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

  • Yu-Zhu Wang1Email author,

    Affiliated with

    • Liping Hu2 and

      Affiliated with

      • Yin-Xia Wang1

        Affiliated with

        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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq2_HTML.gif take the following form:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ1_HTML.gif
        (1.1)

        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq3_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq4_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq5_HTML.gif and http://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. http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq7_HTML.gif is the kinematic viscosity, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq8_HTML.gif is the vortex viscosity, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq9_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq10_HTML.gif are spin viscosities, and http://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 http://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 http://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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq17_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq19_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq22_HTML.gif and http://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, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq24_HTML.gif . Without loss of generality, we take http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq26_HTML.gif is smooth up to time http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq27_HTML.gif under the assumption that http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq30_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq31_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq32_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq33_HTML.gif . Assume that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq35_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq36_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq37_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq38_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq39_HTML.gif satisfies
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ3_HTML.gif
        (1.3)

        then the solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq40_HTML.gif can be extended beyond http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq42_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq43_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq44_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq45_HTML.gif . Assume that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq47_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq48_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq49_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq50_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq51_HTML.gif is the maximal existence time, then
        http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq53_HTML.gif , its Fourier transform http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq54_HTML.gif is defined by
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ5_HTML.gif
        (2.1)
        and for any given http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq55_HTML.gif , its inverse Fourier transform http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq56_HTML.gif is defined by
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq57_HTML.gif , supported in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq58_HTML.gif such that
        http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq59_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq60_HTML.gif , the homogeneous Triebel-Lizorkin space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq61_HTML.gif as the set of tempered distributions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq62_HTML.gif such that
        http://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
        http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq64_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq65_HTML.gif , then
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ11_HTML.gif
        (2.7)

        hold, where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq66_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq67_HTML.gif are positive constants independent of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq68_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq70_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ12_HTML.gif
        (2.8)

        holds for all http://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:
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq72_HTML.gif , such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ14_HTML.gif
        (2.10)

        holds for all vectors http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq73_HTML.gif with http://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
        http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq75_HTML.gif in terms of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq76_HTML.gif as
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ17_HTML.gif
        (2.13)
        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq77_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq78_HTML.gif denote the Riesz transforms. Since http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ18_HTML.gif
        (2.14)
        with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq80_HTML.gif . Taking
        http://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
        http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq81_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq82_HTML.gif . Consequently, we get
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ22_HTML.gif
        (2.18)
        For any given http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq83_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq84_HTML.gif , let
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ24_HTML.gif
        (2.20)
        which is equivalent to
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ25_HTML.gif
        (2.21)

        Taking http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq85_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq86_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq89_HTML.gif on http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ26_HTML.gif
        (3.1)

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

        Integrating with respect to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq93_HTML.gif , we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ27_HTML.gif
        (3.2)
        Applying http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq94_HTML.gif to (1.2) and taking the http://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 http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq97_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq98_HTML.gif that
        http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq99_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq100_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ31_HTML.gif
        (3.6)
        Let
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ33_HTML.gif
        (3.8)

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

        Applying http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq106_HTML.gif , using integration by parts, we get
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ34_HTML.gif
        (3.9)
        Similarly, we obtain
        http://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), http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq107_HTML.gif , and integration by parts, we have
        http://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 http://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
        http://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
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ39_HTML.gif
        (3.14)
        in 3D and
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ41_HTML.gif
        (3.16)
        in 3D and
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ43_HTML.gif
        (3.18)
        provided that
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ45_HTML.gif
        (3.20)
        Similarly, we obtain
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ47_HTML.gif
        (3.22)

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

        Integrating (3.22) with respect to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq110_HTML.gif from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq111_HTML.gif to http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ48_HTML.gif
        (3.23)
        which implies
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ49_HTML.gif
        (3.24)
        For all http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_Equ50_HTML.gif
        (3.25)

        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq114_HTML.gif depends on http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq116_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F128614/MediaObjects/13661_2011_Article_24_IEq117_HTML.gif , we know that http://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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