Regularity criterion for 3D MHD fluid passing through the porous medium in terms of gradient pressure

Journal of Computational and Applied Mathematics 270 (2014) 88–99

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Regularity criterion for 3D MHD fluid passing through the porous medium in terms of gradient pressure S. Rahman Department of Applied Mathematics, Northwestern Polytechnical University, 710129, Xi’an, Shaanxi, PR China

article

abstract

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Article history: Received 11 July 2013 Received in revised form 13 February 2014

The aim of the paper is to establish regularity criteria for the weak solution of fluid passing through the porous media in R3 . Our results show that if ∇ P ∈ Lα,γ with α2 + γ3 ≤ 3, 1 ≤

γ ≤ ∞, then the weak solution is regular and unique; if P ∈ Lα,γ with γ ≤ ∞, then the weak solution is regular and unique.

Keywords: Regularity criterion Porous medium Incompressible MHD fluid

2

α

+

3

γ

≤ 2,

3 2

≤

© 2014 Elsevier B.V. All rights reserved.

1. Introduction The literature study reveals that viscoelastic fluids have many practical applications in chemical engineering such as plastics, pulps and emulsions. The viscoelastic fluids passing through the porous medium in the presence of a magnetic field are most frequent in nature and the research of such flows is important in many scientific and engineering applications. A few examples are the flow of mercury amalgams, handling of biological fluids and flow of plasma. The porous medium was first started by Darcy [1] and Forchheimer [2]. The study of above MHD system has a long history. Sermange and Temam [3] are considered as pioneer researchers who proved the local well-posedness of weak solutions for any given initial datum u0 , b0 ∈ H s (R3 ), s ≥ 3. But whether this unique local solution can exists globally is an outstanding challenging problem. Fundamental Serrien-type regularity was given by C. He, Z. Xin [4] and Y. Zhou [5] in terms of the velocity. Chen et al. [6] proved regularity by adding the condition on ∇j (∇ × u) and some further improvement was done by He and Wang [7]. It is also mentioned that recently logarithmically improved regularity criteria was established in [8] and [9]. S. Gala, Y. Sawano, H. Tanaka [10] established Serrien’s uniqueness results of the Leray weak solution for the 3D incompressible MHD equations in Orlicz–Morrey spaces. The norm of the space Lα,γ = Lα (0, T ; Lγ (R3 )) is defined by

∥ u∥ =

    

t

α

 α1

∥u(., τ )∥Lγ dτ 0   ess sup ∥u(·, τ )∥Lγ 0<τ
if 1 ≤ α < ∞, if α = ∞,

where

∥u(·, τ )∥Lγ =

    R3

 γ1 |u(x, τ )|γ dx

  ess sup |u(·, τ )|Lγ x∈R3

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.cam.2014.02.019 0377-0427/© 2014 Elsevier B.V. All rights reserved.

if 1 ≤ γ < ∞, if γ = ∞.

S. Rahman / Journal of Computational and Applied Mathematics 270 (2014) 88–99

89

Regularity criteria for the 3D MHD equations in terms of pressure are obtained by Y. Zhou [11]. He derived that if ∇ P ∈ Lα,γ with α2 + γ3 ≤ 3 provided u0 , b0 ∈ H s (R3 ) for s ≥ 3, then the solution remains smooth on [0, T ]. Later on H. Duan [12] 2γ

also improved the results of Y. Zhou [11] and proved that for any given u0 , b0 ∈ H s (R3 ) with s ≥ 3 and ∇ P ∈ L 2γ −3 (0, T ; Lγ (R3 )) for 1 < γ ≤ ∞, and then (u, b) can be extended smoothly beyond t = T . The objective of current paper is to establish regularity of weak solutions of 3D incompressible, MHD fluid passing through the porous medium, in terms of pressure. Our result shows that for any initial velocity and magnetic field u0 , ifb0 ∈ H s (R3 ) for s ≥ 3 and ∇ P ∈ Lα,γ with α2 + γ3 ≤ 3, 1 ≤ γ ≤ ∞, then the weak solution is regular and strong. Let us consider the 3D flow of an incompressible fluid passing through the porous medium. Let u1 , u2 and u3 be the components of velocity u. It is well known when the fluid passes through the porous medium, there exists a pressure drop. This pressure drop is given by using Darcy’s law

φ

u, K where φ is the porosity of the medium and K is the permeability of the medium. The fundamental equations which governs the 3D equations under the assumption of an incompressible, unsteady MHD fluid passing through the porous medium are R=−

∂u + u · ∇ u = ν1 △u − ▽P + b · ∇ b − mu, ∂t ∂b + u · ∇ b = ν2 △b + b · ∇ u, ∂t ∇ · u = ∇ · b = 0, u(x, 0) = u0 (x), b(x, 0) = b0 (x),

(1.1) (1.2) (1.3) (1.4)

where u is the velocity, b is the magnetic field, ν1 is the kinematic viscosity, ν2 is the magnetic diffusivity, and P is the φ pressure of the medium. For simplicity we let m = K , also take ν1 = ν2 = 1. Our main results are mentioned in the following theorem. Theorem 1. Suppose that u0 , b0 ∈ H s (R3 ), s ≥ 3 and ∇ · u0 = 0 = ∇ · b0 , in the sense of distribution. Assume that a pair of (u, b) is the Leray–Hopf weak solution satisfying (1.1)–(1.4) on [0, T ]. If ∇ P ∈ Lα,γ and b ∈ L3α,3γ with α2 + γ3 ≤ 3, 1 ≤ γ ≤ ∞, and ∥∇ P ∥L1 , ∥∇ P ∥ 2 ,∞ , ∥b∥L3 , and ∥b∥L2,∞ are sufficiently small, then the solution remains smooth on [0, T ]. L3

