REGULARITY CRITERIA FOR WEAK SOLUTIONS TO 3D INCOMPRESSIBLE MHD EQUATIONS WITH HALL TERM

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1 Electronic Journal of Differential Equations, Vol (2018), No. 10, pp ISSN: UL: or EGULAITY CITEIA FO WEAK SOLUTIONS TO D INCOMPESSIBLE MHD EQUATIONS WITH HALL TEM JAE-MYOUNG KIM Communicated by Jesus Ildefonso Diaz Abstract. We study the regularity conditions for a weak solution to the incompressible D magnetohydrodynamic equations with Hall term in the whole space. In particular, we show the regularity criteria in view of gradient vectors in various spaces. 1. Introduction We consider the incompressible D magneto hydro dynamic (MHD) equations with Hall term t u u + u u + π = b b, (1.1) t b b + u b b u + (( b) b) = 0, (1.2) div u = div b = 0, (1.) Here u : Q T := [0, T ) is the flow velocity vector, b : Q T is the magnetic vector, π = p + b 2 2 : Q T is the total pressure. We consider the initial value problem of (1.1) (1.), which requires initial conditions u(x, 0) = u 0 (x) and b(x, 0) = b 0 (x) x (1.4) The initial conditions satisfy the compatibility condition, i.e. div u 0 (x) = 0, and div b 0 (x) = 0. Definition 1.1. A weak solution pair (u, b) of the incompressible D MHD equations with the Hall term (1.1) (1.4) is regular in Q T provided that u L (Q T ) + b L (Q T ) <. For a long time, the effects of Hall current on fluids has been a subject of great interest to researchers. A current induced in a direction normal to the electric and magnetic fields is commonly called Hall current [22]. In particular, the effects of Hall current are very important if the strong magnetic field is applied The mathematical derivations of the incompressible D MHD equations with the Hall term could be given in [1] from either two-fluids or kinetic models. It is wellknown that the global existence of weak solutions, local existence and uniqueness 2010 Mathematics Subject Classification. 5B65, 5Q5, 76W05. Key words and phrases. Magnetohydrodynamics equation; weak solution; regularity condition. c 2018 Texas State University. Submitted July 0, Published May 7,

2 2 J.-M. KIM EJDE-2018/10 of smooth solutions to the system (1.1) (1.4) were established in [5, 6]. ecently, various results for this equation were proved in view of partial regularity, temporary decay and regularity or blow-up conditions (see [5, 6, 7, 8, 9, 12, 11, 21, 26, 27] and references therein.) We list only some results relevant to our concerns. In view of the regularity conditions in Lorentz space, He and Wang [1] proved that a weak solution pair (u, b) becomes regular in the presence of a certain type of the integral conditions, typically referred to as Serrin s condition, namely, u L q, (0, T ; L p, ( )) with /p + 2/q 1, < p, or u L q, (0, T ; L p, ( )) with /p + 2/q 2, 2 < p, (also see [, 4, 17]). Also, Wang proved in [25] that a weak solution pair (u, b) become regular if u satisfies u L 2 (0, T ; BMO( ). On the other hand, recently, Zhang [27] obtained the regularity criterion u L 2 1 α (0, T ; Ḃ α, ), b L 2 1 β (0, T ; Ḃ β, ) (1.5) with 1 < α < 1 and 0 < β < 1. Our study is motivated by these viewpoints, we obtain the regularity conditions for a weak solution to the incompressible D MHD equations with the Hall term (1.1) (1.4) in a whole space. Our proof of main results is based on a priori estimate for the gradient of the velocity field. Our main results reads as follows. Theorem 1.2. Suppose that (u, b) is a weak solution of (1.1) (1.4) with initial condition u 0, b 0 H 2 ( ). If (u, b) satisfies one of the following cases: T 0 ( u ql p, + b ml l, ) dt < (1.6) with the relations p + 2 q = 2, 2 < p and l + 2 m = 1, < l. or T 0 2 ) ql 1 β ( u p, + b dt < (1.7) Ḃ, β with the relations p + 2 q = 2, 2 < p and 0 < β < 1, then (u, b) is regular in Q T. Theorem 1.. Suppose that (u, b) is a weak solution of (1.1) (1.4) with initial condition u 0, b 0 H 2 ( ). If (u, b) satisfies one of the following two conditions: or T T then (u, b) is regular in Q T. 0 0 u BMO( ) + b 2 BMO( ) dt <, (1.8) u 2 BMO 1 ( ) + 2 b 2 BMO 1 ( ) dt <. (1.9)

