Open Access

A Beale-Kato-Madja Criterion for Magneto-Micropolar Fluid Equations with Partial Viscosity

Boundary Value Problems20112011:128614

DOI: 10.1155/2011/128614

Received: 18 February 2011

Accepted: 7 March 2011

Published: 15 March 2011

Abstract

We study the incompressible magneto-micropolar fluid equations with partial viscosity in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq1_HTML.gif . A blow-up criterion of smooth solutions is obtained. The result is analogous to the celebrated Beale-Kato-Majda type criterion for the inviscid Euler equations of incompressible fluids.

1. Introduction

The incompressible magneto-micropolar fluid equations in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq2_HTML.gif take the following form:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq3_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq4_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq5_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq6_HTML.gif denote the velocity of the fluid, the microrotational velocity, magnetic field, and hydrostatic pressure, respectively. https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq7_HTML.gif is the kinematic viscosity, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq8_HTML.gif is the vortex viscosity, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq9_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq10_HTML.gif are spin viscosities, and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq11_HTML.gif is the magnetic Reynold.

The incompressible magneto-micropolar fluid equation (1.1) has been studied extensively (see [17]). In [2], the authors have proven that a weak solution to (1.1) has fractional time derivatives of any order less than https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq12_HTML.gif in the two-dimensional case. In the three-dimensional case, a uniqueness result similar to the one for Navier-Stokes equations is given and the same result concerning fractional derivatives is obtained, but only for a more regular weak solution. Rojas-Medar [4] established local existence and uniqueness of strong solutions by the Galerkin method. Rojas-Medar and Boldrini [5] also proved the existence of weak solutions by the Galerkin method, and in 2D case, also proved the uniqueness of the weak solutions. Ortega-Torres and Rojas-Medar [3] proved global existence of strong solutions for small initial data. A Beale-Kato-Majda type blow-up criterion for smooth solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq13_HTML.gif to (1.1) that relies on the vorticity of velocity https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq14_HTML.gif only is obtained by Yuan [7]. For regularity results, refer to Yuan [6] and Gala [1].

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq15_HTML.gif , (1.1) reduces to micropolar fluid equations. The micropolar fluid equations was first developed by Eringen [8]. It is a type of fluids which exhibits the microrotational effects and microrotational inertia, and can be viewed as a non-Newtonian fluid. Physically, micropolar fluid may represent fluids consisting of rigid, randomly oriented (or spherical particles) suspended in a viscous medium, where the deformation of fluid particles is ignored. It can describe many phenomena that appeared in a large number of complex fluids such as the suspensions, animal blood, and liquid crystals which cannot be characterized appropriately by the Navier-Stokes equations, and that it is important to the scientists working with the hydrodynamic-fluid problems and phenomena. For more background, we refer to [9] and references therein. The existences of weak and strong solutions for micropolar fluid equations were proved by Galdi and Rionero [10] and Yamaguchi [11], respectively. Regularity criteria of weak solutions to the micropolar fluid equations are investigated in [12]. In [13], the authors gave sufficient conditions on the kinematics pressure in order to obtain regularity and uniqueness of the weak solutions to the micropolar fluid equations. The convergence of weak solutions of the micropolar fluids in bounded domains of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq16_HTML.gif was investigated (see [14]). When the viscosities tend to zero, in the limit, a fluid governed by an Euler-like system was found.

If both https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq17_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq18_HTML.gif , then (1.1) reduces to be the magneto-hydrodynamic (MHD) equations. There are numerous important progresses on the fundamental issue of the regularity for the weak solution to MHD systems (see [1523]). Zhou [18] established Serrin-type regularity criteria in term of the velocity only. Logarithmically improved regularity criteria for MHD equations were established in [16, 23]. Regularity criteria for the 3D MHD equations in term of the pressure were obtained [19]. Zhou and Gala [20] obtained regularity criteria of solutions in term of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq19_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq20_HTML.gif in the multiplier spaces. A new regularity criterion for weak solutions to the viscous MHD equations in terms of the vorticity field in Morrey-Campanato spaces was established (see [21]). In [22], a regularity criterion https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq21_HTML.gif for the 2D MHD system with zero magnetic diffusivity was obtained.