Theorem 2. Suppose that u0 , b0 ∈ H s (R3 ), s ≥ 3 and ∇ · u0 = 0 = ∇ · b0 , in the sense of distribution. Assume that a pair of (u, b) is the Leray–Hopf weak solution satisfying (1.1)–(1.4) on [0, T ]. If P ∈ Lα,γ and b ∈ L2α,2γ with α2 + γ3 ≤ 2, 32 ≤ γ ≤ ∞, or ∥P ∥ 3 , ∥P ∥L1,∞ , ∥b∥L3 , and ∥b∥L2,∞ are sufficiently small, then the solution remains smooth on [0, T ]. L2

Before going to the proof of Theorems 1 and 2, we recall the definition of Leray–Hopf weak solutions. Definition. A pair of (u, b) is called a Leray–Hopf weak solution to equations (1.1)–(1.4), if (u, b) satisfies the following properties (i) u and b are weakly continuous from [0, ∞) to L2 (R3 ). (ii) u and b verify (1.1)–(1.4), in the sense of distribution, i.e., ∞



  R3

0

∞



    ∞ ∂φ + (u · ∇φ) u − (b · ∇φ)b dxdt + u0 φ(x, 0)dx − m uφ dxdt ∂t R3 0 R3



= 0

R3

∇ u · ∇φ dxdt ,

and ∞



  R3

0

(iii)

    ∞ ∂φ + (u · ∇φ) b − (b · ∇φ)u dxdt + u0 φ(x, 0)dx = ∇ u · ∇φ dxdt , ∂t R3 0 R3

for all φ ∈ C0∞ (R3 × [0, ∞)) with ∇ · φ = 0. ∞





0

∞

 u · ∇φ dxdt = 0



and

R3

0

R3

b · ∇φ dxdt = 0,

for every φ ∈ C0∞ (R3 × [0, ∞)). (iv) The energy inequality for t ≥ 0 is true, i.e.,

∥u(t )∥2L2 + ∥b(t )∥2L2 + 2

t

 0

  ∥∇ u(s)∥2L2 + ∥∇ u(s)∥2L2 ds + 2m

t

 0

∥u(s)∥2L2 ds ≤ ∥u0 ∥2L2 + ∥b0 ∥2L2 .

(1.5)

90

S. Rahman / Journal of Computational and Applied Mathematics 270 (2014) 88–99

By the strong solution we mean a weak solution u such that u ∈ L∞ (0, T ; H 1 )



L2 (0, T ; H 2 ).

It is well known that strong solutions are regular and unique in the class of weak solutions. 2. Proof of Theorem 2 In order to prove Theorem 1, first we show the following theorem. Lemma 1. Suppose that the initial velocity and the magnetic field u0 , b0 ∈ Ls (R3 ) for s ≥ 3 with ∇ · u0 = 0 = ∇ · b0 . Assume that (u, b) is a Leray–Hopf weak solution of (1.1)–(1.4) and P is smooth on R3 × (0, T ). If ∇ P ∈ Lα,γ and b ∈ L3α,3γ with α2 + γ3 ≤ 3, 1 < γ ≤ ∞, and ∥∇ P ∥L1 , ∥∇ P ∥ 2 ,∞ , ∥b∥L3 , and ∥b∥L2,∞ are sufficiently small, then L3

sup (∥u(t )∥Ls + ∥b(t )∥Ls ) ≤ C (∥u0 ∥Ls + ∥b0 ∥Ls ) ,

(2.1)

0≤t ≤T

where C depends on the m, T , norms of P and b. Proof. It is sufficient to prove (2.1) for α2 + γ3 = 3 because the proof of (2.1) for α2 + γ3 < 3 is similar. Taking ∇ div on both sides of (1.1) for smooth (u, P ), we have

−△(∇ P ) =

3 

∂i ∂j (∇(ui uj ) − ∇(bi bj )).

i,j=1

Therefore, according to the Calderon Zygmund inequality, it follows that for 1 < q < ∞,

∥∇ P ∥Lq ≤ C1 (∥|u| |∇ u|∥Lq + ∥|b| |∇ b|∥Lq ) . For s ≥ 3, multiplying (1.1) by su|u|

s−2

(2.2) 3

and integrating over R , we obtain



 ∥u(·, t )∥sLs + s |∇ u|2 |u|s−2 dx + s(s − 2) u · ∇ u|u|s−3 ∇|u|dx 3 3 dt R R   s −2 s−2 = −s ∇ P · u|u| dx + s (b · ∇ b)u|u| dx − sm ∥u∥sLs 3 R3   R     s−1 ≤s |∇ P | |u| dx + s (b · ∇ b)u|u|s−2 dx + sm ∥u∥sLs . 3 3 d

R

R

1− 2s

s 2

s

s

|u| ∇|u| and u · ∇ u = 2s |u|2− 2 ∇|u| 2 , we have 2  s   d 4(s − 2)  s 2   ∥u(·, t )∥sLs + s |u| 2 −1 |∇ u| 2 + ∇|u| 2  2 L L dt s       ≤s |∇ P | |u|s−1 dx + s (b · ∇ b)u|u|s−2 dx + sm ∥u∥sLs .

Now using ∇|u| =

R3

2 s

(2.3)

R3

Similarly we have d dt

  s 2   4(s − 2)  s 2   ∥b(·, t )∥sLs + s |b| 2 −1 |∇ b| 2 + ∇|b| 2  2 = s (b · ∇ u)b|b|s−2 dx. L

s

L

(2.4)