3 EJDE-2018/10 D INCOMPESSIBLE MHD EQUATIONS WITH HALL TEM Theorem 1.2 extends the result by He and Wang [1] with respect to the gradient of the velocity field. Moreover, using the estimate in [24, Lemma A.5], we obtain BMO 1 ( )-regularity condition. This article is organized as follows: In Section 2 we recall the notion of weak solutions and review some known results. In Section, we present the proofs of the Theorem 1.2 and Preliminaries In this section we introduce the notation and definitions to be used in this paper. We also recall the well-known results for our analysis. For 1 q, W k,q ( ) indicates the usual Sobolev space with standard norm k,q, i.e. W k,q ( ) = {u L q ( ) : D α u L q ( ), 0 α k}. When q = 2, we denote W k,q ( ) by H k. All generic constants will be denoted by C, which may vary from line to line BMO and Lorentz spaces. The John-Nirenberg space or the Bounded Mean Oscillation space (in short BMO space) [14] consists of all functions f which are integrable on every ball B (x) and satisfy: f 2 1 BMO = sup sup x >0 B(x, ) B(x,) f(y) f B (y) dy <. Here, f B is the average of f over all ball B (x) in. It will be convenient to define BMO in terms of its dual space, H 1. On the other hand, following [16] let w be the solution to the heat equation w t w = 0 with initial data v. Then v BMO = sup sup w 2 dt dy. x >0 B(x, ) B(x,) and define the BMO 1 -norm by v 2 BMO = sup 1 2 sup w 2 dt dy. 1 x >0 B(x, ) B(x,) 0 We note that if u is a tempered distribution. Then u BMO 1 if and only if there exist f i BMO with u = i f i in [16, Theorem 1]. Let m(ϕ, t) be the Lebesgue measure of the set {x : ϕ(x) > t}, i.e. m(ϕ, t) := m{x : ϕ(x) > t}. We denote by the Lorentz space L p,q ( ) with 1 p, q with the norm [2] ( ) 1/q t ϕ L p,q ( ) = q q/p dt (m(ϕ, t)) 0 t <, for 1 q, (2.1) sup t 0 {t(m(ϕ, t)) 1/p }, for q =. Followed in [2], Lorentz space L p,q ( ) may be defined by real interpolation methods L p,q ( ) = (L p1 ( ), L p2 ( )) α,q, (2.2) with 1 p = 1 α p 1 we note that + α p 2, 1 p 1 < p < p 2. From the interpolation method above, L 2p p 1,2 ( ) = ( L 2 ( ), L 6 ( ) ) 2p,2. (2.) 0

4 4 J.-M. KIM EJDE-2018/10 We also need the Hölder inequality in Lorentz spaces (see [20]) for our proof. Lemma 2.1. Assume 1 p 1, p 2, 1 q 1, q 2 and u L p1,q1 ( ), v L p2,q2 ( ). Then uv L p,q ( ) with 1 p = 1 p p 2 and 1 q 1 q q 2, and uv L p,q ( ) C u L p 1,q 1 ( ) v L p 2,q 2 ( ). (2.4) 2.2. Besov space. Following [2], let B = {ξ d, ξ 4 } and C = {ξ d : /4 ξ 8/}. Choose two nonnegative smooth radial function χ, ϕ supported, respectively, in B and C such that χ(ξ) + j 0 ϕ(2 j ξ) = 1, ξ d, ϕ(2 j ξ) = 1, ξ d \ {0}. j Z We denote ϕ j = ϕ(2 j ξ), h = F 1 ϕ and h = F 1 χ, where F 1 stands for the inverse Fourier transform. Then the dyadic blocks j and S j can be defined as follows S j f = j f = ϕ(2 j D)f = 2 jd d h(2 j y)f(x y)dy, k j 1 k f = χ(2 j D)f = 2 jd d h(2 j y)f(x y)dy. Formally, j = S j S j 1 is a frequency projection to annulus {C 1 2 j ξ C 2 2 j }, and S j is a frequency projection to the ball { ξ C2 j }. One can easily verify that with our choice of ϕ, j k f = 0 if j k 2 and j (S k 1 f k f) = 0 if j k 5. With the introduction of j and S j, let us recall the definition of the Besov space. Let s, p, q [1, ], the homogeneous space is defined as where f Ḃs p,q = Ḃp,q s = {f S : f Ḃs < }, p,q { ( j Z 2sjq j f q L p ) 1/q, for 1 q <, sup j Z 2 sj j f L p, for q =, In particular, when p = q = 2, the Besov space and Sobolev space are equivalence; that is Ḣ s Ḃs 2,2, H s B s 2,2. Now we recall first the definition of weak solutions. Definition 2.2. Let u 0, b 0 L 2 ( ) with the divergence free conditions. We say that (u, b) is a weak solution of Hall-MHD equations (1.1) (1.) with initial condition u 0, b 0 L 2 ( ), if u and b satisfy the following: T 0 (i) u L ([0, T ); L 2 ( )) L 2 ([0, T ); H 1 ( )), and b L ([0, T ); L 2 ( )) L 2 ([0, T ); H 1 ( )). (ii) (u, b) satisfies (1.1) (1.2) in the sense of distribution; that is ( φ ) T t + φ + (u )φ u dx dt + u 0 φ(x, 0) dx = (b )φ b dx dt 0

