On partial regularity for the steady Hall magnetohydrodynamics system arXiv:1505.01254v1 [math.AP] 6 May 2015
Dongho Chae∗ and J¨org Wolf
†
∗Department of Mathematics Chung-Ang University Seoul 156-756, Republic of Korea e-mail:
[email protected] and †Department of Financial Engineering Ajou University Suwon 443-749, Republic of Korea e-mail:
[email protected] Abstract We study partial regularity of suitable weak solutions of the steady Hall magnetohydrodynamics equations in a domain Ω ⊂ R3 . In particular we prove that the set of possible singularities of the suitable weak solution has Hausdorff dimension at most one. Moreover, in the case Ω = R3 , we show that the set of possible singularities is compact. AMS Subject Classification Number: 35Q35, 35Q85,76W05 keywords: steady Hall-MHD equations, parial regularity
1
Introduction
The resistive incompressible Hall magnetohydrodynamics(Hall-MHD) is described by he following equations: ∂u + u · ∇u + ∇p = (∇ × B) × B + ν∆u + f , ∂t ∂B − ∇ × (u × B) + ∇ × ((∇ × B) × B) = µ∆B + ∇ × g, ∂t ∇ · u = 0, ∇ · B = 0,
where 3D vector fields u = u(x, t), B = B(x, t) are the fluid velocity and the magnetic field respectively. The scalar field p = p(x, t) is the pressure, while the positive 1
constants ν and µ represent the viscosity and the magnetic resistivity respectively. The given vector fields f and ∇ × g are external forces on the magnetically charged fluid flows. Historically, the Hall-MHD system was first considered by Lighthill([15]). Compared with the usual MHD system, the Hall-MHD system contains the extra term ∇ × ((∇ × B) × B), called the Hall term. The inclusion of this term is essential in understanding the problem of magnetic reconnection, which corresponds to the change of the topology of magnetic field lines. This phenomena of magnetic reconnection is really observed, for example, in space plasma([10, 12]), star formation([22]) and neutron star([19]). For the other physical features related to the Hall-MHD we refer [20, 21], while for a comprehensive review of the physical aspect of the equations we refer [17]. Since the Hall term involves the second order derivative of the magnetic field, it becomes important when the magnetic shear is very large, and this occurs during the reconnection procedure. In the laminar flows this term is small compared with the other term, and can be neglected, which is the case of the usual MHD. Since the Hall term is quadratically nonlinear, containing the second order derivative, it causes major difficulties in the mathematical study of the Hall-MHD system, and only recently the rigorous results on the Cauchy problem appeared. In [1] the authors proved the global existence of weak solutions, while the local in time wellposedness as well as the global in time well-posedness for small initial data was proved in [4]. This later result was refined in [5]. In the case of µ = 0 it is proved in [7] that the Cauchy problem is not globally in time well-posed, rigorously verifying the numerical experiment of [8]. For a special axially symmetric initial data the authors of [9] proved the global well-posedness of the system. In [6] the long time behaviors of the solution were also studied. Since the Hall-MHD system has more complicated structure than the usual MHD system and the Navier-Stokes equations, the study of full regularity of weak solutions would be extremely difficult. Therefore, it might be reasonable to begin with the partial regularity, similarly to the case of the Navier-Stokes equations, the partial regularity of which was studied e.g. in [18, 2, 14, 16, 23]. In the time-dependent problem, mainly due to the difficulty of defining the correct localized energy inequality for a suitable weak solutions we concentrate the partial regularity problem of the steady Hall-MHD system. Contrary to the case of the Navier-Stokes equations and the usual MHD system the full regularity of the steady weak solutions is difficult to deduce. Instead, we prove that the set of possible singularity of the steady suitable weak solution of the Hall-MHD system has Hausdorff dimension at most 1(see Remark 5.2 and Theorem 5.3 below). Moreover, for a steady suitable weak solution on R3 the set of possible singularity is a compact set(see Corollary 7.3 below). The partial regularity of the time dependent problem will be studied elsewhere.
