Open Access

Blow-up criteria for smooth solutions to the generalized 3D MHD equations

Boundary Value Problems20132013:187

DOI: 10.1186/1687-2770-2013-187

Received: 22 April 2013

Accepted: 5 August 2013

Published: 21 August 2013

Abstract

In this paper, we focus on the generalized 3D magnetohydrodynamic equations. Two logarithmically blow-up criteria of smooth solutions are established.

MSC:76D03, 76W05.

Keywords

generalized MHD equations blow-up criteria

1 Introduction

We study blow up criteria of smooth solutions to the incompressible generalized magnetohydrodynamics (GMHD) equations in R 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq1_HTML.gif
{ t u + u u B B + ( Δ ) α u + ( p + 1 2 | B | 2 ) = 0 , t B + u B B u + ( Δ ) β B = 0 , u = 0 , B = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ1_HTML.gif
(1.1)
with the initial condition
t = 0 : u = u 0 ( x ) , B = B 0 ( x ) , x R 3 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ2_HTML.gif
(1.2)

Here u = ( u 1 , u 2 , u 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq2_HTML.gif, B = ( B 1 , B 2 , B 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq3_HTML.gif and P = p + 1 2 | B | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq4_HTML.gif are non-dimensional quantities corresponding to the flow velocity, the magnetic field and the total kinetic pressure at the point ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq5_HTML.gif, while u 0 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq6_HTML.gif and B 0 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq7_HTML.gif are the given initial velocity and initial magnetic field with u 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq8_HTML.gif and B 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq9_HTML.gif, respectively.

The GMHD equations is a generalized model of MHD equations. It has important physical background. Therefore, the GMHD equations are also mathematically significant. For 3D Navier-Stokes equations, whether there exists a global smooth solution to 3D impressible GMHD equations is still an open problem. 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. Fundamental mathematical issues such as the global regularity of their solutions have generated extensive research and many interesting results have been established (see [15]).

When α = β = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq10_HTML.gif, (1.1) reduces to MHD equations. There are numerous important progresses on the fundamental issue of the regularity for the weak solution to (1.1), (1.2) (see [618]). A criterion for the breakdown of classical solutions to (1.1) with zero viscosity and positive resistivity in R 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq1_HTML.gif was derived in [9]. Some sufficient integrability conditions on two components or the gradient of two components of u + B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq11_HTML.gif and u B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq12_HTML.gif in Morrey-Campanato spaces were obtained in [10]. A logarithmal improved blow-up criterion of smooth solutions in an appropriate homogeneous Besov space was obtained by Wang et al. [11]. Zhou and Fan [15] established various logarithmically improved regularity criteria for the 3D MHD equations in terms of the velocity field and pressure, respectively. These regularity criteria can be regarded as log in time improvements of the standard Serrin criteria established before. Two new regularity criteria for the 3D incompressible MHD equations involving partial components of the velocity and magnetic fields were obtained by Jia and Zhou [17].

When α = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq13_HTML.gif, B = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq14_HTML.gif, (1.1) reduces to Navier-Stokes equations. Leray [19] and Hopf [20] constructed weak solutions to the Navier-Stokes equations, respectively. The solution is called the Leray-Hopf weak solution. Later on, much effort has been devoted to establish the global existence and uniqueness of smooth solutions to the Navier-Stokes equations. Different criteria for regularity of the weak solutions have been proposed and many interesting results have been obtained [2125].

In the paper, we obtain two logarithmically blow-up criteria of smooth solutions to (1.1), (1.2) in Morrey-Campanato spaces. We hope that the study of equations (1.1) can improve the understanding of the problem of Navier-Stokes equations and MHD equations.

Now we state our results as follows.

Theorem 1.1 Let u 0 , B 0 H m ( R 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq15_HTML.gif, m 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq16_HTML.gif, with u 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq17_HTML.gif, B 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq9_HTML.gif and 3 4 < α = β 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq18_HTML.gif. Assume that ( u , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq19_HTML.gif is a smooth solution to (1.1), (1.2) on [ 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq20_HTML.gif. If u satisfies
0 T u ( t ) M ˙ p , q r 1 + ln ( e + u ( t ) L ) d t < , 2 α r + 3 p + 1 = 2 α , 3 2 α 1 < p , 1 < p q , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ3_HTML.gif
(1.3)

then the solution ( u , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq21_HTML.gif can be extended beyond t = T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq22_HTML.gif.