5 EJDE-2018/10 D INCOMPESSIBLE MHD EQUATIONS WITH HALL TEM 5 T 0 T = ( φ ) t + φ + (u )φ b dx dt + b 0 φ(x, 0) dx T (b )φ u dx dt + ( b) b ( φ) dx dt, 0 0 for all φ C0 ( [0, T )) with div φ = 0, and u ψdx = 0, b ψdx = 0, for every ψ C 0 ( ).. Proof of main results Proof of Theorem 1.2. (L 2 -estimate or energy estimate): By the standard energy estimate, we obtain ( u 2 + b 2 ) dx + ( u 2 + b 2 ) dx = 0. (.1) 2 dt (H 1 -estimate): Testing u and b to the fluid equation and by the magnetic equation of (1.1) and (1.2), respectively, using the integrating by parts, integrating on domain, we have 2 dt ( u(τ) 2 L 2 ( ) + b(τ) 2 L 2 ( ) ) + ( u 2 + b 2 )dx [(u )u] : udx + [(b )b] : u dx (.2) [(u )b] b dx + [(b )u] : bdx + (( b) b) b dx := I 1 + I 2 + I + I 4 + I 5. We estimate separately the terms in the right hand side of (.2). The first term I 1 is computed as follows: I 1 u L, (.) where the divergence free condition of u is used. On the other hand, we observe that I 2 + I 4 u b 2. since (b ) b u dx + (b ) u b dx ( b = u dx + u ) b dx x j x j j=1 = b j j=1 b j ( ( b u) ) dx = 0, x j where we use the product rule and div b = 0.

6 6 J.-M. KIM EJDE-2018/10 Note that f 2 L 4 f Ḣ1+β C f Ḃ β, f Ḣ 1+β with 0 < β < 1, (.4) C f 1 β L 2 f β L 2 with 0 < β < 1, (.5) (e.g. see [18, 19] and [2, Theorem 2.42]). First of all, using the interpolation (2.2), Lemma 2.1, Hölder and Young s inequalities, we estimate I as follows: I b 2 u dx u L p, b 2 p L p 1,1 = u L p, b 2 L 2p p 1,1 where θ = 1 2p. Similarly, from (.), we have C u L p, b 2θ L 2 2 b 2(1 θ) L 2 2p 2p C u L p, b 2 L b 2 L 2, I 1 u L C u 2p 2p L p, u 2 L u 2 L 2. (.6) Case 1: Again, using the interpolation (2.2), Lemma 2.1, Hölder and Young s inequalities, we bound I 5 as follows. I 5 C b L l, b L 2l l 2,2 b L 2,2 C b L l, b l l L b l+ 2 l L 2 C b 2l l L l, b 2 L b 2 L 2. Summing the terms I 1 I 5, inequality (.2) becomes 2 dt ( u 2 L + 2 b 2 L 2) + 1 ( 2 u b 2 ) dx 2 ( (.7) 2p 2l 2p l C u p, + b ( u 2 L + 2 b 2 L 2). L L l, ) Case 2: Using (.4) and (.5), we bound I 5 as follows: I 5 = i ( b i b) i bdx C b 2 L 4 b L 2 b C b Ḃ β, Ḣ b 1+β L 2 C b Ḃ β b 1 β, L b 1+β 2 L b 2 L + C b 2 1 β b 2 2 Ḃ, β L 2. Summing the terms I 1 I 5, the (.2) becomes 2 dt ( u 2 L + 2 b 2 L 2) + 1 ( 2 u b 2 ) dx 2 ( (.8) 2p 2 2p 1 β C u p, + b )( u 2 L + 2 b 2 L 2). L Ḃ β,

