Uniqueness of weak solution to the generalized ... - Springer Link

Annali di Matematica (2014) 193:699–722 DOI 10.1007/s10231-012-0298-2

Uniqueness of weak solution to the generalized magneto-hydrodynamic system Qiao Liu · Jihong Zhao · Shangbin Cui

Received: 7 August 2012 / Accepted: 10 October 2012 / Published online: 30 October 2012 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2012

Abstract We study uniqueness of weak solution for the generalized incompressible magneto-hydrodynamic (GMHD) system with suitable β, and we prove that the weak solu2β

r tions are unique in the class L 2β−1+r (0, T ; B∞,∞ ) with r ∈ (1 − 2β, 1].

Keywords Generalized magneto-hydrodynamic system · Weak solution · Uniqueness · Besov spaces Mathematics Subject Classification (2010)

35B65 · 35Q35 · 76W05

1 Introduction In this paper, we consider the following generalized incompressible magneto-hydrodynamic (GMHD) system in Rn × (0, T ), n ≥ 3: u t + (−)β u + (u · ∇)u − (b · ∇)b + ∇ P = 0,

(x, t) ∈ Rn × (0, T ),

bt + (−)β b + (u · ∇)b − (b · ∇)u = 0, (x, t) ∈ Rn × (0, T ), n div u = 0, div b = 0, (x, t) ∈ R × (0, T )

(1.1) (1.2) (1.3)

Research supported by the National Natural Science Foundation of China (11171357). Q. Liu (B) Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, People’s Republic of China e-mail: [email protected] J. Zhao Institute of Applied Mathematics, College of Science, Northwest A&F University, Shaanxi, Yangling 712100, People’s Republic of China e-mail: [email protected] S. Cui Department of Mathematics, Sun Yat-sen University, Guangzhou, Guangdong 510275, People’s Republic of China e-mail: [email protected]

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Q. Liu et al.

with the initial conditions u(x, 0) = u 0 (x), b(x, 0) = b0 (x),

x ∈ Rn ,

(1.4)

where u = u(x, t) = (u 1 (x, t), . . . , u n (x, t)), b = b(x, t) = (b1 (x, t), . . . , bn (x, t)) and P = P(x, t) stand for the fluid velocity, the magnetic field, and the total kinetic pressure, respectively, and β ∈ ( 21 , 1). The fractional Laplace operator (−)β with respect to space variable x is a Riesz potential operator defined as usual through Fourier transform as F ((−)β f )(ξ ) = |ξ |2β F f (ξ ), where F f (ξ ) = f (ξ ) = √ 1 n Rn e−i xξ f (x)dx. The 2π initial velocity field u 0 and the initial magnetic field d0 satisfy div u 0 = 0 and div b0 = 0. The GMHD system (1.1)–(1.4) describes the macroscopic behavior of the electrically conducting incompressible fluids in a magnetic field; it includes the well-known Navier– Stokes equations (β = 1, b ≡ 0) and the standard MHD equations (β = 1). In the last several decades, there have been numerous studies on the GMHD problems by many physicists and mathematicians due to its physical importance, complexity and mathematical challenges, see for example, [3,4,7,8,13,22,25,27–33] and the references therein. In [27], Wu proved that the system (1.1)–(1.4) has a global-in-time weak solution for any given divergence free initial value (u 0 , b0 ) ∈ L 2 (Rn ). Yuan [31] obtained the local-in-time existence and uniqueness of smooth solution for any given sufficient smooth initial data (u 0 , b0 ). However, whether the global weak solution is regular and unique or the unique local smooth solution can exist globally is an outstanding challenge problem, just as the situation for the Navier–Stokes equations and the MHD equations. So, a lot of literatures are devoted to find regularity criteria for the local smooth solutions or to find the uniqueness criteria for the weak solutions for these equations, we refer the reader to see [2,6,7,10,11,15,17,19–21,23] for Navier–Stokes equations, and [4,8,13,32,34] for MHD equations. For the Navier–Stokes equations, it is well-known the Leray-Hopf weak solutions are unique and regular in the class L q (0, T ; L p )

n 2 + ≤ 1, n ≤ p ≤ ∞, see [10, 12, 15, 21, 23] q p 2 n n with + ≤ 2, < p ≤ ∞, see [1, 2]. q p 2

with

L q (0, T ; W 1, p )

Recently, by means of the Fourier localization technique and Bony’s paraproduct decomposition, Chen et al. [7], Chemin and Lerner [6], Lemarié [17] and May [20] extended the condition to L q (0, T ; B rp,∞ )

with

2 n n + = 1 + r, < p ≤ ∞, r ∈ (−1, 1] and ( p, r ) = (∞, 1); q p 1+r

−1 ) [7, 20] or C([0, T ]; B∞,∞

see [6, 17].