2
2
Weak solution and higher regularity of u
We consider the following steady Hall-MHD system in R3 . (2.1) (2.2) (2.3)
(u · ∇)u − ∆u = −∇p + (∇ × B) × B + f , ∇ × (B × u) − ∆B = −∇ × ((∇ × B) × B) + ∇ × g, ∇ · u = 0, ∇ · B = 0.
We note that we set µ = ν = 1 for convenience. As (u · ∇)u = (∇ × u) × u + 12 |u|2 , (2.1) turns into |u|2 + (∇ × B) × B + f in R3 . (2.4) (∇ × u) × u − ∆u = −∇ p + 2 Applying ∇× to the both sides of the above, we get (2.5)
∇ × (ω × u) − ∆ω = ∇ × ((∇ × B) × B) + ∇ × f
in R3 ,
where ω stands for the vorticity ∇ × u. Taking the sum of (2.2) and (2.5), we are led to (2.6)
∇ × (V × u) − ∆V = ∇ × (f + g) in R3 ,
where (2.7)
V = B + ω.
As ∇ · V = 0, there exists a solenoidal potential v such that ∇ × v = V . From (2.6) we deduce that v solves the system in R3 , (2.8) (2.9)
∇ · v = 0, (v · ∇)v − ∆v = −∇π + (∇ × v) × b + f + g,
where b = v − u. Clearly, ∇ × b = B. ˙ 1, 2 is a ˙ 1, 2 × L2 × W Definition 2.1. Let f ∈ L6/5 and g ∈ L2 . We say (u, p, B) ∈ W loc weak solution to (2.3)–(2.2) if Z ∇u : ∇ϕ − u ⊗ u : ∇ϕdx R3
=
(2.10)
Z
p∇ · ϕdx +
R3
Z
Z
((∇ × B) × B) · ϕdx +
R3
Z
f · ϕdx,
R3
∇B : ∇ϕ + B × u · ∇ × ϕdx
R3
(2.11)
=−
Z
((∇ × B) × B) · ∇ × ϕdx +
R3
Z
g · ∇ × ϕdx
R3
˙ 1, 2 stands for the homogeneous Sobolev space. for all ϕ ∈ Cc∞ . Here W 3
˙ 1, 2 ֒→ L6 , by the Calder´on-Zygmund inequality and Remark 2.2. Since B ∈ W 2 Sobolev’s embedding theorem we infer b ∈ L∞ loc . In particular, having V ∈ Lloc we 1, 2 2 easily deduce that v ∈ Wloc and π ∈ Lloc . Furthermore, (v, π) satisfies the following integral identity for all ϕ ∈ Cc∞ Z ∇v : ∇ϕ − v ⊗ v : ∇ϕdx R3
(2.12)
=
Z
π∇ · ϕdx +
R3
Z
((∇ × v) × b) · ϕdx +
R3
Z
(f + g) · ϕdx.
R3
By using standard regularity methods we get the following ˙ 1, 2 × L2 × W ˙ 1, 2 be a Theorem 2.3. Let f ∈ L6/5 and g ∈ L2 . Let (u, p, B) ∈ W loc weak solution to (2.3)–(2.2). Suppose, f , g ∈ Lqloc (Ω) for some 56 < q < +∞ and for an open set Ω ⊂ R3 . Then (2.13)
1, q V ∈ Wloc (Ω),
2, q v ∈ Wloc (Ω),
2, q∧2 u ∈ Wloc (Ω),
where q ∧ 2 = min{q, 2}. Proof First, assume 32 ≤ q < +∞. Since u ∈ L6 and V ∈ L2loc , we see that −V × u + 3/2 f + g ∈ Lloc (Ω). Observing (2.6), by the aid of Calder´on-Zygmund’s inequality and 1, 3/2 Sobolev’s embedding theorem we find V ∈ Wloc (Ω) ⊂ L3loc (Ω). As ω = V − B, this 1, 3 implies ω ∈ L3loc (Ω). Taking into account ∇ · u = 0, we get u ∈ Wloc (Ω). Once more s appealing to Sobolev’s embedding therorem, we obtain u ∈ Lloc (Ω) for all 1 ≤ s < +∞, and thus −V × u + f + g ∈ Lqloc (Ω). Again applying Calder´on-Zygmund’s inequality, 1, q 2, q we see that V ∈ Wloc (Ω). As ∇ · v = 0 from the last statement we infer v ∈ Wloc (Ω). 1, 2 1, q∧2 Finally, recalling ω = V − B and B ∈ Wloc (Ω), we obtain ω ∈ Wloc (Ω). In case 56 < q < 23 , we immediately get −V × u + f + g ∈ Lqloc (Ω), and the assertion can be proved as in the previous case.