7 EJDE-2018/10 D INCOMPESSIBLE MHD EQUATIONS WITH HALL TEM 7 For Cases 1 and 2 with the given conditions, we apply the Grownwall s inequality to estimates (.7) and (.8), respectively, to find T sup ( u(τ) 2 L + 2 b(τ) 2 L 2) + ( 2 u b 2 ) dx dt 0<τ T 0 C( u(0) 2 L 2 + b(0) 2 L 2). (H 2 -estimate) Applying the operator to (1.1) (1.2), then multiplying it by u and b, respectively, and integrating on domain, we obtain 2 dt ( u 2 + b 2 )dx + ( u 2 L + 2 b 2 L 2) = (u u) u dx + (b b) u dx (u b) b dx + (b u) b dx (( b) b) b dx := J 1 + J 2 + J + J 4 + J 5 By the commutator estimate in [10, Theorem 2.1 or Corollary 2.1] or [15], we note that [(u )u], u dx C u H 2 u 2 H 2, [(u )b], b dx C u H 2 b 2 H 2, [(b )u], b dx C u H 2 b 2 H 2. Also, integrating by parts we obtain the estimate for the remaining convection term follows as: [(b )b], u dx C [b b], u dx C u H 2 b 2 H 2. Thus J 1 + J 2 + J + J 4 C u H 2( u 2 H 2 + b 2 H 2) C( u 4 H 2 + b 4 H 2) u 2 H 2 C( u 2 H 2 + b 2 H 2)( u 2 H 2 + b 2 H 2) u 2 H 2 (.9) Case 1: For the term J 5, using the chain rule, we note that J 5 = ( b b + 2 i ( b) i b) b dx (.10) And thus, we have J 5 b Ll, b L 2l l 2,2 b L 2 C b 2l l L l, b 2 L b 2 L 2

8 8 J.-M. KIM EJDE-2018/10 Summing the estimate of terms J 1 J 5 with the energy estimate and H 1 -estimates, we obtain 2 dt ( u 2 H + 2 b 2 H 2) ( u 2 H + b 2 H ) ( 2q 2l ) (.11) 2q l C u Lq, + b L l, + u 2 H + 2 b 2 H ( u 2 2 H + 2 b 2 H 2) Case 2: Using (.4) and (.5), we bound J 5 as follows: J 5 C b Ḃ β, b Ḣ 1+β b L 2 C b Ḃ β 2 b 1 β, L b 1+β 2 L 2 C b 2 1 β b 2 Ḃ, β H b 2 L 2. As in case 1, summing the estimate of terms J 1 J 5 with the energy estimate and H 1 -estimates, we obtain 2 dt ( u 2 H + 2 b 2 H 2) ( u 2 H + b 2 H ) ( 2p 2 2p 1 β C u p, + b L Ḃ β, + u 2 H 2 + b 2 H 2 ) ( u 2 H 2 + b 2 H 2) (.12) Under the assumption, we apply Grown s inequality to the estimates (.11) and (.12), respectively, we finally obtain sup ( u(τ) 2 H + 2 b(τ) 2 H 2) + u 2 H + b 2 H ( u 0 2 H + b H 2) 0 τ T Th proof is complete. Proof of Theorem 1.. (H 1 -estimate): Testing u and b to the fluid equation and the magnetic equation of (1.1) and (1.2), respectively, using the integrating by parts, integrating on domain, we have 2 dt ( u(τ) 2 L 2 ( ) + b(τ) 2 L 2 ( ) ) + ( u 2 + b 2 )dx [(u )u] : udx + [(b )b] : udx (.1) [(u )b] : b dx + [(b )u] : bdx + (( b) b) : b dx := I 1 + I 2 + I + I 4 + I 5. Case 1: By the Hölder, Young inequalities and the space duality BMO-H 1, we have b 2 u dx u BMO b 2 H 1 u BMO b L 2 b L 2 Similarly, we obtain = u BMO b 2 L 2. u dx C u BMO u 2 L 2.