The above regularity results have been proved to still hold for the MHD equations (see e.g., [8,13,25,32]). For the 3D GMHD equations, Wu [27–30] obtained some regularity criteria only relying on the velocity u. Recent result obtained by Zhou [33] (see also [18]) states that if the weak solution (u, b) satisfies u ∈ L q (0, T ; L p )

123

with

3 3 2β + ≤ 2β − 1, < p ≤ ∞, q p 2β − 1

Uniqueness of weak solution to the generalized magneto-hydrodynamic system

701

then (u, b) is regular on (0, T ]. Yuan in [31] extended this result to the case u ∈ L q (0, T ; B sp,∞ )

with

2β 3 + ≤ 2β − 1 + s, q p

3 < p ≤ ∞. − 1 < s ≤ 1 and ( p, s) = (∞, 1). 2β − 1 + s The purpose of this paper is to uniqueness conditions of weak solutions for the GMHD system (1.1)–(1.4) in some Besov spaces. The tools we will use are mainly the LittlewoodPaley theory, the Bonys paraproduct decomposition and the Chemin-Lerner spaces. First we recall the definition of weak solutions to the GMHD system (1.1)–(1.4). Definition 1.1 (weak solution) A measurable vector function (u, b) is called a weak solution to the GMHD system on the interval [0, T ) with initial value (u 0 , b0 ) ∈ L 2 (Rn ), if it satisfies the following properties (i) (u, b) ∈ L ∞ (0, T ; L 2 (Rn )) ∩ L 2 (0, T ; H β (Rn )). (ii) div u = div b = 0 in the sense of distributions, that is, T

T u · ∇φdxdt =

0 Rn

b · ∇φdxdt = 0, 0 Rn

for all φ ∈ C0∞ (Rn × (0, T )). (u, b) verifies system (1.1)–(1.4) in the sense of distribution, that is,

T (∂t φ + (u · ∇)φ)udxdt + 0 Rn

T u 0 φ(x, 0)dx =

Rn

T

0 Rn

T

(∂t φ + (u · ∇)φ)bdxdt +

0 Rn

(u2β φ + (b · ∇)φb)dxdt,

b0 φ(x, 0)dx = Rn

(b2β φ + (b · ∇)φu)dxdt, 0 Rn 1

for all φ ∈ C0∞ (Rn × (0, T )) with div φ = 0, where = (−) 2 . (iii) (u, b) satisfies the energy inequality, that is, t u(t)2L 2

+ b(t)2L 2

+2

(β u(τ )2L 2 + β b(τ )2L 2 )dτ ≤ u 0 2L 2 + b0 2L 2 .

0

The main results of this paper are as follows: Theorem 1.2 Let β ∈ ( 21 , 1], T > 0 and (u 0 , b0 ) ∈ L 2 (Rn ) with div u 0 = div b0 = 0. Let (u 1 , b1 ) and (u 2 , b2 ) be two weak solutions of the GMHD system (1.1)–(1.3) with the same initial data (u 0 , b0 ) satisfying one of the following two conditions: 2β

2β

r1 r2 (a) (u 1 , b1 ) ∈ L 2β−1+r1 (0, T ; B∞,∞ ) and (u 2 , b2 ) ∈ L 2β−1+r2 (0, T ; B∞,∞ ) for some 1 − 2β < r1 , r2 < 1 such that r1 + r2 > 1 − 2β. 1 (b) (u 1 , b1 ), (u 2 , b2 ) ∈ L 1 (0, T ; B∞,∞ ).

Then (u 1 , b1 ) ≡ (u 2 , b2 ) a.e. on Rn × (0, T ).