We have the following corollary immediately.

Corollary 1.1 Let u 0 , B 0 H m ( R 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq15_HTML.gif, m 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq16_HTML.gif, with u 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq17_HTML.gif, B 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq9_HTML.gif and 3 4 < α = β 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq23_HTML.gif. Assume that ( u , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq19_HTML.gif is a smooth solution to (1.1), (1.2) on [ 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq20_HTML.gif. Suppose that T is the maximal existence time, then
0 T u ( t ) M ˙ p , q r 1 + ln ( e + u ( t ) L ) d t = , 2 α r + 3 p + 1 = 2 α , 3 2 α 1 < p , 1 < p q . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ4_HTML.gif
(1.4)
Theorem 1.2 Let u 0 , B 0 H m ( R 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq15_HTML.gif, m 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq16_HTML.gif, with u 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq17_HTML.gif, B 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq9_HTML.gif and 3 4 < α = β 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq23_HTML.gif. Assume that ( u , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq19_HTML.gif is a smooth solution to (1.1), (1.2) on [ 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq20_HTML.gif. If
0 T u ( t ) M ˙ p , q r 1 + ln ( e + u ( t ) L ) d t < , 2 α r + 3 p = 2 α , 3 2 α < p , 1 < p q , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ5_HTML.gif
(1.5)

then the solution ( u , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq21_HTML.gif can be extended beyond t = T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq22_HTML.gif.

We have the following corollary immediately.

Corollary 1.2 Let u 0 , B 0 H m ( R 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq15_HTML.gif, m 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq16_HTML.gif, with u 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq17_HTML.gif, B 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq9_HTML.gif and 3 4 < α = β 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq23_HTML.gif. Assume that ( u , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq19_HTML.gif is a smooth solution to (1.1), (1.2) on [ 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq20_HTML.gif. Suppose that T is the maximal existence time, then
0 T u ( t ) M ˙ p , q r 1 + ln ( e + u ( t ) L ) d t = , 2 α r + 3 p = 2 α , 3 2 α < p , 1 < p q . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ6_HTML.gif
(1.6)

The paper is organized as follows. We first state some preliminaries on function spaces and some important inequalities in Section 2. Then we prove main results in Section 3 and Section 4, respectively.

2 Preliminaries

Before stating our main results, we recall the definition and some properties of the homogeneous Morrey-Campanato space.

Definition 2.1 For 1 < p q + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq24_HTML.gif, the Morrey-Campanato space M ˙ p , q ( R 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq25_HTML.gif is defined by
M ˙ p , q = { f L loc p ( R 3 ) : f M ˙ p , q = sup R > 0 sup x R 3 R 3 q 3 p f L p ( B ( x , R ) ) < } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equa_HTML.gif

where B ( x , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq26_HTML.gif denotes the ball of center x with radius R.

Let 1 q p < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq27_HTML.gif, we define the homogeneous space N ˙ p , q ( R 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq28_HTML.gif by
N ˙ p , q ( R 3 ) = { f L p ( R 3 ) | f = k N g k , where  ( g k ) L comp p ( R 3 )  and k R 3 d k 3 ( 1 q 1 p ) g k L p < , where for any  k , d k = diam ( Supp g k ) < , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equb_HTML.gif
where L comp p ( R 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq29_HTML.gif is the space of all L p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq30_HTML.gif functions in R 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq31_HTML.gif with compact support. N ˙ p , q ( R 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq32_HTML.gif is a Banach space when it is equipped with the norm
f N ˙ p , q = inf { k N d k 3 ( 1 q 1 p ) g k L p } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equc_HTML.gif

where the infimum is taken over all possible decompositions.

Lemma 2.1 Let 1 < q p < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq33_HTML.gif and p, q satisfy 1 p + 1 p = 1 q + 1 q = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq34_HTML.gif. Then M ˙ p , q ( R 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq35_HTML.gif is the dual space of N ˙ p , q ( R 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq32_HTML.gif.

Lemma 2.2 Let 1 < q p < 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq36_HTML.gif, m 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq37_HTML.gif and 1 q + 1 q = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq38_HTML.gif. Set
γ = 3 2 + 3 q + 3 m ( 0 , 1 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equd_HTML.gif
Then there exists a constant C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq39_HTML.gif such that for any f H ˙ γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq40_HTML.gif and g L m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq41_HTML.gif,
f g N ˙ p , q C f H ˙ γ g L m . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Eque_HTML.gif

The following lemma comes from [21].