Regularity criteria for the generalized viscous MHD equations were also obtained in [24]. Logarithmically improved regularity criteria for two related models to MHD equations were established in [25] and [26], respectively. Lei and Zhou [27] studied the magneto-hydrodynamic equations with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq22_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq23_HTML.gif . Caflisch et al. [28] and Zhang and Liu [29] obtained blow-up criteria of smooth solutions to 3-D ideal MHD equations, respectively. Cannone et al. [30] showed a losing estimate for the ideal MHD equations and applied it to establish an improved blow-up criterion of smooth solutions to ideal MHD equations.

In this paper, we consider the magneto-micropolar fluid equations (1.1) with partial viscosity, that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq24_HTML.gif . Without loss of generality, we take https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq25_HTML.gif . The corresponding magneto-micropolar fluid equations thus reads
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ2_HTML.gif
(1.2)

In the absence of global well-posedness, the development of blow-up/non blow-up theory is of major importance for both theoretical and practical purposes. For incompressible Euler and Navier-Stokes equations, the well-known Beale-Kato-Majda's criterion [31] says that any solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq26_HTML.gif is smooth up to time https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq27_HTML.gif under the assumption that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq28_HTML.gif . Beale-Kato-Majdas criterion is slightly improved by Kozono and Taniuchi [32] under the assumption https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq29_HTML.gif . In this paper, we obtain a Beale-Kato-Majda type blow-up criterion of smooth solutions to the magneto-micropolar fluid equations (1.2).

Now we state our results as follows.

Theorem 1.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq30_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq31_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq32_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq33_HTML.gif . Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq34_HTML.gif is a smooth solution to (1.2) with initial data https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq35_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq36_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq37_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq38_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq39_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ3_HTML.gif
(1.3)

then the solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq40_HTML.gif can be extended beyond https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq41_HTML.gif .

We have the following corollary immediately.

Corollary 1.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq42_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq43_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq44_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq45_HTML.gif . Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq46_HTML.gif is a smooth solution to (1.2) with initial data https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq47_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq48_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq49_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq50_HTML.gif . Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq51_HTML.gif is the maximal existence time, then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ4_HTML.gif
(1.4)

The paper is organized as follows. We first state some preliminaries on functional settings and some important inequalities in Section 2 and then prove the blow-up criterion of smooth solutions to the magneto-micropolar fluid equations (1.2) in Section 3.

2. Preliminaries

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq52_HTML.gif be the Schwartz class of rapidly decreasing functions. Given https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq53_HTML.gif , its Fourier transform https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq54_HTML.gif is defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ5_HTML.gif
(2.1)
and for any given https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq55_HTML.gif , its inverse Fourier transform https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq56_HTML.gif is defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ6_HTML.gif
(2.2)
Next, let us recall the Littlewood-Paley decomposition. Choose a nonnegative radial functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq57_HTML.gif , supported in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq58_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ7_HTML.gif
(2.3)
The frequency localization operator is defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ8_HTML.gif
(2.4)
Let us now define homogeneous function spaces (see e.g., [33, 34]). For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq59_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq60_HTML.gif , the homogeneous Triebel-Lizorkin space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq61_HTML.gif as the set of tempered distributions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq62_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ9_HTML.gif
(2.5)
BMO denotes the homogenous space of bounded mean oscillations associated with the norm
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ10_HTML.gif
(2.6)

Thereafter, we will use the fact https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq63_HTML.gif .

In what follows, we will make continuous use of Bernstein inequalities, which comes from [35].

Lemma 2.1.

For any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq64_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq65_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ11_HTML.gif
(2.7)

hold, where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq66_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq67_HTML.gif are positive constants independent of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq68_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq69_HTML.gif .

The following inequality is well-known Gagliardo-Nirenberg inequality.

Lemma 2.2.

There exists a uniform positive constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq70_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ12_HTML.gif
(2.8)

holds for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq71_HTML.gif .

The following lemma comes from [36].

Lemma 2.3.

The following calculus inequality holds:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ13_HTML.gif
(2.9)

Lemma 2.4.

There is a uniform positive constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq72_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ14_HTML.gif
(2.10)

holds for all vectors https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq73_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq74_HTML.gif .

Proof.

The proof can be found in [37]. For completeness, the proof will be also sketched here. It follows from Littlewood-Paley decomposition that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ15_HTML.gif
(2.11)
Using (2.7) and (2.11), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ16_HTML.gif
(2.12)
By the Biot-Savard law, we have a representation of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq75_HTML.gif in terms of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq76_HTML.gif as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ17_HTML.gif
(2.13)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq77_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq78_HTML.gif denote the Riesz transforms. Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq79_HTML.gif is a bounded operator in BMO, this yields
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ18_HTML.gif
(2.14)
with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq80_HTML.gif . Taking
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ19_HTML.gif
(2.15)

It follows from (2.12), (2.14), and (2.15) that (2.10) holds. Thus, the lemma is proved.