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Theorem 1.3 Let β ∈ [ 78 , 1], 3 ≤ n ≤ 4(2β − 1), T > 0 and (u 0 , b0 ) ∈ L 2 (Rn ) with div u 0 = div b0 = 0. Let (u 1 , b1 ) and (u 2 , b2 ) be two weak solutions of the GMHD system (1.1)–(1.3) with the same initial data (u 0 , b0 ) satisfying 2β

2β

−r1 −r2 ) and (u 2 , b2 ) ∈ L 2β−1−r2 (0, T ; B∞,∞ ) (u 1 , b1 ) ∈ L 2β−1−r1 (0, T ; B∞,∞

for some 0 < r1 , r2 ≤ 2β − 1. Then, (u 1 , b1 ) ≡ (u 2 , b2 ) a.e. on Rn × (0, T ). r− n

p , Theorem 1.2 is still valid when the Remarks 1. Due to the embedding B rp,∞ → B∞,∞ assumption (a) is substituted by the following assumption:

(u 1 , b1 ) ∈ L q1 (0, T ; B rp11 ,∞ ) and (u 2 , b2 ) ∈ L q2 (0, T ; B rp22 ,∞ ), n where 2β qi + pi = 2β − 1 + ri , 1 − 2β < ri < 1, r1 + r2 > 1 − 2β. 2. Due to the inequality (see [7] for its proofs)

n 2β−1+ri

< pi ≤ ∞, (i = 1, 2) and

≤ C( f L 2 + curl f B∞,∞ ), f B∞,∞ 1 0 the assertion related to the assumption (b) of Theorem 1.2 implies that the following BealeKato-Majda-type uniqueness criterion (see [1]) holds: if two weak solutions (u 1 , b1 ) and (u 2 , b2 ) of the GMHD system with the same initial data satisfy 0 ). (curl u 1 , curl b1 ), (curl u 2 , curl b2 ) ∈ L 1 (0, T ; B∞,∞

Then (u 1 , b1 ) ≡ (u 2 , b2 ) a.e. on Rn × (0, T ). 3. For the Navier–Stokes equations, Chemin [5] and Lemarié [17] obtained the uniqueness −1 ). It is a nature question whether the condition r + r > of weak solutions in C(0, T ; B∞,∞ 1 2 1 − 2β in the assumption (a) can be removed. Theorem 1.3 shows that the answer to this question is affirmative in the case β ∈ [ 78 , 1] and n ≤ 4(2β − 1). However, it seems very difficult for the rest cases. The rest part of this paper is organized as follows. In Sect. 2, we collect some preliminaries materials, including the Littlewood–Paley decomposition, the definition of Besov spaces and some useful lemmas. In Sect. 3, we give the proof of Theorem 1.2. In the last section, we give the proof of Theorem 1.3. Throughout this paper, we denote by C an universal positive constant whose value may change from line to line, and the notation A B means that A ≤ C B. If X is a Banach space, T is a positive real number and p ∈ [1, +∞], we denote p p by L T (X ) or L T X the space L p (0, T ; X ). The norm of the space X is denoted by · X . 2 Preliminaries In this section, we are going to recall some basic facts on the Littlewood–Paley theory, the definition of Besov space and some useful lemmas. Part of the materials presented here can be found in [6,9,16,24,26,30]. Let S (Rn ) be the space of Schwartz class of rapidly decreasing functions. Given f ∈ S (Rn ), the Fourier transform of f is defined by 1 F f = f := e−i x·ξ f (x)dx. n (2π) 2 n R

123


703

We choose two nonnegative functions χ, ϕ ∈ S (Rn ), respectively, support in B = {ξ ∈ Rn , |ξ | ≤ 43 } and C = {ξ ∈ Rn , 43 ≤ |ξ | ≤ 83 } such that χ(ξ ) + ϕ(2− j ξ ) = 1 for all ξ ∈ Rn ;

j≥0

ϕ(2− j ξ ) = 1

for all ξ ∈ Rn \{0}.

j∈Z

Setting ϕ j = ϕ(2− j ξ ), then suppϕ j ∩ suppϕ j = φ if | j − j | ≥ 2 and suppχ ∩ suppϕ j = φ if j ≥ 1. Let h = F −1 ϕ and h = F −1 χ. Define the frequency localization operators j f = 0 for j ≤ −2; −1 f = S0 f = χ(D) f ; j f = ϕ(2− j D) f = 2n j h(2 j y) f (x − y)dy, for j ≥ 0; Rn

S j f = χ(2

−j

D) f =

k f = 2

−1≤k≤ j−1

nj

h(2 j y) f (x − y)dy.