Lemma 2.3 Assume that 1 < p < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq42_HTML.gif. For f , g W m , p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq43_HTML.gif, and 1 < q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq44_HTML.gif, 1 < r < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq45_HTML.gif, we have
α ( f g ) f α g L p C ( f L q 1 α 1 g L r 1 + g L q 2 α f L r 2 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ7_HTML.gif
(2.1)

where 1 α m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq46_HTML.gif and 1 p = 1 q 1 + 1 r 1 = 1 q 2 + 1 r 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq47_HTML.gif.

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

Lemma 2.4 Let j, m be any integers satisfying 0 j < m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq48_HTML.gif, and let 1 q , r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq49_HTML.gif, and p R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq50_HTML.gif, j m θ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq51_HTML.gif be such that
1 p j n = θ ( 1 r m n ) + ( 1 θ ) 1 q . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equf_HTML.gif
Then, for all f L q ( R n ) W m , r ( R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq52_HTML.gif, there is a positive constant C depending only on n, m, j, q, r, θ such that the following inequality holds:
j f L p C f L q 1 θ m f L r θ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ8_HTML.gif
(2.2)

with the following exception: if 1 < r < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq45_HTML.gif and m j n r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq53_HTML.gif is a nonnegative integer, then (2.2) holds only for a satisfying j m θ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq54_HTML.gif.

3 Proof of Theorem 1.1

Proof Let = ( Δ ) 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq55_HTML.gif. We multiply the first equation of (1.1) by Δ u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq56_HTML.gif and use integration by parts. This yields
1 2 d d t u ( t ) L 2 2 + 1 + α u ( t ) L 2 2 = R 3 u u Δ u d x R 3 B B Δ u d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ9_HTML.gif
(3.1)
Similarly, we obtain
1 2 d d t B ( t ) L 2 2 + 1 + α B ( t ) L 2 2 = R 3 u B Δ B d x R 3 B u Δ B d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ10_HTML.gif
(3.2)
Summing up (3.1) and (3.2), we deduce that
d d t ( u ( t ) L 2 2 + B ( t ) L 2 2 ) + ( 1 + α u ( t ) L 2 2 + 1 + α B ( t ) L 2 2 ) = R 3 u u Δ u d x R 3 B B Δ u d x + R 3 u B Δ B d x R 3 B u Δ B d x I 1 + I 2 + I 3 + I 4 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ11_HTML.gif
(3.3)
By using Lemmas 2.1, 2.2 and (2.2), we have
I 1 C u M ˙ p , q u Δ u N ˙ p , q C u M ˙ p , q u H ˙ α Δ u L m C u M ˙ p , q 1 + α u L 2 u L 2 1 5 2 α + 3 m α 1 + α u L 2 5 2 α 3 m α C u M ˙ p , q u L 2 1 5 2 α + 3 m α 1 + α u L 2 1 + 5 2 α 3 m α 1 8 1 + α u L 2 2 + C u M ˙ p , q r u L 2 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ12_HTML.gif
(3.4)

where r = 2 1 5 2 α + 3 m α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq57_HTML.gif.

We apply integration by parts, B = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq58_HTML.gif, Lemmas 2.1, 2.2 and (2.2). This gives
I 2 + I 4 = R 3 k 2 B i i B j u j d x 2 R 3 k B i i k B j u j d x = C u M ˙ p , q B Δ B N ˙ p , q C u M ˙ p , q B H ˙ α Δ B L m C u M ˙ p , q 1 + α B L 2 B L 2 1 5 2 α + 3 m α 1 + α B L 2 5 2 α 3 m α C u M ˙ p , q B L 2 1 5 2 α + 3 m α 1 + α B L 2 1 + 5 2 α 3 m α 1 8 1 + α B L 2 2 + C u M ˙ p , q r B L 2 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ13_HTML.gif
(3.5)

where r = 2 1 5 2 α + 3 m α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq57_HTML.gif.

Similarly, we obtain
I 3 1 8 1 + α B L 2 2 + C u M ˙ p , q r B L 2 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ14_HTML.gif
(3.6)

where r = 2 1 5 2 α + 3 m α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq57_HTML.gif.