In order to prove Theorem 1.1, we need the following interpolation inequalities in two and three space dimensions.

Lemma 2.5.

In three space dimensions, the following inequalities
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ20_HTML.gif
(2.16)
hold, and in two space dimensions, the following inequalities
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ21_HTML.gif
(2.17)

hold.

Proof.

(2.16) and (2.17) are of course well known. In fact, we can obtain them by Sobolev embedding and the scaling techniques. In what follows, we only prove the last inequality in (2.16) and (2.17). Sobolev embedding implies that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq81_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq82_HTML.gif . Consequently, we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ22_HTML.gif
(2.18)
For any given https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq83_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq84_HTML.gif , let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ23_HTML.gif
(2.19)
By (2.18) and (2.19), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ24_HTML.gif
(2.20)
which is equivalent to
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ25_HTML.gif
(2.21)

Taking https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq85_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq86_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq87_HTML.gif , respectively. From (2.21), we immediately get the last inequality in (2.16) and (2.17). Thus, we have completed the proof of Lemma 2.5.

3. Proof of Main Results

Proof of Theorem 1.1.

Multiplying (1.2) by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq88_HTML.gif , respectively, then integrating the resulting equation with respect to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq89_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq90_HTML.gif and using integration by parts, we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ26_HTML.gif
(3.1)

where we have used https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq91_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq92_HTML.gif .

Integrating with respect to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq93_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ27_HTML.gif
(3.2)
Applying https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq94_HTML.gif to (1.2) and taking the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq95_HTML.gif inner product of the resulting equation with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq96_HTML.gif , with help of integration by parts, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ28_HTML.gif
(3.3)
It follows from (3.3) and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq97_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq98_HTML.gif that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ29_HTML.gif
(3.4)
By Gronwall inequality, we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ30_HTML.gif
(3.5)
Thanks to (1.3), we know that for any small constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq99_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq100_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ31_HTML.gif
(3.6)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ32_HTML.gif
(3.7)
It follows from (3.5), (3.6), (3.7), and Lemma 2.4 that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ33_HTML.gif
(3.8)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq101_HTML.gif depends on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq102_HTML.gif , while https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq103_HTML.gif is an absolute positive constant.

Applying https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq104_HTML.gif to the first equation of (1.2), then taking https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq105_HTML.gif inner product of the resulting equation with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq106_HTML.gif , using integration by parts, we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ34_HTML.gif
(3.9)
Similarly, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ35_HTML.gif
(3.10)
Using (3.9), (3.10), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq107_HTML.gif , and integration by parts, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ36_HTML.gif
(3.11)

In what follows, for simplicity, we will set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq108_HTML.gif .

From Hölder inequality and Lemma 2.3, we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ37_HTML.gif
(3.12)
Using integration by parts and Hölder inequality, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ38_HTML.gif
(3.13)
By Lemma 2.5, Young inequality, and (3.8), we deduce that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ39_HTML.gif
(3.14)
in 3D and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ40_HTML.gif
(3.15)

in 2D.

From Lemmas 2.2 and 2.5, Young inequality, and (3.8), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ41_HTML.gif
(3.16)
in 3D and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ42_HTML.gif
(3.17)

in 2D.

Consequently, we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ43_HTML.gif
(3.18)
provided that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ44_HTML.gif
(3.19)
It follows from (3.13) and (3.18) that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ45_HTML.gif
(3.20)
Similarly, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ46_HTML.gif
(3.21)
Combining (3.11), (3.12), (3.20), and (3.21) yields
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ47_HTML.gif
(3.22)

for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq109_HTML.gif .

Integrating (3.22) with respect to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq110_HTML.gif from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq111_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq112_HTML.gif and using Lemma 2.4, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ48_HTML.gif
(3.23)
which implies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ49_HTML.gif
(3.24)
For all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq113_HTML.gif , from Gronwall inequality and (3.24), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ50_HTML.gif
(3.25)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq114_HTML.gif depends on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq115_HTML.gif .

Noting that (3.2) and the right hand side of (3.25) is independent of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq116_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq117_HTML.gif , we know that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq118_HTML.gif . Thus, Theorem 1.1 is proved.

Declarations

Acknowledgment

This work was supported by the NNSF of China (Grant no. 10971190).

Authors’ Affiliations

(1)
School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power
(2)
College of Information and Management Science, Henan Agricultural University

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© Yu-Zhu Wang et al. 2011

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