R3

Case 1. For 1 < γ < ∞: Using Hölder’s inequality, Young’s inequality, Interpolation’s inequality and (2.2) on the first term of the right hand side of (2.3), we get

 s R3

1 |∇ P | |u|s−1 dx ≤ s ∥∇ P ∥La1 ∥u∥sL− a2 1 ≤ s ∥∇ P ∥1Lγ−θ ∥∇ P ∥θLq ∥u∥sL− a2   1 ≤ C1 ∥∇ P ∥1Lγ−θ ∥|u| |∇ u|∥θLq + ∥|b| |∇ b|∥θLq ∥u∥sL− a2    θ  s s  s θ  s  1 ≤ C1 ∥∇ P ∥1Lγ−θ |u| 2 −1 |∇ u||u|− 2 +2  q + |b| 2 −1 |∇ b| |b|− 2 +2  q ∥u∥sL− a2 L L   θ  s θ (4−s)θ (4−s)θ  s    1 ≤ C1 ∥∇ P ∥1Lγ−θ |u| 2 −1 |∇ u| 2 ∥u∥ (42−s)q + |b| 2 −1 |∇ b| 2 ∥b∥ (42−s)q ∥u∥sL− a2 . L

L 2−q

L

L 2−q

S. Rahman / Journal of Computational and Applied Mathematics 270 (2014) 88–99

If a2 =

 s

(4−s)q , 2 −q

91

then the above equation becomes

  θ  s θ (4−s)θ (4−s)θ +s−1 +s−1  s    |∇ P | |u|s−1 dx ≤ C1 ∥∇ P ∥1Lγ−θ |u| 2 −1 |∇ u| 2 ∥u∥La22 + |b| 2 −1 |∇ b| 2 ∥b∥La22 L L R3  2 2   s s  s −1    ≤ |u| 2 |∇ u| 2 + |b| 2 −1 |∇ b| 2 2

L

2(1−θ) 2−θ

+ C2 ∥∇ P ∥Lγ

L

  2(s−1)+θ(4−s) 2(s−1)+θ(4−s) ∥u∥La2 2−θ + ∥b∥La2 2−θ ,

(2.5)

where C2 is a constant and a1 , a2 , q and θ satisfy 1 a1 1

+

s−1 a2 1−θ

= 1,

(2.6)

θ + , γ q (4 − s)q a2 = . 2−q For estimating ∥u∥sLa2 we also need a1

=

(2.7) (2.8)

2(s − 1) + θ (4 − s)

≤ s. 2−θ Solving (2.6)–(2.9), we obtain θ=

1 2

and

a2 =

3sγ 3γ − 2

(2.9)

.

Using θ and a2 in (2.5), applying Interpolation’s inequality and Young’s inequality, it yields



   2  s 2  2  2s −1    |∇ u| 2 + |b| 2 −1 |∇ b| 2 + C2 ∥∇ P ∥L3γ ∥u∥s 3γ s + ∥b∥s 3γ s |u| L L 2 L 3γ −2 L 3γ −2  2  s 2  s  s −1    − 1 ≤ |u| 2 |∇ u| 2 + |b| 2 |∇ b| 2 L L 2   (γ −1)s (γ −1)s s s 2 γ γ γ γ 3 ∥u∥L3s + ∥b∥Ls ∥b∥L3s + C2 ∥∇ P ∥Lγ ∥u∥Ls      s  s −1  s 2 2 ≤ |u| 2 |∇ u| 2 + |b| 2 −1 |∇ b| 2

|∇ P ||u|s−2 dx ≤

s R3

s

2

L

L

2γ

    + C3 ∥∇ P ∥L3γ(γ −1) ∥u∥sLs + ∥b∥sLs + ϵ ∥u∥sL3s + ∥b∥sL3s   s 2 2  s  s −1    ≤ |u| 2 |∇ u| 2 + |b| 2 −1 |∇ b| 2 L L 2      2γ   s −2  s 2 s 2   + C3 ∥∇ P ∥L3γ(γ −1) ∥u∥sLs + ∥b∥sLs + ∇|u| 2  2 + ∇|b| 2  2 , s

L

(2.10)

L

where ϵ is a small constant such that the following Sobolev’s inequality holds

  s 2 (s − 2)  s 2    ϵ ∥u∥sL3s = ϵ |u| 2  6 ≤ ∇|u| 2  2 , L L s  s 2   ( s − 2 ) s 2    ϵ ∥b∥sL3s = ϵ |b| 2  6 ≤ ∇|b| 2  2 .

L L s Now we estimate the second term of the right hand side of (2.3) and according to Hölder’s inequality, Young’s inequality and Interpolation’s inequality, and obtain

          s (b · ∇ b)u|u|s−2 dx = −s (b · ∇(u|u|s−2 ))bdx  3    3 R   R    = s (b · ∇(u|u|s−2 ))bdx 3  R    s −4 s−3  ≤ s b · (u · (u · ∇ u)|u| + (s − 2)u|u| ∇|u|)bdx . R3

92

S. Rahman / Journal of Computational and Applied Mathematics 270 (2014) 88–99 2 s

1− 2s

s

s

s

∇|u| 2 and u · ∇ u = 2s |u|2− 2 ∇|u| 2 , it follows       s s s (b · ∇ b)u|u|s−2 dx ≤ 2(s − 1) |b|2 |u| 2 −1 |∇|u | 2 |dx  3  R R3   s s  −1  ≤ 2(s − 1) ∥b∥2La1 ∥u∥L2a2 ∇|u| 2  2 L   s−2  s 2 2 4 ≤ ∇|u| 2  2 + C4 ∥b∥La1 ∥u∥sL− a2 L s   s−2  s 2 2 ≤ ∇|u| 2  2 + C4 ∥b∥2L3γ ∥b∥2La2 ∥u∥sL− a2

Since ∇|u| =

|u|

≤

s  s−2 

L

2    ∇|u|  2 + C4 ∥b∥2L3γ ∥u∥sLa2 + ∥b∥sLa2 ,

s

s 2

(2.11)

L

where C4 is a constant and a1 and a2 satisfy 2 a1 4 a1

+ =

s 2

−1 a2

2 3γ

+

= 2 a2

1 2

,

(2.12)

.

(2.13)

The simultaneous solution of (2.12) and (2.13) yields a2 =

3sγ 3γ − 2

.