Substituting (3.4)-(3.6) into (3.3) yields
d d t ( u ( t ) L 2 2 + B ( t ) L 2 2 ) + ( 1 + α u ( t ) L 2 2 + 1 + α B ( t ) L 2 2 ) C u M ˙ p , q r ( u L 2 2 + B L 2 2 ) C u M ˙ p , q r 1 + ln ( e + u L ) ( u L 2 2 + B L 2 2 ) C u M ˙ p , q r 1 + ln ( e + u L ) ( u L 2 2 + B L 2 2 ) [ 1 + ln ( e + u L ) ] C u M ˙ p , q r 1 + ln ( e + u L ) ( u L 2 2 + B L 2 2 ) [ 1 + ln ( e + u H 3 ) ] C u M ˙ p , q r 1 + ln ( e + u L ) ( u L 2 2 + B L 2 2 ) [ 1 + ln ( e + u H 3 2 ) ] C u M ˙ p , q r 1 + ln ( e + u L ) ( u L 2 2 + B L 2 2 ) [ 1 + ln ( e + 3 u L 2 2 + 3 B L 2 2 ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ15_HTML.gif
(3.7)
Owing to (1.3), we know that for any small constant ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq59_HTML.gif, there exists T 0 < T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq60_HTML.gif such that
T 0 t u ( τ ) M ˙ p , q r 1 + ln ( e + u L ) d τ < ε 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equg_HTML.gif
For any T 0 t < T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq61_HTML.gif, let
Θ ( t ) = sup T 0 τ t ( 3 u ( τ ) L 2 2 + 3 B ( τ ) L 2 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equh_HTML.gif
By (3.7), we obtain
d d t ( u ( t ) L 2 2 + B ( t ) L 2 2 ) + ( 1 + α u ( t ) L 2 2 + 1 + α B ( t ) L 2 2 ) C u M ˙ p , q r 1 + ln ( e + u L ) ( u L 2 2 + B L 2 2 ) [ 1 + ln ( e + Θ ( t ) ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ16_HTML.gif
(3.8)
It follows from (3.8) and Gronwall’s inequality that
u ( t ) L 2 2 + B ( t ) L 2 2 + T 0 t ( 1 + α u ( τ ) L 2 2 + 1 + α B ( τ ) L 2 2 ) d τ ( u ( T 0 ) L 2 2 + B ( T 0 ) L 2 2 ) exp { C ( 1 + ln ( e + Θ ( t ) ) ) T 0 t u ( τ ) M ˙ p , q r 1 + ln ( e + u L ) d τ } C 0 exp { C ε [ 1 + ln ( e + Θ ( t ) ) ] } C 0 exp { 2 C ε [ ln ( e + Θ ( t ) ) ] } C 0 ( e + Θ ( t ) ) 2 C ε , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ17_HTML.gif
(3.9)

where C 0 = u ( T 0 ) L 2 2 + B ( T 0 ) L 2 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq62_HTML.gif.

Applying m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq63_HTML.gif to the first equation of (1.1), then taking L 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq64_HTML.gif inner product of the resulting equation with m u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq65_HTML.gif and using integration by parts, we get
1 2 d d t m u ( t ) L 2 2 + m + α u ( t ) L 2 2 = R 3 m ( u u ) m u d x + R 3 m ( B B ) m u d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ18_HTML.gif
(3.10)
Similarly, we have
1 2 d d t m B ( t ) L 2 2 + m + α B ( t ) L 2 2 = R 3 m ( u B ) m B d x + R 3 m ( B u ) m B d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ19_HTML.gif
(3.11)
Combining (3.10)-(3.11), using u = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq66_HTML.gif, B = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq58_HTML.gif and integration by parts yields
1 2 d d t ( m u ( t ) L 2 2 + m B ( t ) L 2 2 ) + m + α u ( t ) L 2 2 + m + α B ( t ) L 2 2 = R 3 [ m ( u u ) u m u ] m u d x + R 3 [ m ( B B ) B m B ] m u d x = R 3 [ m ( u B ) u m B ] m B d x + R 3 [ m ( B u ) B m u ] m B d x J 1 + J 2 + J 3 + J 4 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ20_HTML.gif
(3.12)

In what follows, for simplicity, we set m = 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq67_HTML.gif.