9 EJDE-2018/10 D INCOMPESSIBLE MHD EQUATIONS WITH HALL TEM 9 Again, by the vector identity, the Hölder and Young inequalities, we have I 5 = [( b) b] ( b)dx ( ) = ( b) b ( b) b ( b)dx (.14) C b BMO b 2 b H 1 C b 2 BMO b 2 L b 2 L 2. Summing the estimates above, the (.1) becomes d dt ( u 2 L + 2 b 2 L 2) + ( 2 u b 2 ) dx (.15) C( u BMO + b 2 BMO)( u 2 L + 2 b 2 L 2). Case 2: Following [24, Lemma A.5], we note that u 2 L = uu 4 L 2 C u L 2 u BMO 1. Using this estimate, we have b 2 u dx u L 2 b 2 L C u 4 L 2 2 b L 2 b BMO 1 C u 2 L 2 b 2 BMO b 2 L 2. Similarly, we obtain u dx C u 2 L 2 u 2 BMO u 2 L 2. By the vector identity, the Hölder and Young inequalities, we have I 5 = [( b) b] ( b)dx ( ) = ( b) b ( b) b ( b)dx (.16) C ( b) b L 2 2 b L 2 C 2 b 2 BMO 1 b 2 L b 2 L 2 Using the estimates above, (.1) becomes d dt ( u 2 L 2 + b 2 L 2) + ( 2 u b 2 ) dx C( u 2 BMO 1 + b 2 BMO 1)( u 2 L 2 + b 2 L 2). (.17)

10 10 J.-M. KIM EJDE-2018/10 (H 2 -estimate) Taking to (1.1) (1.2), then multiplying it by u and b, respectively, and integrating on domain, we derive 2 dt ( 2 u b 2 )dx + ( u 2 L + 2 b 2 L 2) = (u u) u dx + (b b) u dx (u b) b dx (.18) + (b u) b dx (( b) b) : b dx := J 1 + J 2 + J + J 4 + J 5 As in the proof of Theorem 1.2, namely (.9), we note that J 1 + J 2 + J + J 4 C( u 2 H 2 + b 2 H 2)( u 2 H 2 + b 2 H 2) u 2 H 2. Case 1. From (.10) with the space duality BMO-H 1, we have J 5 b BMO 2 b L 2 b L 2 C b 2 BMO 2 b 2 L b 2 L 2. Summing the estimates J 1 J 5 with the energy estimate and H 1 -estimates, the (.18) becomes d dt ( u 2 H + 2 b 2 H 2) + ( u 2 H + b 2 H ) (.19) C ( u BMO + b 2 BMO + u 2 H + 2 b 2 H )( u 2 2 H + 2 b 2 H 2). Case 2. From (.9), we note that J 1 + J 2 + J + J 4 C( u 2 H 2 + b 2 H 2)( u 2 H 2 + b 2 H 2) u 2 H 2. Following [24, Lemma A.5], we note that uu L 2 C u L 2 u BMO 1. As in the previous proof, namely, from (.10) with the space duality BMO-H 1, we have J 5 C 2 b BMO 1 2 b L b 2 L 2. Summing J 1 J 5 with the energy estimate and H 1 -estimate, the (.18) becomes d dt ( u 2 H 2 + b 2 H 2) + ( u 2 H + b 2 H ) C( u 2 BMO b 2 BMO 1 + u 2 H 2 + b 2 H 2)( u 2 H 2 + b 2 H 2). (.20) Under the assumption, we apply Gronwall s inequality to the estimates (.19) and (.20), respectively, we finally obtain sup ( u(τ) 2 H + 2 b(τ) 2 H 2) + u 2 H + b 2 H ( u 0 2 H + b H 2). 0 τ T The proof is complete. Acknowledgments. J.-M. Kim was supported by grants NF-20151A5A and NF-20161D1A1B

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12 12 J.-M. KIM EJDE-2018/10 Jae-Myoung Kim Department of Mathematical Sciences, Seoul National University, Seoul, Korea address: