 2 ) + ( u ˙ L 2 2 + w ˙ L 2 2 + 2 w ˙ × u ˙ L 2 2 ) C ( 1 + u L 6 3 + w L 6 3 + H t L 2 2 + H t L 2 2 + H H 1 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ44_HTML.gif
(2.35)
To estimate H t L 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq73_HTML.gif, one can differentiate (1.1)4 with respect to t, multiply the resulting equations by H t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq63_HTML.gif in L 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq57_HTML.gif, and integrate by parts over R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq76_HTML.gif to get
1 2 d d t | H t | 2 d x + σ | H t | 2 d x = ( H u t u t H H div u t ) H t d x + ( H t u u H t H t div u ) H t d x : = K 1 + K 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ45_HTML.gif
(2.36)
Integrating by parts and using (1.1)5, (2.1), (2.10) and (2.11), then we deduce
K 1 = ( H u ˙ u ˙ H H div u ˙ ) H t d x + ( H i i H t j H k j H t k ) ( u u j ) d x C H L 6 H t L 3 u ˙ L 2 + C u ˙ L 6 H L 2 H t L 3 + C H L 12 H t L 2 u L 4 u L 6 C ( H t L 2 1 2 H t L 2 1 2 u ˙ L 2 + H t L 2 u L 4 ) ε 1 H t L 2 2 + ε 2 u ˙ L 2 2 + C ( ε 1 , ε 2 ) ( H t L 2 2 + u L 4 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equi_HTML.gif
for some positive constants ε 1 , ε 2 ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq77_HTML.gif. For the second term on the right-hand side of (2.36), integrating by parts and using (2.1) give
K 2 = ( H t u 1 2 H t div u ) H t d x C u L 2 H t L 2 C H t L 2 1 2 H t L 2 3 2 ε 1 H t L 2 2 + C ( ε 1 ) H t L 2 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equj_HTML.gif
Putting the estimates of K 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq78_HTML.gif, K 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq79_HTML.gif into (2.36) and choosing ε 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq80_HTML.gif small enough, one has
1 2 d d t | H t | 2 d x + σ 2 | H t | 2 d x ε 2 u ˙ L 2 2 + C ( ε 2 ) ( H t L 2 2 + u L 4 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ46_HTML.gif
(2.37)
Then, combining (2.35) and (2.37), using Young’s inequality, and choosing ε 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq81_HTML.gif suitably small yield that
d d t ( ρ u ˙ L 2 2 + ρ w ˙ L 2 2 + H t L 2 2 ) + ( u ˙ L 2 2 + w ˙ L 2 2 + H t L 2 2 + 2 w ˙ × u ˙ L 2 2 ) C ( 1 + u L 6 3 + w L 6 3 + H t L 2 2 + H H 1 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ47_HTML.gif
(2.38)
Firstly, we use (2.4)-(2.6), (2.9), (2.10), (2.11), (2.1) and (2.2) to infer from the standard L p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq56_HTML.gif-estimate that
u L 6 + w L 6 C ( div u L 6 + × u L 6 + div w L 6 + × w L 6 ) C ( 1 + G 1 L 6 + G 2 L 6 + V 1 L 6 + V 2 L 6 + | H | 2 L 6 ) C ( 1 + G 1 L 2 + G 2 L 2 + V 1 L 2 + V 2 L 2 ) C ( 1 + ρ u ˙ L 2 + ρ w ˙ L 2 + H H 1 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ48_HTML.gif
(2.39)
and
ρ u ˙ L 2 + ρ w ˙ L 2 C ( ρ u t L 2 + ρ w t L 2 ) + C ( ρ u u L 2 + ρ w w L 2 ) C ( ρ u t L 2 + ρ w t L 2 ) + C ( u L u L 2 + w L w L 2 ) C ( ρ u t L 2 + ρ w t L 2 ) + C ( u L 6 1 2 + w L 6 1 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ49_HTML.gif
(2.40)
Moreover, by the standard L 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq57_HTML.gif-estimate of an elliptic system, we infer from (1.1)4, (2.1), (2.2) and (2.11) that
H H 1 C ( H t L 2 + u H L 2 + H u L 2 + u L 2 ) C ( H t L 2 + u L 6 H L 3 + H L u L 2 + 1 ) C ( H t L 2 + H L 2 1 2 H L 6 1 2 + 1 ) C ( H t L 2 + H H 1 1 2 + 1 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equk_HTML.gif
and hence,
H H 1 C ( H t L 2 + 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ50_HTML.gif
(2.41)
Combining (2.39)-(2.41), we obtain
u L 6 + w L 6 C ( 1 + ρ u t L 2 + ρ w t L 2 + H t L 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ51_HTML.gif
(2.42)
Now, putting (2.41) and (2.42) into (2.38), one has
d d t ( ρ u ˙ L 2 2 + ρ w ˙ L 2 2 + H t L 2 2 ) + ( u ˙ L 2 2 + w ˙ L 2 2 + H t L 2 2 + 2 w ˙ × u ˙ L 2 2 ) C ( 1 + ρ u ˙ L 2 2 + ρ w ˙ L 2 2 + H t L 2 2 ) ( 1 + ρ u t L 2 + ρ w t L 2 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equl_HTML.gif