Using a2 in (2.11) and applying Interpolation’s inequality, Young’s inequality and Sobolev’s inequality, we have

         s 2 2 s s s (b · ∇ b)u|u|s−2  dx ≤ (s − 2)  2 ∥ ∥ ∥ ∥ ∥ ∥ u | + C b u + b ∇| 4 3γ s 3γ s  3  L3γ L2 s R L γ −2 L 3γ −2    (γ −1)s (γ −1)s s s (s − 2)  s 2  γ ∥u∥Lγ3s + ∥b∥Ls γ ∥b∥Lγ3s ≤ ∇|u| 2  2 + C5 ∥b∥2L3γ ∥u∥Ls s

L

 2γ     (s − 2)  s 2  γ −1 ≤ ∇|u| 2  2 + C5 ∥b∥L3γ ∥u∥sLs + ∥b∥sLs + ϵ ∥u∥sL3s + ∥b∥sL3s

L s    2γ   s−2  2(s − 2)  s 2 s 2  γ −1 ≤ ∇|u| 2  2 + C5 ∥b∥L3γ ∥u∥sLs + ∥b∥sLs + ∇|b| 2  2 , L L s s

(2.14)

where C5 is a constant. Now from (2.10) and (2.14), we see that (2.3) becomes s−2  s 2 ∥u(·, t )∥ + ∇|u| 2  2 ≤ L dt s d

s Ls





2γ 3(γ −1)



C3 ∥∇ P ∥Lγ

2γ γ −1 3 L γ



+ C5 ∥b∥

2  s  s  s  ∥u∥Ls + ∥b∥sLs + |b| 2 −1 |∇ b| 2

  2(s − 2)  s 2 + ∇|b| 2  2 + sm ∥u∥sLs . s

2

L

(2.15)

L

The right hand side of (2.4) can be estimated in a similar way:

 s

 (b · ∇ u)b|b|s−2 dx = −s (b · ∇(b|b|s−2 ))udx 3 R3   R    s−2  ≤ s (b · ∇(b|b| ))udx 3 R  s s ≤ 2(s − 1) |b| 2 |u||∇|b | 2 |dx R3   s s   ≤ 2(s − 1) ∥b∥L2a1 ∥u∥La2 ∇|b| 2  2 L   s−2  s 2 s ≤ ∇|b| 2  2 + C6 ∥b∥La1 ∥u∥sLa2 L s   s−2  s 2 2 2 ≤ ∇|b| 2  2 + C6 ∥b∥2L3γ ∥b∥sL− a2 ∥u∥La2 ≤

s  s−2  s

L

2    ∇|b|  2 + C6 ∥b∥2L3γ ∥u∥sLa2 + ∥b∥sLa2 , s 2

L

(2.16)

S. Rahman / Journal of Computational and Applied Mathematics 270 (2014) 88–99

93

where C6 is a constant, a1 and a2 satisfy s 2

a1 s a1

+ =

1 a2 2 3γ

1

=

2

+

,

(2.17)

s−2 a2

.

(2.18)

Solving (2.17) and (2.18), it leads to a2 =

3sγ

γ −2

.

Using a2 in (2.16), applying Interpolation’s inequality and Young’s inequality, we obtain

         s 2 s 2 s s (b · ∇ b)b|b|s−2 dx ≤ (s − 2)  2 ∇|b|  2 + C6 ∥b∥L3γ ∥u∥ 3γ s + ∥b∥ 3γ s  3  L s R L 3γ −2 L 3γ −2     (γ −1)s (γ −1)s s s (s − 2)  s 2 γ γ γ γ 2 2 ∥u∥L3s + ∥b∥Ls ∥b∥L3s ≤ ∇|b|  2 + C7 ∥b∥L3γ ∥u∥Ls s

L

   2γ   (s − 2)  2(s − 2)  s 2 s 2  γ −1 ≤ ∇|b| 2  2 + C7 ∥b∥L3γ ∥u∥sLs + ∥b∥sLs + ∇|u| 2  2 , s

s

L

(2.19)

L

where C7 is a constant. Now by (2.19) and (2.4), it shows d dt

∥b(·, t )∥sLs +

   2γ   (s − 2)  2(s − 2)  s 2 s 2  γ −1 ∇|b| 2  2 ≤ C7 ∥b∥L3γ ∥u∥sLs + ∥b∥sLs + ∇|u| 2  2 . L L s s

(2.20)

Adding (2.15) and (2.20), we get

dt

2γ 3(γ −1)



d 

 ∥u(·, t )∥sLs + ∥b(·, t )∥sLs ≤

C3 ∥∇ P ∥Lγ

2γ 3(γ −1)



C3 ∥∇ P ∥Lγ

≤

2γ

+ (C5 + C7 ) ∥b∥Lγ3−γ 1 + (C5 + C7 ) ∥b∥

2γ γ −1 3 L γ



  s ∥u∥Ls + ∥b∥sLs + sm ∥u∥sLs 

+ sm

 s  ∥u∥Ls + ∥b∥sLs .

(2.21)

Now we use Gronwall’s inequality and know sup 0 ≤t ≤T

  ∥u(·, t )∥sLs + ∥b(·, t )∥sLs ≤ exp

T



2γ 3(γ −1)



C3 ∥∇ P ∥Lγ 0

  2γ   + (C5 + C7 ) ∥b∥Lγ3−γ 1 + sm dt ∥u0 ∥sLs + ∥b0 ∥sLs .

2γ

As α2 + γ3 = 3, 3(γ −1) = α and the above inequality becomes sup 0≤t ≤T

     ∥u(·, t )∥sLs + ∥b(·, t )∥sLs ≤ exp C3 ∥∇ P ∥αLα,γ + (C5 + C7 ) ∥b∥3L3αα,3γ + smT ∥u0 ∥sLs + ∥b0 ∥sLs .