Rn

Informally, j = S j+1 − S j is a frequency projection to the annulus {|ξ | ≈ 2 j }, while S j is the frequency projection to the ball {|ξ | 2 j }. One easily verifies that with the above choice of ϕ, j k f ≡ 0 if | j − k| ≥ 2 and j (Sk−1 f k f ) ≡ 0 if | j − k| ≥ 5. We recall the Bony’s paraproduct decomposition. Let u and v be two temperate distributions, the paraproducts between u and v are defined by Tu v := S j−1 u j v and Tv u := S j−1 v j u. j

j

Define the remainder of the paraproduct R(u, v) as R(u, v) := j u j v. | j− j |≤1

Then, we have the following Bony’s decomposition: uv = Tu v + Tv u + R(u, v).

(2.1)

We shall sometimes also use the following simplified decomposition uv = Tu v + Tv u with Tv u = Tv u + R(u, v) = S j+2 v j u. j

Now we introduce the definition of inhomogeneous Besov spaces by means of the Littlewood–Paley projection j and S j . Definition 2.1 Let r ∈ R, 1 ≤ p, q ≤ ∞, the inhomogeneous Besov space B rp,q := B rp,q (Rn ) is defined by B rp,q (Rn ) := { f ∈ S (Rn ); f B rp,q < ∞},

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where f B rp,q =

⎧ ⎪ ⎪ ∞ ⎨

1 q

2 jrq

q j f L p

for q < ∞,

j=−1 ⎪ ⎪ ⎩ sup js j≥−1 2 j f L p

for q = ∞.

We introduce the well–known Bernstein’s Lemma; its proofs can be found in Chemin [5] or Danchin [9]. Lemma 2.2 (Bernstein’s Lemma) Let 1 ≤ p ≤ q ≤ ∞. Assume that f ∈ L p (Rn ), then there exist constants C, C1 independent of f, j such that sup ∂ α f L q ≤ C2

jk+n j ( 1p − q1 )

|α|=k

for supp f ⊂ {|ξ | 2 j },

f L p

for supp f ⊂ {|ξ | ≈ 2 j }.

C1−1 2 jk f L p ≤ sup ∂ α f L p ≤ C1 2 jk f L p |α|=k

We also recall the definition of a class of spaces introduced by Chemin and Lerner [6]. L T B rp,∞ is the space of Definition 2.3 Let T > 0, r ∈ R and p, q ∈ [1, +∞]. The space distributions u ∈ S (Rn × R) such that q

q u L Br T

p,∞

:= sup 2r j j u L q L xp < ∞. T

j≥−1

β

The following proposition gives some other properties of the semigroup (e−t (−) )t>0 . Proposition 2.4 Let T > 0, r, r1 , r2 ∈ R and p, q ∈ [1, +∞], we have following assertions: n

( 1p − q1 ) −t (−)β r2 −r1 −t (−)β 2β

(1) If p ≤ q, then the family (t 2β (2) If r1 < r2 , then the family (t

e

)t>0 is continuous from L q to L p ;

1 2 )0 0 and 1 ≤ p ≤ ∞. This property still holds for the semigroup β

(e−t (−) )t≥0 ; we have the following proposition in a particular case of this characterization. Proposition 2.5 Let 1 ≤ p ≤ ∞ and r > 0. Then, f ∈ B −r p,∞ if and only if β

r

sup τ 2β e−τ (−) f L p < ∞

0 0, we set Aλj (t) = 2− js e−λφ j (t) ( j u, j b) L 2 , where φ j (t) is defined by

φ j (t) =

t 0

2 j (( j u 1 , j b1 )(τ ) L ∞ + ( j u 2 , j b2 )(τ ) L ∞ )dτ.