Caccioppoli-type inequality for B
3
Suppose f , g ∈ L2loc (Ω) for some open set Ω ⊂ R3 . As it has been proved in Section 2 2, 2 we get u ∈ Wloc (Ω) if (u, p, B) is a weak solution to the steady Hall-MHD system. By means of Sobolev’s embedding theorem this implies u ∈ L∞ loc (Ω). Accordingly, 2 −B × u + g ∈ Lloc (Ω). Thus, for the sake of generality in the present and next section we study the local regularity for the following general model. Let Ω ⊂ R3 be a domain. We consider the system (3.1)
−∆B = −∇ × ((∇ × B) × B) + ∇ × F
in Ω.
We start our discussion with the following notion of a weak solution to (3.1).
4
1, 2 Definition 3.1. Let F ∈ L2loc (Ω). (i) A vector function B ∈ Wloc (Ω) is said to be a weak solution to (3.1) if Z Z Z (3.2) ∇B : ∇ϕdx = − (∇ × B) × B · ∇ × ϕdx + F · ∇ × ϕdx Ω
Ω
Ω
for all ϕ ∈ Cc∞ (Ω). (ii) A weak solution B to (3.1) is called a suitable weak solution to (3.1) if, in addition, the following local energy inequality holds: Z φ|∇B|2 dx Ω
1 ≤ 2
Z
2
∆φ|B| dx −
Ω
(3.3)
Z
(∇ × B) × B · (B × ∇φ)dx
Ω
+
Z
(φF · ∇ × B + F · B × ∇φ)dx
Ω
for all non-negative φ ∈ Cc∞ (Ω). Remark 3.2. Let B be a suitable weak solution to (3.1). Then for every constant vector Λ ∈ R3 there holds Z φ|∇B|2 dx Ω
1 ≤ 2
Z
2
∆φ|B − Λ| dx −
Ω
(∇ × B) × B · ((B − Λ) × ∇φ)dx
Ω
+
(3.4)
Z
Z
{φF · ∇ × B + F · (B − Λ) × ∇φ} dx
Ω
for all non-negative φ ∈ Cc∞ (Ω). This can be readily seen by combining (3.3) and (3.2) with ϕ = φΛ. Now, we state the following Caccioppoli-type inequality 1, 2 Lemma 3.3. Let F ∈ L2loc (Ω). Let B ∈ Wloc (Ω) be a suitable weak solution to (3.1). Then for every ball Br = Br (x0 ) ⊂⊂ Ω and 0 < ρ < r there holds Z 1 |∇B|2 dx r Bρ
(3.5)
Z Z cr 2 cr 2 2 2 (1 + |B r,x0 | ) − |B − B r,x0 | dx + − |B − B r,x0 |4 dx ≤ 2 (r − ρ)2 (r − ρ) Br Br c 2 + kF k2,Br , r 5
where B r,x0 stands for the mean value Z Z 1 − Bdx := Bdx, meas(Br ) Br Br (x0 )
and c = const > 0 denotes a universal constant. Proof Let Br = Br (x0 ) ⊂⊂ Ω be a fixed ball. Given ρ ∈ (0, r), we consider a cut-off function ζ defined as ζ ∈ Cc∞ (Br ) such that 0 ≤ ζ ≤ 1 in R3 , ζ ≡ 1 on Bρ and |∇ζ|2 + |∇2 ζ| ≤ c(r − ρ)−2 in R3 . From (3.4) with φ = ζ 2 and Λ = B r,x0 we obtain the following Caccioppoli-type inequality Z ζ 2 |∇B|2 dx Br
c ≤ (r − ρ)2
(3.6)
Z
2
|B − B r,x0 | dx + c
Z
Br
B
Br
Br
|F |2 dx
r Z c ζ|∇B| |B| |B − B r,x0 |dx + r−ρ B Z r Z c 2 = |B − B r,x0 | dx + c |F |2 dx + J. (r − ρ)2
Applying H¨older’s and Young’s inequality, we estimate Z Z 1 c 2 2 ζ 2|∇B|2 dx |B| |B − B r,x0 | dx + J≤ 2 (r − ρ) 2 Br Br Z Z c c 2 2 ≤ |B r,x0 | |B − B r,x0 | dx + |B − B r,x0 |4 dx 2 2 (r − ρ) (r − ρ) Br Br Z 1 ζ 2 |∇B|2 dx. + 2 Br
Inserting the above estimate of J into the right-hand side of (3.6), and dividing the resulting estimate by r, we are led to Z 1 |∇B|2 dx r Bρ
(3.7)
Z Z cr 2 cr 2 2 2 − |B − B r,x0 |4 dx (1 + |B r,x0 | ) − |B − B r,x0 | dx + ≤ 2 (r − ρ)2 (r − ρ) Br Br c 2 + kF k2,Br . r
Whence, the claim. 6
Remark 3.4. Setting Z E(r) = E(r, x0 ) = −
1/4 |B − B r;x0 | dx 4
Br (x0 )
0 < r < dist(x0 , ∂Ω),
(3.5) becomes Z 1/2 o c 1 cr n 2 (1 + |B r,x0 |)E(r) + E(r)2 + 1/2 kF k2,Br . (3.8) |∇B| dx ≤ r r−ρ r Bρ
4
Blow-up
We begin our discussion with the following fundamental estimate for solutions to the model problem, which will be used in the blow-up lemma below. 1, 2 Lemma 4.1. Let Λ ∈ R3 . Let W ∈ L4 (B1 ) ∩ Wloc (B1 ) be a weak solution to
(4.1)
−∆W = −∇ × ((∇ × W ) × Λ)
in B1 ,
i. e. (4.2)
Z
B1
∇W : ∇Φdx = −
Z
((∇ × W ) × Λ) · ∇ × Φdx
B1
for all Φ ∈ W 1, 2 (B1 ) with supp(Φ) ⊂⊂ B1 . Then, Z 1/4 1/4 Z 4 4 3 (4.3) − |W −W Bτ | dx ≤ C0 τ (1+|Λ| ) − |W −W B1 | dx Bτ
∀ 0 < τ < 1,
B1
where C0 > 0 denotes a universal constant. Proof Since the assertion is trivial for 21 < τ < 1, we may assume that 0 < τ < 21 . Let ζ ∈ Cc∞ (B1 ) be a suitable cut-off function for B1/2 , i. e. 0 ≤ ζ ≤ 1 in B1 and ζ ≡ 1 on B1/2 . In (4.2) inserting the admissible test function Φ = ζ 2m (W − W B1 ) (m ∈ N), by using Cauchy-Schwarz’s inequality along with Young’s inequality we obtain Z Z 2m 2 2 (4.4) ζ |∇W | dx ≤ c(1 + |Λ| ) ζ 2m−2 |W − W B1 |2 dx. B1
B1
If W is smooth in B1 , since (4.1) is a linear system, the same inequality holds for D α W in place of W for any multi-index α. By a standard mollifying argument together with Sobolev’s embedding theorem we see that W is smooth. In particular, in (4.4) putting m = 3 it follows that Z Z 6 α 2 6 (4.5) ζ |D W | dx ≤ c(1 + |Λ| ) |W − W B1 |2 dx ∀ |α| ≤ 3. B1
B1
7
By means of Sobolev’s embedding theorem and Jensen’s inequality we get Z 4 12 (4.6) k∇W k∞,B1/2 ≤ c(1 + |Λ| ) |W − W B1 |4 dx. B1
Applying Poincar´e’s inequality, we obtain Z (4.7) − |W − W Bτ |4 dx ≤ cτ 4 k∇W k4∞,B1/2 . Bτ