(2.22)

As ∇ P ∈ Lα,γ and b ∈ L3α,3γ , C3 ∥∇ P ∥αLα,γ + (C5 + C7 ) ∥b∥3L3αα,3γ + smT ≤ C8 ,

(2.23)

and (2.22) becomes sup 0 ≤t ≤T

    ∥u(·, t )∥sLs + ∥b(·, t )∥sLs ≤ C8 ∥u0 ∥sLs + ∥b0 ∥sLs ,

where C8 depends on m, T , norms of P and b. Case 2. γ = 1, α = ∞: in this case we have θ = d dt

∥u(·, t )∥sLs +

3(s − 2)  s 2 ∇|u| 2  2 ≤ L s







1 2

(2.24)

and a2 = 3s; therefore (2.3) and (2.4) become 2

C9 ∥∇ P ∥L31 + C10 ∥b∥2L3

      s 2 s 2   ∇|u| 2  2 + ∇|b| 2  2 L

L

s  s 2 + |b| 2 −1 |∇ b| 2 + sm ∥u∥sLs , L 2



d dt



 s 2   3(s − 2)  s 2   ∥b(·, t )∥sLs + s |b| 2 −1 |∇ b| 2 + ∇|b| 2  2 L

s

L

(2.25)

     s 2 s 2   ≤ C11 ∥b∥2L3 ∇|u| 2  2 + ∇|b| 2  2 . L

L

(2.26)

94

S. Rahman / Journal of Computational and Applied Mathematics 270 (2014) 88–99

Combining (2.25) and (2.26), we get

      3(s − 2)  s 2 s 2   ∥u(·, t )∥sLs + ∥b(·, t )∥sLs + ∇|u| 2  2 + ∇|b| 2  2 L L dt s        2 s 2 s 2   ≤ C9 ∥∇ P ∥L31 + (C10 + C11 ) ∥b∥2L3 ∇|u| 2  2 + ∇|b| 2  2 + sm ∥u∥sLs . d 

L

L

2

Since ∥∇ P ∥L31 and ∥b∥2L3 are sufficiently small, we can choose 2

C9 ∥∇ P ∥L31 + (C10 + C11 ) ∥b∥2L3 ≤

3(s − 2) s

,

and derive d 

 ∥u(·, t )∥sLs + ∥b(·, t )∥sLs ≤ sm ∥u∥sLs dt   ≤ sm ∥u∥sLs + ∥b∥sLs . Applying Gronwall’s inequality, it shows sup



0≤t ≤T

   ∥u(·, t )∥sLs + ∥b(·, t )∥sLs ≤ C12 ∥u0 ∥sLs + ∥b0 ∥sLs ,

where C12 depends on m and T . Case 3. γ = ∞, α = 32 : Now θ = d  dt

 ∥u∥sLs + ∥b∥sLs ≤

1 2

and a2 = s; therefore we have



2



2

C13 ∥P ∥L3∞ + (C14 + C15 ) ∥b∥2L∞

≤

(2.27)



C13 ∥P ∥L3∞ + (C14 + C15 ) ∥b∥2L∞

 ∥u∥sLs + ∥b∥sLs + sm ∥u∥sLs    + sm ∥u∥sLs + ∥b∥sLs . 

Applying Gronwall’s inequality, it follows sup



∥u(·, t )∥ + ∥b(·, t )∥ s Ls

0≤t ≤T

s Ls



T

 ≤ exp



2 3

C13 ∥P ∥L∞ + (C14 + C15 ) ∥b∥

2 L∞

 + sm dt



  ∥u0 ∥sLs + ∥b0 ∥sLs

0

 = exp

2

C13 ∥P ∥ 32 L

2

From the fact that ∥P ∥ 32 L3 2

C13 ∥P ∥ 32 L3

,∞

,∞

,∞ 3

+ (C14 + C15 ) ∥b∥2L2,∞ + smT



  ∥u0 ∥sLs + ∥b0 ∥sLs .

(2.28)

and ∥b∥2L2,∞ are sufficiently small, we can choose

+ (C14 + C15 ) ∥b∥2L2,∞ + smT ≤ C16 ,

and (2.28) becomes sup 0≤t ≤T



   ∥u(·, t )∥sLs + ∥b(·, t )∥sLs ≤ C16 ∥u0 ∥sLs + ∥b0 ∥sLs

(2.29)

where C16 depends on m, T , norms of P and b. This finishes the proof. Proof of Theorem 1. Since u0 , b0 ∈ Ls (R3 ) for 3 ≤ s ≤ 4, following the results of Giga [13] (see also [14] for the Navier– Stokes equations), for a maximal interval [0, T∗ ), there exists a unique solution u˜ ∈ BC ([0, T∗ ; Ls (R3 ))) and b˜ ∈ BC ([0, T∗ ; Ls (R3 ))). As a pair of (u, b) is a Leray–Hopf weak solution which satisfies (1.5) and according to the uniqueness criterion of Serrin–Masuda (see [3]), u ≡ u˜ and b ≡ b˜ on [0, T∗ ). According to the same argument for the priori estimates (2.1), the local smooth solution would be extended to time T . This completes the proof of Theorem 1. 3. Proof of Theorem 2 In order to prove Theorem 2, first we show the following lemma.

S. Rahman / Journal of Computational and Applied Mathematics 270 (2014) 88–99

95

Lemma 2. Suppose that the initial velocity and magnetic field u0 , b0 ∈ Ls (R3 ) where s ≥ 3 with ∇ · u0 = 0 = ∇ · b0 . Assume that (u, P ) is a smooth solution of (1.1)–(1.4) in R3 ×(0, T ). If P ∈ Lα,γ and b ∈ L2α,2γ with α2 + γ3 ≤ 2, 32 < γ ≤ ∞, or ∥P ∥ 3 , L2

∥P ∥L1,∞ , ∥b∥L3 , and ∥b∥L2,∞ are sufficiently small then sup (∥u(t )∥Ls + ∥b(t )∥Ls ) ≤ C (∥u0 ∥Ls + ∥b0 ∥Ls ) ,

(3.1)

0 ≤t ≤T

where C depends on the m, T , norms of P and b. Proof. It is sufficient to consider the equality α2 + γ3 = 2. Taking ∇ div on both sides of (1.1) for smooth (u, P ), we have 3 

−△P =

∂i ∂j (ui uj − bi bj ).

i ,j = 1

Therefore, according to the Calderon Zygmund, for 1 < q < ∞, we get

  ∥P ∥Lq ≤ C ∥u∥2L2q + ∥b∥2L2q .