j ≤ j+4

We get by (3.3) and (3.4) that d λ A (t) + λφ j (t)Aλj (t) + a j 22 jβ Aλj (t) dt j 2− js e−λφ j (t) ( j (u · ∇u 1 ) L 2 + [u 2 , j ]∇ − ∂i j u j u i2 L 2 j > j

+ j (b · ∇b1 ) L 2 + j (b2 · ∇b) L 2 + j (u · ∇b1 ) L 2 + [u 2 , j ]∇u − ∂i j b j u i2 L 2 + j (b · ∇u 1 ) L 2 + j (b2 · ∇u) L 2 )(τ )dτ j > j

≡ J1 + J2 + · · · + J8 ,

(3.32)

where we have used the facts ∂i j u j u i2 , j u = − j ∂i u i2 j u, j u = 0 and ∂i j b j u i2 , j b = − j ∂i u i2 j b, j b = 0. Using the equality of Bony decomposition, we have j (u · ∇u 1 ) L 2 ≤ j (Tu i ∂i u 1 ) L 2 + j (T∂i u 1 u i ) L 2 + j R(u i , ∂i u 1 ) L 2 . (3.33) By the equality j (Tu i ∂i u 1 ) = we get j (Tu i ∂i u 1 )(τ ) L 2

| j − j|≤4 j (S j −1 u

| j − j|≤4

| j − j|≤4

2j

2j

j

≤ j −2

and Bernstein’s Lemma,

j

u L 2 j u 1 L ∞

j

≤ j −2

j ≤ j+2

i∂ u ) i j 1

2 j s eλφ j

(τ ) Aλj

j u 1 L ∞

2 j s eλφ j (τ ) Aλj (τ )φ j (τ ).

(3.34)

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By the equalities (3.7) and (3.9), we can estimate the last two terms of (3.33) as

j (T∂i u 1 u i )(τ ) L 2

2 j j

u 1 L ∞ j u L 2

| j − j|≤4 j

≤ j −2

2 j s eλφ j (τ ) Aλj (τ )

| j − j|≤4

j R(u i , ∂i u 1 )(τ ) L 2

2 j j

u 1 L ∞

j

≤ j −2

2

j ≤ j+2

j s

eλφ j (τ ) Aλj (τ )φ j (τ );

(3.35)

2 j j u L 2 j

u 1 L ∞

j , j

≥ j−3;| j − j

|≤1

2 j 2 j s eλφ j (τ ) Aλj (τ ) j

u 1 L ∞

j , j

≥ j−3;| j − j

|≤1

2j

2 j (s−1) eλφ j (τ ) Aλj (τ )φ j (τ ).

(3.36)

j ≥ j−3

Combining (3.33)–(3.36) together, we can estimate J1 as t

t J1 (τ )dτ =

0

2− js e−λφ j (τ ) j (u · ∇u 1 )(τ ) L 2 dτ

0

2

( j − j)s

j ≤ j

+

t

eλ(φ j (τ )−φ j (τ )) Aλj (τ )φ j (τ )dτ

0

2

−( j − j)(1−s)

j ≥ j

t

eλ(φ j (τ )−φ j (τ )) Aλj (τ )φ j (τ )dτ.

(3.37)

0

Similarly, t (J3 + J4 + J5 + J7 + J8 )(τ )dτ

j ≤ j

0

+

j ≥ j

2−( j − j)(1−s)

t

2

( j − j)s

t


0


(3.38)

0

To estimate the term J2 , we first notice that there holds ∂i j u j u i2 L 2 ≤ [Tu i , j ]∂i u L 2 +T j ∂i u u i2 − ∂i j u j u i2 L 2 [u 2 , j ]∇ − j > j

2

+ j (T∂i u u i2 ) L 2

123

j > j

+ j R(u i2 , ∂i u) L 2 .

(3.39)


715

Similar to derivation of estimate (3.18), we have ∇ S j −1 u 2 L ∞ j u L 2 [Tu i , j ]∂i u(τ ) L 2 2

| j − j|≤4

2( j −1) u 2 L ∞ 2 j s eλφ j (τ ) Aλj (τ )

| j − j|≤4

2 j s eλφ j (τ ) Aλj (τ )φ j (τ ).