Using Hölder’s inequality and (2.1), (2.2), we have
J 1 3 ( u u ) u 3 u L 3 2 3 u L 3 C u L 3 3 u L 3 2 C u L 2 6 α + 1 2 ( 2 + α ) 3 + α u L 2 11 2 ( 2 + α ) 1 8 3 + α u L 2 2 + C u L 2 2 ( 6 α + 1 ) 4 α 3 1 8 3 + α u L 2 2 + C ( e + Θ ( t ) ) 2 ( 6 α + 1 ) 4 α 3 C ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ21_HTML.gif
(3.13)
Similarly, we have
J 2 3 ( B B ) B 3 B L 3 2 3 u L 3 C B L 3 3 B L 3 3 u L 3 C B L 2 4 α + 2 2 ( 2 + α ) 3 + α B L 2 6 2 ( 2 + α ) u L 2 2 α 1 2 ( 2 + α ) 3 + α u L 2 5 2 ( 2 + α ) 1 8 3 + α u L 2 2 + C B L 2 ( 8 α + 4 ) 4 α + 3 3 + α B L 2 12 4 α + 3 u L 2 2 ( 2 α 1 ) 4 α + 3 1 8 3 + α u L 2 2 + 1 6 3 + α B L 2 2 + C B L 2 8 α + 4 4 α 3 u L 2 2 ( 2 α 1 ) 4 α 3 1 8 3 + α u L 2 2 + 1 6 3 + α B L 2 2 + C ( e + Θ ( t ) ) 2 ( 6 α + 1 ) 4 α 3 C ε , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ22_HTML.gif
(3.14)
J 3 3 ( u B ) u 3 B L 3 2 3 B L 3 C u L 3 3 B L 3 2 + C B L 3 3 B L 3 3 u L 3 1 8 3 + α u L 2 2 + 1 6 3 + α B L 2 2 + C ( e + Θ ( t ) ) 2 ( 6 α + 1 ) 4 α 3 C ε , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ23_HTML.gif
(3.15)
and
J 4 3 ( B u ) B 3 u L 3 2 3 B L 3 C u L 3 3 B L 3 2 + C B L 3 3 B L 3 3 u L 3 1 8 3 + α u L 2 2 + 1 6 3 + α B L 2 2 + C ( e + Θ ( t ) ) 2 ( 6 α + 1 ) 4 α 3 C ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ24_HTML.gif
(3.16)
Inserting (3.13)-(3.16) into (3.12) yields
d d t ( 3 u ( t ) L 2 2 + 3 B ( t ) L 2 2 ) + 3 + α u ( t ) L 2 2 + 3 + α B ( t ) L 2 2 C ( e + Θ ( t ) ) 2 ( 6 α + 1 ) 4 α 3 C ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equi_HTML.gif

Gronwall’s inequality implies the boundedness of H 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq68_HTML.gif-norm of u and B provided that 2 ( 6 α + 1 ) 4 α 3 C ε 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq69_HTML.gif, which can be achieved by the absolute continuous property of integral (1.3). We have completed the proof of Theorem 1.1. □

4 Proof of Theorem 1.2

Proof Let = ( Δ ) 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq55_HTML.gif. Multiplying the first equation of (1.1) by Δ u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq56_HTML.gif and using integration by parts, we obtain
1 2 d d t u ( t ) L 2 2 + 1 + α u ( t ) L 2 2 = R 3 u u Δ u d x R 3 B B Δ u d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ25_HTML.gif
(4.1)
Similarly, we get
1 2 d d t B ( t ) L 2 2 + 1 + α B ( t ) L 2 2 = R 3 u B Δ B d x R 3 B u Δ B d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ26_HTML.gif
(4.2)
Summing up (4.1) and (4.2), we deduce that
d d t ( u ( t ) L 2 2 + B ( t ) L 2 2 ) + ( 1 + α u ( t ) L 2 2 + 1 + α B ( t ) L 2 2 ) = R 3 u u Δ u d x R 3 B B Δ u d x + R 3 u B Δ B d x R 3 B u Δ B d x = R 3 u u u d x R 3 B B u d x + R 3 u B B d x R 3 B u B d x I 1 + I 2 + I 3 + I 4 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ27_HTML.gif
(4.3)
Using Lemmas 2.1, 2.2 and (2.2), we obtain
I 1 C u M ˙ p , q u u N ˙ p , q C u M ˙ p , q u H ˙ α u L m C u M ˙ p , q 1 + α u L 2 u L 2 1 3 2 α + 3 m α 1 + α u L 2 3 2 α 3 m α C u M ˙ p , q u L 2 1 3 2 α + 3 m α 1 + α u L 2 1 + 3 2 α 3 m α 1 8 1 + α u L 2 2 + C u M ˙ p , q r u L 2 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ28_HTML.gif
(4.4)

where r = 2 1 5 2 α + 3 m α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq57_HTML.gif.