(3.2) s −2

For any s ≥ 3, multiplying (1.1) by su|u|

3

and integrating over R , we obtain



 ∥u(·, t )∥sLs + s |∇ u|2 |u|s−2 dx + s(s − 2) u · ∇ u|u|s−3 ∇|u|dx 3 3 dt R R   s−4 s −3 =s P ((u · ∇ u)u|u| + (s − 2)u|u| ∇|u|)dx + s (b · ∇ b)u|u|s−2 dx − sm ∥u∥sLs . d

R3

R3

2 s

s

2− 2s

2 s

∇|u| and u · ∇ u = |u| ∇|u| 2 in the above equation, we have    d 4(s − 2)  s 2 ∥u(·, t )∥sLs + s |∇ u|2 |u|s−2 dx + ∇|u| 2  2 L dt s R3   s s = 2(s − 1) Pu|u| 2 −2 ∇|u| 2 dx + s (b · ∇ b)u|u|s−2 dx − sm ∥u∥sLs .

Now we use ∇|u| =

|u|

s 2

1− 2s

R3

Since

4 s2

s 2

R3

s−2

2

|∇|u | | ≤ |u|

2

|∇ u| , we obtain

4(s − 1)  s 2 ∥u(·, t )∥ + ∇|u| 2  2 ≤ 2(s − 1) L dt s d



s Ls





s 2

|P | |u| |∇|u | R3

s −1 2

    s−2  |dx + s  (b · ∇ b)u|u| dx + sm ∥u∥sLs . (3.3) 3 R

Similarly we have d dt

∥b(·, t )∥sLs +

4(s − 1)  s 2 ∇|b| 2  2 = s L s





 R3

(b · ∇ u)b|b|s−2 dx.

(3.4)

Case 1. For 32 < γ < ∞, using Holder’s inequality, Young’s inequality, Interpolation’s inequality and (3.2) on the first term of the right hand side of (3.4), we get 2(s − 1)

 R3

  s s s s  −1  |P | |u| 2 −1 |∇|u | 2 |dx ≤ 2(s − 1) ∥P ∥La1 ∥u∥L2a2 ∇|u| 2  2 L

 (s − 1)  s 2  2 ≤ ∇|u| 2  2 + C1 ∥P ∥2La1 ∥u∥sL− a2 L 2s  (s − 1)  s 2  2(1−θ ) 2 ∥P ∥2θa2 ∥u∥sL− ≤ ∇|u| 2  2 + C2 ∥P ∥Lγ a2 L 2s L 2   4θ  (s − 1)  s 2  2(1−θ ) 2 ∥u∥La2 + ∥b∥4Laθ2 ∥u∥sL− ≤ ∇|u| 2  2 + C2 ∥P ∥Lγ a2 2s

L

  4θ+s−2  (s − 1)  s 2  2(1−θ ) ∥u∥La2 + ∥b∥L4aθ2+s−2 , ≤ ∇|u| 2  2 + C2 ∥P ∥Lγ 2s

(3.5)

L

where C3 is constant and a1 , a2 and θ satisfy the following equations 1 a1 1 a1

+ =

s 2

−1 a2

1−θ

γ

= +

1 2

,

2θ a2

(3.6)

.

(3.7)

96

S. Rahman / Journal of Computational and Applied Mathematics 270 (2014) 88–99

We want to estimate ∥u∥sLs ; therefore let on the right hand side the power of ∥u∥Ls be 4θ + s − 2 ≤ s.

(3.8)

Solving (3.6)–(3.8), we obtain

θ=

1

and a2 =

2

sγ . γ −1

Using θ and a2 in (3.5), applying Interpolation’s inequality and Young inequality, we obtain 2(s − 1)



s

|P | |∇|u | 2 |dx ≤ R3

 (s − 1)  s 2  ∇|u| 2  2 + C2 ∥P ∥Lγ ∥u∥s 2s

+ C2 ∥P ∥Lγ ∥b∥s

γs L γ −1 (2γ −3)s 2γ Ls

L

γs L γ −1

 (2γ −3)s 3s 3s (s − 1)  s 2  ∥u∥L23sγ + C3 ∥P ∥Lγ ∥b∥Ls 2γ ∥b∥L23sγ ∇|u| 2  2 + C3 ∥P ∥Lγ ∥u∥ L 2s 2 2γ 2γ (s − 1)  s   2γ −3 2γ −3 ≤ ∇|u| 2  2 + C4 ∥P ∥Lγ ∥u∥sLs + ϵ ∥u∥sL3s + C4 ∥P ∥Lγ ∥b∥sLs + ϵ ∥b∥sL3s L 2s   2γ (s − 1)  (s − 1)  s 2 s 2   2γ −3 ≤ ∇|u| 2  2 + C4 ∥P ∥Lγ ∥u∥sLs + ∇|u| 2  2 L L 2s s   2γ ( s − 1 ) s 2  2γ −3 + C4 ∥P ∥Lγ ∥b∥sLs + ∇|b| 2  2 L s   2γ 2γ 3(s − 1)  s 2 2γ −3 2γ −3 ≤ ∇|u| 2  2 + C4 ∥P ∥Lγ ∥u∥sLs + C4 ∥P ∥Lγ ∥b∥sLs ≤

2s

L

 (s − 1)  s 2  + ∇|b| 2  2 , s

(3.9)

L

where ϵ is a small constant such that the following Sobolev’s inequality holds

 s 2  (s − 1)  s 2    ϵ ∥u∥sL3s = ϵ |u| 2  6 ≤ ∇|u| 2  2 , L L s   s 2  ( s − 1 ) s 2    ϵ ∥b∥sL3s = ϵ |b| 2  6 ≤ ∇|b| 2  2 .