(3.40)

| j − j|≤4

Notice that T j ∂i u u i2 −

∂i j u j u i2 =

j > j

S j +2 j ∂i u j u 2 ,

j−2≤ j ≤ j

it gives by Bernstein’s Lemma 2.2 that (T j ∂i u u i2 − ∂i j u j u i2 )(τ ) L 2 j > j

S j +2 j ∂i u L 2 j u 2 L ∞

j−2≤ j ≤ j

2 j j u L 2 j u 2 L ∞

j−2≤ j ≤ j

2 js eλφ j (τ ) Aλj (τ )φ j (τ ).

(3.41)

The last two terms can be estimate similar to (3.8) and (3.10),

2 j s eλφ j (τ ) Aλj (τ )φ j (τ ); j (T∂i u u i2 )(τ ) L 2

(3.42)

j ≤ j+2

j R(u i2 , ∂i u)(τ ) L 2 2 j

2 j (s−1) eλφ j (τ ) Aλj (τ )φ j (τ ).

(3.43)

j ≥ j−3

We can estimate the term [u 2 j ]∇u −

j > j

∂i j b j u i2 L 2 in the same way. Hence, com-

bining (3.39)–(3.43), we get t (J2 + J6 )(τ )dτ

2

( j − j)s

j ≤ j

0

+

t


0

2

−( j − j)(1−s)

j ≥ j

t

eλ(φ j (τ )−φ j (τ )) Aλj (τ )φ j (τ )dτ. (3.44)

0

Inserting (3.37), (3.38) and (3.44) into (3.32), it follows that Aλj (t) + λ

t

φ j (τ )Aλj (τ )dτ

0

+

j ≥ j

+ aj2

Aλj (τ )dτ

0

t

2( j − j)s

j ≤ j

t 2 jβ


0

t

2

−( j − j)(1−s)


(3.45)

0

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Q. Liu et al.

Notice that φ j (τ ) = φ j (τ ) + (φ j (τ ) − φ j (τ )) and φ j (τ ) − φ j (τ ) ≥ 0 for j ≥ j imply that t 1 eλ(φ j (τ )−φ j (τ )) (φ j (τ ) − φ j (τ ))dτ for j ≥ j . λ 0

Hence

2( j − j)s

j ≤ j


0

2( j − j)s

j ≤ j

t j≥−1

t

Aλj (τ )φ j (τ )dτ +

1 ( j − j)s 2 sup Aλj (τ ) λ τ ∈[0,t] j ≤j

0

1 ( j − j)s 2 sup Aλj (τ ). λ τ ∈[0,t]

Aλj φ j (τ )dτ +

sup

( j

t

(3.46)

j ≤j

0

Notice that φ j (t) − φ j (t) is an increasing function with t for j ≥ j, and φ j (t) − φ j (t) ≤ − j)((u 1 , b1 ) L 1 (0,t;B∞,∞ 1 1 ) + (u 2 , b2 ) L 1 (0,t;B∞,∞ ) ), we have

2

−( j − j)(1−s)

j ≥ j

t


0

2

−( j − j)(1−s) λ(φ j (t))−φ j (t)

t

Aλj (τ )φ j (τ )dτ

e

j ≥ j

0

2−( j − j)(1−s) e

λ( j − j)((u 1 ,b1 ) L 1 (0,t;B 1

∞,∞ )

+(u 2 ,b2 ) L 1 (0,t;B 1

∞,∞ )

j ≥ j

)

t


0

2−( j − j)(1−s)

j ≥ j

t


0

t sup

j ≥−1

Aλj (τ )φ j (τ )dτ,

(3.47)

0

where we have used the assumption λ((u 1 , b1 ) L 1 (0,t;B∞,∞ 1 1 ) + (u 2 , b2 ) L 1 (0,t;B∞,∞ ) ) ≤ (1 − s)ln2. Summing up (3.45)–(3.47) together, it follows that sup

j≥−1;τ ∈[0,t]

t ≤ C sup j≥−1

123

0

Aλj (τ ) + λ

Aλj (τ )dτ +

t sup j≥−1

C λ


0

sup

j≥−1;τ ∈[0,t]

Aλj (τ ).

t + sup a j 2 j≥−1

2β j 0

Aλj (τ )dτ

(3.48)


717

Taking λ big enough and using the Gronwall inequality yield that sup

j≥−1;τ ∈[0,t]

Aλj (τ ) = 0.

