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Blow-up criterion of smooth solutions for magneto-micropolar fluid equations with partial viscosity

Yu-Zhu Wang*, Yifang Li and Yin-Xia Wang

Author Affiliations

School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou 450011, China

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Boundary Value Problems 2011, 2011:11  doi:10.1186/1687-2770-2011-11

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/11

Received:4 April 2011
Accepted:15 August 2011
Published:15 August 2011

© 2011 Wang et al; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


In this paper, we investigate the Cauchy problem for the incompressible magneto-micropolar fluid equations with partial viscosity in ℝn(n = 2, 3). We obtain a Beale-Kato-Majda type blow-up criterion of smooth solutions.

MSC (2010): 76D03; 35Q35.

magneto-micropolar fluid equations; smooth solutions; blow-up criterion

1 Introduction

The incompressible magneto-micropolar fluid equations in ℝn(n = 2, 3) takes the following form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M1">View MathML</a>


where u(t, x), v(t, x), b(t, x) and p(t, x) denote the velocity of the fluid, the micro-rotational velocity, magnetic field and hydrostatic pressure, respectively. μ, χ, γ, κ and ν are constants associated with properties of the material: μ is the kinematic viscosity, χ is the vortex viscosity, γ and κ are spin viscosities, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M2">View MathML</a> is the magnetic Reynold. The incompressible magneto-micropolar fluid equations (1.1) has been studied extensively (see [1-8]). Rojas-Medar [5] established the local in time existence and uniqueness of strong solutions by the spectral Galerkin method. Global existence of strong solution for small initial data was obtained in [4]. Rojas-Medar and Boldrini [6] proved the existence of weak solutions by the Galerkin method, and in 2D case, also proved the uniqueness of the weak solutions. Wang et al. [2] obtained a Beale-Kato-Majda type blow-up criterion for smooth solution (u, v, b) to the magneto-micropolar fluid equations with partial viscosity that relies on the vorticity of velocity ∇ × u only (see also [8]). For regularity results, refer to Yuan [7] and Gala [1].

If b = 0, (1.1) reduces to micropolar fluid equations. The micropolar fluid equations was first proposed by Eringen [9]. It is a type of fluids which exhibits the micro-rotational effects and micro-rotational inertia, and can be viewed as a non-Newtonian fluid. Physically, micropolar fluid may represent fluids that 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 appeared in a large number of complex fluids such as the suspensions, animal blood, 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 [10] and references therein. The existences of weak and strong solutions for micropolar fluid equations were treated by Galdi and Rionero [11] and Yamaguchi [12], respectively. The global regularity issue has been thoroughly investigated for the 3D micropolar fluid equations and many important regularity criteria have been established (see [13-19]). The convergence of weak solutions of the micropolar fluids in bounded domains of ℝn was investigated (see [20]). When the viscosities tend to zero, in the limit, a fluid governed by an Euler-like system was found.

If both v = 0 and χ = 0, then Equations 1.1 reduces to be the magneto-hydrodynamic (MHD) equations. The local well-posedness of the Cauchy problem for the incompressible MHD equations in the usual Sobolev spaces Hs(ℝ3) is established in [21] for any given initial data that belongs to Hs(ℝ3), s ≥ 3. But whether this unique local solution can exist globally is a challenge open problem in the mathematical fluid mechanics. There are numerous important progresses on the fundamental issue of the regularity for the weak solution to (1.1), (1.2) (see [22-34]). In this paper, we consider the magneto-micropolar fluid equations (1.1) with partial viscosity, i.e., μ = χ = 0. Without loss of generality, we take γ = κ = ν = 1. The corresponding magneto-micropolar fluid equations thus reads

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M3">View MathML</a>


We obtain a blow-up criterion of smooth solutions to (1.2), which improves our previous result (see [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 [35] says that any solution u is smooth up to time T under the assumption that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M4">View MathML</a>. Beale-Kato-Majda's criterion is slightly improved by Kozono et al. [36] under the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M5">View MathML</a>. In this paper, we obtain a Beale-Kato-Majda type blow-up criterion of smooth solutions to Cauchy problem for the magneto-micropolar fluid equations (1.2).

Now, we state our results as follows.

Theorem 1.1 Assume that u0, v0, b0 Hm(ℝn)(n = 2, 3), m ≥ 3 with ∇ · u0 = 0, ∇ · b0 = 0. Let (u, v, b) be a smooth solution to Equations 1.2 with initial data u(0, x) = u0(x), v(0, x) = v0(x), b(0, x) = b0(x) for 0 ≤ t < T . If u satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M6">View MathML</a>


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

We have the following corollary immediately.

Corollary 1.1 Assume that u0, v0, b0 Hm(ℝn)(n = 2, 3), m ≥ 3 with ∇ · u0 = 0, ∇ · b0 = 0. Let (u, v, b) be a smooth solution to Equations 1.2 with initial data u(0, x) = u0(x), v(0, x) = v0(x), b(0, x) = b0(x) for 0 ≤ t < T . Suppose that T is the maximal existence time, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M7">View MathML</a>


The plan of the paper is arranged as follows. We first state some preliminary 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M68">View MathML</a> be the Schwartz class of rapidly decreasing functions. Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M69">View MathML</a>, its Fourier transform <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M8">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M9">View MathML</a>

and for any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M70">View MathML</a>, its inverse Fourier transform <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M10">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M11">View MathML</a>

In what follows, we recall the Littlewood-Paley decomposition. Choose a non-negative radial functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M71">View MathML</a>, supported in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M12">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M13">View MathML</a>

The frequency localization operator is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M14">View MathML</a>

Next, we recall the definition of homogeneous function spaces (see [37]). For (p, q) ∈ [1, ∞]2 and s ∈ ℝ, the homogeneous Besov space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M15">View MathML</a> is defined as the set of f up to polynomials such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M16">View MathML</a>

In what follows, we shall make continuous use of Bernstein inequalities, which comes from [38].

Lemma 2.1 For any s ∈ ℕ, 1 ≤ p q ≤ ∞ and f Lp(ℝn), then the following inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M17">View MathML</a>



<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M18">View MathML</a>


hold, where c and C are positive constants independent of f and k.

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

Lemma 2.2 Let j, m be any integers satisfying 0 ≤ j < m, and let 1 ≤ q, r ≤ ∞, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M19">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M20">View MathML</a>

Then for all f Lq(ℝn) ∩Wm,r(ℝn), there is a positive constant C depending only on n, m, j, q, r, θ such that the following inequality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M21">View MathML</a>


with the following exception: if 1 < r < 1 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M22">View MathML</a> is a nonnegative integer, then (2.3) holds only for a satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M23">View MathML</a>.

The following lemma comes from [39].

Lemma 2.3 Assume that 1 < p < ∞. For f, g Wm,p, and 1 < q1, q2 ≤ ∞, 1 < r1, r2 < 1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M24">View MathML</a>


where 1 ≤ α m and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M25">View MathML</a>.

Lemma 2.4 There exists a uniform positive constant C, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M26">View MathML</a>


holds for all vectors f H3(ℝn)(n = 2, 3) with ∇ · f = 0.

Proof. The proof can be founded in [36]. For the convenience of the readers, the proof will be also sketched here. It follows from Littlewood-Paley composition that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M27">View MathML</a>


Using (2.1), ( 2.2) and (2.6), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M28">View MathML</a>



<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M29">View MathML</a>


It follows from (2.7), (2.8) and Calderon-Zygmand theory that (2.5) holds. Thus, we have completed the proof of lemma. □

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M30">View MathML</a>


hold, and in two space dimensions, the following inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M31">View MathML</a>



Proof. (2.9) and (2.10) 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 (2.9) and (2.10). Sobolev embedding implies that H3(ℝn), ↪ L4(ℝn) for n = 2, 3. Consequently, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M32">View MathML</a>


For any given 0 ≠ f H3(ℝn) and δ > 0, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M33">View MathML</a>


By (2.11) and (2.12), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M34">View MathML</a>


which is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M35">View MathML</a>


Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M36">View MathML</a> and n = 3 and n = 2, respectively. From (2.14), we immediately get the last inequality in (2.9) and (2.10). Thus, we have completed the proof of Lemma 2.5. □

3 Proof of main results

Proof of Theorem 1.1. Adding the inner product of u with the first equation of (1.2), of v with the second equation of (1.2) and of b the third equation of (1.2), then using integration by parts, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M37">View MathML</a>


where we have used ∇ ·· u = 0 and ∇ · b = 0.

Integrating with respect to t, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M38">View MathML</a>


Applying ∇ to (1.2) and taking the L2 inner product of the resulting equation with (∇u, ∇v, ∇b), with help of integration by parts, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M39">View MathML</a>


By (3.3) and ∇ · u = 0, ∇ · b = 0, we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M40">View MathML</a>


Using Gronwall inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M41">View MathML</a>


Owing to (1.3), we know that for any small constant ε > 0, there exists T* < T such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M42">View MathML</a>



<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M43">View MathML</a>


It follows from (3.5), (3.6), (3.7) and Lemma 2.4 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M44">View MathML</a>


where C1 depends on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M45">View MathML</a>, while C0 is an absolute positive constant.

Applying ∇m to the first equation of (1.2), then taking L2 inner product of the resulting equation with ∇mu and using integration by parts, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M46">View MathML</a>


Likewise, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M47">View MathML</a>



<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M48">View MathML</a>


It follows (3.9), (3.10), (3.11), ∇ · u = 0, ∇ · b = 0 and integration by parts that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M49">View MathML</a>


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

With help of Hölder inequality and Lemma 2.3, we derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M50">View MathML</a>


Using integration by parts and Hölder inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M51">View MathML</a>


Thanks to Lemma 2.5, Young inequality and (3.8), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M52">View MathML</a>

in 3D and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M53">View MathML</a>

in 2D.

It follows from Lemmas 2.2, 2.5, Young inequality and (3.8) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M54">View MathML</a>

in 3D and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M55">View MathML</a>

in 2D.

Consequently, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M56">View MathML</a>



<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M57">View MathML</a>


provided that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M58">View MathML</a>

It follows from (3.14), (3.15) and (3.16) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M59">View MathML</a>


Likewise, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M60">View MathML</a>


<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M61">View MathML</a>



<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M62">View MathML</a>


Collecting (3.12), (3.13), (3.17), (3.18), (3.19) and (3.20) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M63">View MathML</a>


for all T* t < T.

Integrating (3.21) with respect to time from T* to τ and using Lemma 2.4, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M64">View MathML</a>


Owing to (3.22), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M65">View MathML</a>


For all T* t < T, with help of Gronwall inequality and (3.23), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M66">View MathML</a>


where C depends on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/11/mathml/M67">View MathML</a>.

Noting that (3.2) and the right-hand side of (3.24) is independent of t for T* t < T , we know that (u(T, ·), v(T, ·), b(T, ·)) ∈ H3(ℝn). Thus, Theorem 1.1 is proved.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

YZW completed the main part of theorem in this paper, YL and YXW revised the part proof. All authors read and approve the final manuscript.


The authors would like to thank the referee for his/her pertinent comments and advice. This work was supported in part by Research Initiation Project for High-level Talents (201031) of North China University of Water Resources and Electric Power.


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