L L s Now we estimate the second term of the right hand of (3.4) and according to Holder’s inequality, Young’s inequality and Interpolation’s inequality, we have

     s (b · ∇ b)u|u|s−2 dx =   3 R

= ≤ ≤ ≤ ≤

     −s (b · ∇(u|u|s−2 ))bdx   3    R   s (b · ∇(u|u|s−2 ))bdx   3 R   s s  −1  2(s − 1) ∥b∥2La1 ∥u∥L2a2 ∇|u| 2  2 L  s s  −1  2 2 2(s − 1) ∥b∥La1 ∥u∥La2 ∇|u| 2  L2   s−1  s 2 2 ∇|u| 2  2 + C5 ∥b∥4La1 ∥u∥sL− a2 L 2s   s−1  s 2 2 ∇|u| 2  2 + C6 ∥b∥2L2γ ∥b∥2La2 ∥u∥sL− a2 2s

L

    s−1  s 2 ≤ ∇|u| 2  2 + C7 ∥b∥2L2γ ∥u∥sLa2 + ∥b∥sLa2 , 2s

(3.10)

L

where C7 is constant, and a1 and a2 satisfy the following equations 2 a1 2 a1

+ =

s 2

−1

=

a2 1

γ

+

1 a2

1 2

,

.

The simultaneous solution of (3.11) and (3.12) yields a2 =

sγ

γ −1

.

(3.11) (3.12)

S. Rahman / Journal of Computational and Applied Mathematics 270 (2014) 88–99

97

Using a2 in (3.10) and applying Interpolation’s inequality, Young’s inequality and Sobolev’s inequality, we obtain

         s 2 s 2 s s (b · ∇ b)u|u|s−2  dx ≤ (s − 1)  2 ∥ ∥ ∥ ∥ u | + C b u ∇| γ s + ∥b ∥ γ s 7 2 γ  3  L L2 2s L γ −1 L γ −1 R     (2γ −3)s (2γ −3)s 3s 3s (s − 1)  s 2 2γ 2γ 2γ 2γ 2 2 ∥u∥L3s + ∥b∥Ls ∥b∥L3s ≤ ∇|u|  2 + C7 ∥b∥L2γ ∥u∥Ls 2s

L

   4γ   (s − 1)  3(s − 1)  s 2 s 2  2γ −3 ≤ ∇|u| 2  2 + C8 ∥b∥L2γ ∥u∥sLs + ∥b∥sLs + ∇|b| 2  2 , 2s

s

L

(3.13)

L

where C8 is constant. Taking (3.9) and (3.13) into account (3.3) becomes d dt

s−1  s 2 ∇|u| 2  2 ≤ L s



∥u(·, t )∥sLs +



4γ

2γ



2γ −3 + C9 ∥b∥L22γγ−3 C4 ∥P ∥Lγ

+



 s  ∥u∥Ls + ∥b∥sLs

2(s − 1)  s 2 ∇|b| 2  2 + sm ∥u∥sLs . L s





(3.14)

The right hand side of (3.4) can be done in a similar way



(b · ∇ u)b|b|

s−2

s R3



(b · ∇(b|b|s−2 ))udx      s (b · ∇(b|b|s−2 ))udx  3  R  s s 2(s − 1) |b| 2 |u||∇|b | 2 |dx 3 R   s s   2(s − 1) ∥b∥L2a1 ∥u∥La2 ∇|b| 2  L2   s−1  s 2 ∇|b| 2  2 + C10 ∥b∥sLa1 ∥u∥sLa2 L s   s−1  s 2 2 2 ∇|b| 2  2 + C11 ∥b∥2L2γ ∥b∥sL− a2 ∥u∥La2

dx = −s

R3

≤ ≤ ≤ ≤ ≤

s

L

    s−1  s 2 ≤ ∇|b| 2  2 + C10 ∥b∥2L2γ ∥u∥sLa2 + ∥b∥sLa2 , 2s

(3.15)

L

where C11 is constant, and a1 and a2 satisfy the following equations s 2

a1 s a1

+ =

1 a2 1

γ

1

= +

2

,

(3.16)

s−2 a2

.

(3.17)

Solving (3.16) and (3.17), we obtain a2 =

sγ

γ −1

.

Using a2 in (3.15), applying Interpolation’s inequality and Young’s inequality, we obtain

         s 2 2 s s s (b · ∇ b)b|b|s−2  dx ≤ (s − 1)  2 ∇|b|  2 + C11 ∥b∥L2γ ∥u∥ γ s + ∥b∥ γ s  3  L s L γ −1 L γ −1 R     ( 2 γ − 3 ) s (2γ −3)s 3s 3s (s − 1)  s 2 2γ ∥u∥L23sγ + ∥b∥Ls 2γ ∥b∥L23sγ ≤ ∇|b| 2  2 + C11 ∥b∥2L2γ ∥u∥Ls s

L

   4γ   (s − 1)  2(s − 1)  s 2 s 2  2γ −3 ≤ ∇|b| 2  2 + C12 ∥b∥L2γ ∥u∥sLs + ∥b∥sLs + ∇|u| 2  2 , s

L

s

(3.18)

L

where C12 is constant. Taking (3.18) into account (3.4) becomes d dt

∥b(·, t )∥sLs +

   4γ   (s − 1)  2(s − 1)  s 2 s 2  2γ −3 ∇|b| 2  2 ≤ C12 ∥b∥L2γ ∥u∥sLs + ∥b∥sLs + ∇|u| 2  2 . L L s s

(3.19)

98

S. Rahman / Journal of Computational and Applied Mathematics 270 (2014) 88–99

Adding (3.14) and (3.19), we get d  dt

∥u(·, t )∥ + ∥b(·, t )∥ s Ls

s Ls



2γ 2γ −3



C4 ∥P ∥Lγ

≤

2γ 2γ −3



C4 ∥P ∥L2γ

≤

4γ 2γ −3 2 L γ

+ (C9 + C12 ) ∥b∥





 ∥u∥sLs + ∥b∥sLs + sm ∥u∥sLs

4γ

+ (C9 + C12 ) ∥b∥L22γγ−3 + sm



 s  ∥u∥Ls + ∥b∥sLs .