Combination of (4.6) and (4.7) yields the desired estimate. In our discussion below we make use of the notion of the Morrey space. We say F ∈ Mp,λ loc (Ω) if for all K ⊂⊂ Ω Z p −λ p [F ]Mp,λ ,K := sup r |F | dx x0 ∈ K, 0 < r ≤ dist(K, ∂Ω) < +∞. Br (x0 )
1 Lemma 4.2. Let F ∈ M2,λ loc (Ω) for some 1 < λ ≤ 3. For every 0 < τ < 2 , 0 < λ−1 M < +∞, K ⊂⊂ Ω and 0 < α < 2 , there exist positive numbers ε0 = ε0 (τ, M, K, α) 1, 2 and R0 = R0 (τ, M, K, α) < dist(K, ∂Ω) such that, if B ∈ Wloc (Ω) is a suitable weak solution to (3.1), and for x0 ∈ K and 0 < R ≤ R0 the condition
(4.8)
|B R,x0 | ≤ M,
E(R, x0 ) + Rα ≤ ε0
is fulfilled, then (4.9)
E(τ R, x0 ) ≤ 2τ C0 (1 + M 3 )(E(R, x0 ) + Rα ),
where C0 > 0 stands for the constant which appears on the right-hand side of (4.3). Proof Assume the assertion of the Lemma is not true. Then there exist 0 < τ < 21 , 0 < 1, 2 together with a sequence B (k) ∈ Wloc (Ω) being M < +∞, K ⊂⊂ Ω and 0 < α < λ−1 2 suitable weak solutions to (3.1) as well as sequences xk ∈ K, 0 < Rk < dist(K, ∂Ω) and εk → 0 as k → +∞ such that (4.10)
(k)
|B Rk ,xk | ≤ M,
Ek (Rk , xk ) + Rkα = εk
and (4.11)
Ek (τ Rk , xk ) > 2τ C0 (1 + M 3 )(Ek (Rk , xk ) + Rkα ).
Here we have used the notation 1/4 Z (k) (k) 4 |B − B r,x0 | dx Ek (r, x0 ) = − , Br (x0 )
Note that (4.10) yields Rk → 0 as k → +∞. 8
x0 ∈ K, 0 < r ≤ dist(K, ∂Ω).
Next, define 1 (k) (B (k) (xk + Rk y) − B Rk ,xk ), εk F k (y) = F (xk + Rk y), y ∈ B1 (0)
W k (y) =
(k ∈ N). Furthermore, set 1/4 Z 4 Ek (σ) = − |W k − (W k )Bσ | dy ,
0 < σ ≤ 1.
Bσ
Then (4.10) and (4.11) turn into (4.12)
(k)
|B Rk ,xk | ≤ M,
Ek (1) +
Rkα = 1, εk
and (4.13)
Rkα = 2τ C0 (1 + M 3 ) Ek (τ ) > 2τ C0 (1 + M ) Ek (1) + εk 3
respectively. Using the chain rule, we find that (3.1) transforms into (k)
(4.14)
−∆W k = −εk ∇ × ((∇ × W k ) × W k ) − ∇ × ((∇ × W k ) × B Rk ,xk ) Rk ∇ × F k in B1 . + εk
Thus, W k ∈ W 1, 2 (B1 ) is a weak solution to (4.14). Let 0 < σ < 1. Using the (k) transformation formula, noticing that |B Rk ,xk | ≤ M, the Caccioppoli-type inequality (3.8) with r = Rk and ρ = σRk turns into (4.15)
k∇W k k2,Bσ ≤ c(1 − σ)
−1
(1 + M)Ek (1) + εk Ek (1)
2
−1/2
cR + k εk
Observing (4.12), and verifying −1/2
(4.16)
(λ−1)/2
Rk R kF kBRk (xk ) ≤ k εk εk
(λ−1)/2−α
[F ]M2,λ ,K ≤ Rk
from (4.15), we get (4.17)
k∇W k k2,Bσ ≤ c(1 − σ)−1 (M + 1) + c[F ]M2,λ ,K .