Hence, (u 1 , b1 ) = (u 2 , b2 ) a.e. Rn × [0, t]. On the other hand, under the assumption (b), we can choose the t > 0 small enough such that (3.48) holds, then by using the standard continuity argument, we obtain (u 1 , b1 ) = (u 2 , b2 ) on Rn ×[0, T ], and the proof of Theorem 1.2 is complete.

4 Proof of Theorem 1.3 In this section, we give the proof of Theorem 1.3. We first introduce a lemma of inhomogeneous Sobolev inequality identified due to P. Gerard, Y. Meyer and F. Oru [9] (see also [13], for another demonstration of these inequalities): Lemma 4.1 Let 0 < α < γ , 1 < p < ∞ and

p q

= (1 − γα ). Then, we have

1− α

α

f L q f H α γ f γ α−γ

B∞,∞

p

α−γ

for all f ∈ H pα ∩ B∞,∞ . Here H pα := { f : f ∈ S (Rn ), f H pα = F −1 {(1 + α 2

|ξ |2 ) F f } L p < ∞}. Notice that whether i = 1 or 2, the classical interpolation inequality implies that the 2β r

i 2 β 2 ri weak solution (u i , bi ) ∈ L ∞ T L ∩ L T H still belongs to L T H . Thus, under the assump2β r

2β

−ri ), tion of Theorem 1.3, we get that (u i , bi ) belongs to L Ti H r1 ∩ L 2β−1−ri (0, T ; B∞,∞ then the above Lemma 4.1 and the classical interpolation inequality imply that (u i , bi ) ∈ 4β

L 2β−1 (0, T ; L 4 (Rn )). Hence, when 3 ≤ n ≤ 4(2β − 1) (here, we need β ≥ 78 ), it follows that Theorem 1.3 will be a straightforward corollary of the following theorem: Theorem 4.2 Let β ∈ [ 78 , 1], 3 ≤ n ≤ 4(2β − 1), T > 0 and (u 0 , b0 ) ∈ L 2 (Rn ) with div u 0 = div b0 = 0. Let (u 1 , b1 ) and (u 2 , b2 ) be two weak solutions of the GMHD system (1.1)–(1.3) with the same initial value (u 0 , b0 ) satisfying 2β

−r1 ) (u 1 , b1 ) ∈ Łq (0, T ; L p ) ∩ L 2β−1−r1 (0, T ; B∞,∞

and 2β

−r2 ) (u 2 , b2 ) ∈ Łq (0, T ; L p ) ∩ L 2β−1−r2 (0, T ; B∞,∞ 2β n with 0 < r1 , r2 ≤ 2β − 1, and for some p ∈ [ 2β−1 , ∞) and q ∈ ( 2β−1 , ∞]. Then, n (u 1 , b1 ) ≡ (u 2 , b2 ) a.e. on R × (0, T ).

In order to prove Theorem 4.2, we need the following two lemmas. n Lemma 4.3 Suppose (u 0 , b0 ) ∈ L s (Rn ), s ≥ 2β−1 . Then, there exit T0 > 0 and a unique solution (u, b) ∈ BC([0, T0 ); L s ) of the GMHD system (1.1)–(1.4) such that

sup t 0 0 independent of T∗ and r . Here, BC denotes the class of bounded and Bσ σ := S (Rn ) ∞,∞ . continuous functions, and B∞,∞ Proof The proof of this lemma is similar to that of Giga [12] (see also [14,16,19,20]).

Lemma 4.4 Let β ∈ [ 78 , 1], 3 ≤ n ≤ 4(2β − 1), T > 0 and (u 0 , b0 ) ∈ L 2 (Rn ) with div u 0 = div b0 = 0. Let (u, b) be a weak solution of the GMHD (1.1)–(1.4) satisfying 2β

−r ) (u, b) ∈ Łq (0, T ; L p ) ∩ L 2β−1−r (0, T ; B∞,∞

with 0 < r ≤ 2β − 1, p ≥ sup t

2β−1 2β

n 2β−1

and q >

2β 2β−1 .

Then, we have

(u, b)(·, t) L ∞ < ∞ and lim t t→0

0