Similarly, we obtain
I 2 C u M ˙ p , q B B N ˙ p , q C u M ˙ p , q B H ˙ α B L m C u M ˙ p , q 1 + α B L 2 B L 2 1 3 2 α + 3 m α 1 + α B L 2 3 2 α 3 m α C u M ˙ p , q B L 2 1 3 2 α + 3 m α 1 + α B L 2 1 + 3 2 α 3 m α 1 8 1 + α B L 2 2 + C u M ˙ p , q r B L 2 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ29_HTML.gif
(4.5)
where r = 2 1 5 2 α + 3 m α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq57_HTML.gif.
I 3 1 8 1 + α B L 2 2 + C u M ˙ p , q r B L 2 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ30_HTML.gif
(4.6)
and
I 4 1 8 1 + α B L 2 2 + C u M ˙ p , q r B L 2 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ31_HTML.gif
(4.7)
Combining (4.3)-(4.7) yields
d d t ( u ( t ) L 2 2 + B ( t ) L 2 2 ) + ( 1 + α u ( t ) L 2 2 + 1 + α B ( t ) L 2 2 ) C u M ˙ p , q r ( u L 2 2 + B L 2 2 ) C u M ˙ p , q r 1 + ln ( e + u L ) ( u L 2 2 + B L 2 2 ) C u M ˙ p , q r 1 + ln ( e + u L ) ( u L 2 2 + B L 2 2 ) [ 1 + ln ( e + u L ) ] C u M ˙ p , q r 1 + ln ( e + u L ) ( u L 2 2 + B L 2 2 ) [ 1 + ln ( e + u H 3 ) ] C u M ˙ p , q r 1 + ln ( e + u L ) ( u L 2 2 + B L 2 2 ) [ 1 + ln ( e + u H 3 2 ) ] C u M ˙ p , q r 1 + ln ( e + u L ) ( u L 2 2 + B L 2 2 ) × [ 1 + ln ( e + 3 u L 2 2 + 3 B L 2 2 ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ32_HTML.gif
(4.8)
Thanks to (1.5), we know that for any small constant ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq59_HTML.gif, there exists T 0 < T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq60_HTML.gif such that
T 0 t u ( τ ) M ˙ p , q r 1 + ln ( e + u L ) d τ < ε 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equj_HTML.gif
For any T 0 t < T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq61_HTML.gif, set
Θ ( t ) = sup T 0 τ t ( 3 u ( τ ) L 2 2 + 3 B ( τ ) L 2 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ33_HTML.gif
(4.9)
By (4.8) and (4.9), we obtain
d d t ( u ( t ) L 2 2 + B ( t ) L 2 2 ) + ( 1 + α u ( t ) L 2 2 + 1 + α B ( t ) L 2 2 ) C u M ˙ p , q r 1 + ln ( e + u L ) ( u L 2 2 + B L 2 2 ) [ 1 + ln ( e + Θ ( t ) ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ34_HTML.gif
(4.10)
Equation (4.10) and Gronwall’s inequality give the estimate
u ( t ) L 2 2 + B ( t ) L 2 2 + T 0 t ( 1 + α u ( τ ) L 2 2 + 1 + α B ( τ ) L 2 2 ) d τ ( u ( T 0 ) L 2 2 + B ( T 0 ) L 2 2 ) × exp { C ( 1 + ln ( e + Θ ( t ) ) ) T 0 t u ( τ ) M ˙ p , q r 1 + ln ( e + u L ) d τ } C 0 exp { C ε [ 1 + ln ( e + Θ ( t ) ) ] } C 0 exp { 2 C ε [ ln ( e + Θ ( t ) ) ] } C 0 ( e + Θ ( t ) ) 2 C ε , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_Equ35_HTML.gif
(4.11)

where C 0 = u ( T 0 ) L 2 2 + B ( T 0 ) L 2 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq62_HTML.gif.

From (4.11), H 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-187/MediaObjects/13661_2013_Article_600_IEq68_HTML.gif estimate for this case is the same as that for Theorem 1.1. Thus, Theorem 1.2 is proved. □

Declarations

Authors’ Affiliations

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

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© Hu and Wang; licensee Springer 2013

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