from which and (2.11), we immediately obtain (2.25) by Gronwall’s inequality, (1.6) and (1.7). As a result of (2.41), we can also deduce the boundedness of H H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq74_HTML.gif. □

The next lemma is used to bound the density gradient and u L 1 ( 0 , T ; L ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq82_HTML.gif.

Lemma 2.8 Under the condition (2.4), it holds that for any 0 T < T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq60_HTML.gif,
sup 0 t T ( ρ L 2 L q + u H 1 + w H 1 ) + 0 T ( u L + u W 1 , q 2 + w W 1 , q 2 ) d t C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ52_HTML.gif
(2.43)

for any q ( 3 , 6 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq24_HTML.gif.

Proof Differentiating (1.1)1 with respect to x i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq83_HTML.gif and multiplying it by | i ρ | q 2 i ρ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq84_HTML.gif ( q 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq85_HTML.gif) in L 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq57_HTML.gif, we obtain, after integrating by parts and summing up, that
d d t | ρ | q d x C ( q ) ( | u | | ρ | q + ρ | ρ | q 1 | div u | ) d x C ( q ) ( u L ρ L q q + div u L q ρ L q q 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ53_HTML.gif
(2.44)
It follows from (2.1), (2.4), (2.6)-(2.9), (2.25) and the interpolation inequality that for any q ( 3 , 6 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq24_HTML.gif,
div u L q C ( G 1 L q + P L q + H L H L q ) C ( 1 + G 1 L 2 + G 1 L 6 + ρ L q ) C ( 1 + u ˙ L 2 + w ˙ L 2 + ρ L q ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equm_HTML.gif
where (2.2) and (2.25) were also used to get that H L C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq86_HTML.gif. So, putting this into (2.44) yields
d d t ρ L q C ( u L + 1 ) ρ L q + C ( 1 + u ˙ L 2 + w ˙ L 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ54_HTML.gif
(2.45)
We now estimate u L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq41_HTML.gif. To do this, we first observe that
( μ + ξ ) u + ( μ + λ ξ ) div u = ρ u ˙ + P 2 ξ × w + ( × H ) × H . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equn_HTML.gif
Hence, using the standard L p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq56_HTML.gif-estimate of an elliptic system leads to
2 u L q C ( ρ u ˙ L q + P L q + w L q + H H L q ) C ( 1 + u ˙ L 2 + ρ L q + w H 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equo_HTML.gif
From (1.1)3 and the standard L 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq57_HTML.gif-estimate of the elliptic system, we have that
w H 1 C ( w H 1 + ρ w ˙ L 2 + u L 2 ) C , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ55_HTML.gif
(2.46)
and then
2 u L q C ( 1 + u ˙ L 2 + ρ L q ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ56_HTML.gif
(2.47)
This, together with Lemmas 2.2 and 2.6, gives
u L C ( u L 2 + 1 ) + C ( div u L + × u L ) log ( e + 2 u L q ) C + C ( div u L + × u L ) log ( e + u ˙ L 2 ) + C ( div u L + × u L ) log ( e + ρ L q ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ57_HTML.gif
(2.48)
Now, if we set f ( t ) = e + ρ L q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq87_HTML.gif and let
g ( t ) = ( 1 + div u L + × u L + u ˙ L 2 + w ˙ L 2 ) log ( e + u ˙ L 2 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equp_HTML.gif
then it is seen from (2.45) and (2.48) that
f ( t ) = C g ( t ) f ( t ) + C g ( t ) f ( t ) ln f ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equq_HTML.gif
due to f ( t ) > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq88_HTML.gif. Thus,
( ln f ( t ) ) C g ( t ) + C g ( t ) ln f ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ58_HTML.gif
(2.49)
On the other hand, since
g ( t ) C ( 1 + div u L 2 + × u L 2 + u ˙ L 2 2 + w ˙ L 2 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equr_HTML.gif
we thus deduce from (2.11), (2.25), (2.4), (2.8), (2.9) and (2.2) that
0 T g ( t ) d t C 0 T ( 1 + div u L 2 + × u L 2 + u ˙ L 2 2 + w ˙ L 2 ) d t C + C 0 T ( div u L 2 + × u L 2 ) d t C + C 0 T ( G 1 L 2 + V 1 L 2 + P L 2 + H L 4 ) d t C + C 0 T ( G 1 L 2 2 + G 1 L 2 2 + V 1 L 2 2 + V 1 L 2 2 ) d t C + C 0 T ( u ˙ L 2 2 + w ˙ L 2 2 + H H 1 4 ) d t C . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ59_HTML.gif
(2.50)
As a result, it follows from (2.49) and Gronwall’s inequality that
f ( t ) C for any  0 t T < T , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equs_HTML.gif
and consequently,
sup 0 t T ρ L q C for any  q ( 3 , 6 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ60_HTML.gif
(2.51)
From this and (2.25), (2.48), (2.50), one obtains
0 T u L d t C . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ61_HTML.gif
(2.52)
Taking q = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq89_HTML.gif in (2.45), we get, by using (2.52) and (2.25) and Gronwall’s inequality, that
sup 0 t T ρ L 2 C . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ62_HTML.gif
(2.53)
Moreover, the standard L 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq57_HTML.gif-estimate of an elliptic system and (1.1)2, together with (2.4), (2.11) and (2.25), implies
2 u L 2 C ( ρ u ˙ L 2 + P L 2 + H H L 2 + u L 2 + w L 2 ) C ( 1 + ρ L 2 ) C . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ63_HTML.gif
(2.54)
Similar to the proof of (2.47), there are
2 u L q + 2 w L q C ( 1 + u ˙ L 2 + ρ w ˙ L q + u L q + w L q ) C ( 1 + u ˙ L 2 + w ˙ L 2 ) , q ( 3 , 6 ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equt_HTML.gif