(3.20)

Now we use Grown wall’s inequality and get



sup

∥u(·, t )∥ + ∥b(·, t )∥ s Ls

0≤t ≤T

s Ls



T



2γ 2γ −3



C4 ∥P ∥Lγ

≤ exp

+ (C9 + C12 ) ∥b∥

0

4γ 2γ −3 2 L γ



 dt



 ∥u0 ∥sLs + ∥b0 ∥sLs .

2γ

As α + γ = 2, 2γ −3 = α and then the above equation becomes as 2

3

sup



0≤t ≤T

    ∥u(·, t )∥sLs + ∥b(·, t )∥sLs ≤ exp C4 ∥P ∥αLα,γ + (C9 + C12 ) ∥b∥2L2αα,2γ ∥u0 ∥sLs + ∥b0 ∥sLs .

(3.21)

As P ∈ Lα,γ and b ∈ L2α,2γ C4 ∥P ∥αLα,γ + (C9 + C11 ) ∥b∥2L2αα,2γ ≤ C13 .

(3.22)

So (3.21) becomes sup ∥u(·, t )∥sLs ≤ C13 ∥u0 ∥sLs + ∥u0 ∥sLs ,





(3.23)

0≤t ≤T

where C13 depends on m, T , norms of P and b. Case 2. If γ = 32 then α = ∞ and a2 = 3s; therefore (3.3) and (3.4) become

         3(s − 1)  s 2 s 2 s 2   2 2 2 2 ∥u(·, t )∥ + ∇|u|  2 ≤ C14 ∥P ∥ 3 + C15 ∥b∥L3 ∇|u|  2 + ∇|b|  2 + sm ∥u∥sLs , L2 L L L dt s d

s Ls

d dt

∥b(·, t )∥sLs +

3(s − 1)  s 2 ∇|b| 2  2 ≤ C16 ∥b∥2L3 L s





     s 2 s 2   ∇|u| 2  2 + ∇|b| 2  2 . L

(3.24)

(3.25)

L

Adding (3.24) and (3.25), we get

      3(s − 1)  s 2 s 2   ∥u(·, t )∥sLs + ∥b(·, t )∥sLs + ∇|u| 2  2 + ∇|b| 2  2 L L dt s        s 2 s 2   ≤ C14 ∥P ∥ 3 + (C15 + C16 ) ∥b∥2L3 ∇|u| 2  2 + ∇|b| 2  2 + sm ∥u∥sLs , d 

L2



≤ C14 ∥P ∥ Since ∥P ∥

L

L

(3.26)

L

and ∥b∥2L3 are sufficiently small so we can choose

3

L2

C13 ∥P ∥

3

L2

L

        s 2 s 2   + (C15 + C16 ) ∥b∥2L3 ∇|u| 2  2 + ∇|b| 2  2 + sm ∥u∥sLs + ∥b∥sLs .

3

L2

+ (C14 + C15 ) ∥b∥2L3 ≤

3(s − 1) s

,

(3.26) becomes d dt

  ∥u(·, t )∥sLs ≤ sm ∥u∥sLs + ∥b∥sLs .

Now we apply Grown wall’s inequality and obtain sup ∥u(·, t )∥sLs ≤ C17 ∥u0 ∥sLs + ∥b0 ∥sLs ,





0≤t ≤T

where C17 depends on m and T . Case 3. For γ = ∞, α = 1 and a2 = s; therefore the inequality reads d 

    ∥u∥sLs + ∥b∥sLs ≤ C18 ∥P ∥L∞ + (C19 + C20 ) ∥b∥2L∞ ∥u∥sLs + ∥b∥sLs + sm ∥u∥sLs dt    ≤ C18 ∥P ∥L∞ + (C19 + C20 ) ∥b∥2L∞ + sm ∥u∥sLs + ∥b∥sLs .

(3.27)

S. Rahman / Journal of Computational and Applied Mathematics 270 (2014) 88–99

99

Now we apply Grown’s wall’s inequality and get sup 0≤t ≤T

  ∥u(·, t )∥sLs + ∥b(·, t )∥sLs ≤ exp



T



C18 ∥P ∥L∞ + (C19 + C20 ) ∥b∥2L∞ + sm dt







∥u0 ∥sLs + ∥b0 ∥sLs



0

= exp



C18 ∥P ∥L1,∞ + (C19 + C20 ) ∥b∥2L2,∞ + smT

   ∥u0 ∥sLs + ∥b0 ∥sLs .

(3.28)

As ∥P ∥L1,∞ and ∥b∥2L2,∞ is sufficiently small so we can choose C18 ∥P ∥L1,∞ + (C19 + C20 ) ∥b∥2L2,∞ + smT ≤ C21 , therefore (3.28) becomes sup 0≤t ≤T

    ∥u(·, t )∥sLs + ∥b(·, t )∥sLs ≤ C22 ∥u0 ∥sLs + ∥b0 ∥sLs

(3.29)

where C22 depends on m, T , norms of P and b. This finishes the proof. Using the same argument as in the proof of Theorem 1, we can prove Theorem 2. Acknowledgments The author would like to express sincere gratitude to Professor Pengcheng Niu for guidance, constant encouragement and providing an excellent research environment. This work was supported by the National Natural Science Foundation of China(Grant No. 11271299), the Mathematical Tianyuan Foundation of China(Grant No. 11126027) and Natural Science Foundation Research Project of Shaanxi Province(2012JM1014). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

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