where we have used (2.1), (2.11), (2.46), (2.51) and (2.54). From this, together with (2.25), (2.46) and (2.51)-(2.54), we can deduce (2.43). □

As a consequence of Lemmas 2.6-2.8, we have the following lemma.

Lemma 2.9 Under the condition (2.4), it holds that for any 0 T < T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq60_HTML.gif,
sup 0 t T ( ρ u t L 2 2 + ρ w t L 2 2 ) + 0 T ( u t L 2 2 + w t L 2 2 ) d t C . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_Equ64_HTML.gif
(2.55)

The proof is the same as that of Lemma 3.6 in [20] and is omitted here.

With the help of Lemmas 2.3, 2.6-2.9 and the local existence theorem, we can complete the proof of Theorem 1.2 by the contradiction arguments. In fact, in view of Lemmas 2.3, 2.6-2.9, it is easy to see that the functions ( ρ , u , w , H ) ( x , T ) = lim t T ( ρ , u , w , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq90_HTML.gif have the same regularities imposed on the initial data (1.5) at the time t = T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq91_HTML.gif. This implies that the compatibility conditions (1.6) and (1.7) are satisfied at the time T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq92_HTML.gif. Thus, we can take ( ρ , u , w , H ) ( x , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq93_HTML.gif as the initial data and apply the local existence theorem to extend the local strong solutions beyond T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq92_HTML.gif. This contradicts the assumption that T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-160/MediaObjects/13661_2013_Article_414_IEq92_HTML.gif is the maximal time of existence.

Declarations

Acknowledgements

This work is partially supported by the Fundamental Research Funds for the Central Universities (Grant No. 11QZR16), the National Natural Science Foundation of China (Grant No. 11001090).

Authors’ Affiliations

(1)
School of Mathematical Sciences, Huaqiao University

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