In addition, in view of (4.12) we estimate (4.18)
kW k k4,B1 = (mes B1 )1/4 Ek (1) ≤ (mes B1 )1/4 .
9
[F ]M2,λ ,K
kF k2,BRk (xk ) .
From (4.17) and (4.18) it follows that (W k ) is bounded in W 1, 2 (Bσ ) for all 0 < σ < 1 and bounded in L4 (B1 ). Thus, by means of reflexivity, eventually passing to subse1, 2 quences, we get W ∈ L4 (B1 ) ∩ Wloc (B1 ) and Λ ∈ R3 such that (4.19) (4.20) (4.21)
Wk → W Wk → W
weakly in L4 (B1 ) as k → +∞, weakly in W 1, 2 (Bσ ) as k → +∞ ∀ 0 < σ < 1,
(k)
B Rk ,xk → Λ in R3
as k → +∞.
On the other hand, by the compactness of the embedding W 1, 2 (Bσ ) ֒→ L4 (Bσ ) from (4.19) we infer (4.22)
Wk → W
strongly in L4 (Bσ ) as k → +∞ ∀ 0 < σ < 1.
Accordingly, (4.23)
lim Ek (σ) = E (σ) ∀ 0 < σ < 1,
k→∞
1/4 Z 4 where E (σ) = − |W − W Bσ | dy . In particular, by the aid of (4.23) (with Bσ
σ = τ ) from (4.13) we get (4.24)
E (τ ) ≥ 2τ C0 (1 + M 3 ).
In view of (4.16) we have −1/2
Rk R (λ−1)/2−α [F ]M2,λ ,K → 0 kF k k2,B1 = k kF kBRk (xk ) ≤ Rk εk εk as k → +∞. Therefore, with the help of (4.19), (4.20), (4.21) and (4.22), letting 1, 2 k → +∞ in (4.14), we see that W ∈ Wloc (B1 ) ∩ L4 (B1 ) is a weak solution to (4.25)
−∆W = −∇ × ((∇ × W ) × Λ) in B1 .
As |Λ| ≤ M appealing to Lemma 4.1, we find (4.26)
E (τ ) ≤ τ C0 (1 + M 3 )E (1).
On the other hand, by virtue of the lower semi continuity of the norm together with (4.13) and (4.23) we get Rα 1 E (1) ≤ lim inf Ek (1) + k ≤ lim Ek (τ ) k→∞ εk 2τ C0 (1 + M 3 ) k→∞ 1 = E (τ ). 2τ C0 (1 + M 3 ) Estimating the right of (4.26) by the inequality we have just obtained, we see that E (τ ) ≤ 12 E (τ ) and hence E (τ ) = 0, which contradicts to (4.24). Whence, the assumption is not true, and this completes the proof of the Lemma. 10
5
Partial regularity
The aim of the present section is to prove the partial regularity of a suitable weak 1, 2 solution B ∈ Wloc (Ω) to system (3.1), which will lead to the partial regularity of a suitable weak solution to the steady Hall-MHD system. As we will see below, the set Σ(B) of possible singularities is given by means of Z 1 2 Σ(B) = x0 ∈ Ω lim inf |∇B| dx > 0 r→0+ r Br (x0 ) n o (5.1) ∪ x0 ∈ Ω sup |B r,x0 | = +∞ . r>0
1, 2 Theorem 5.1. Let F ∈ M2,λ loc (Ω) for some 1 < λ < 3. Let B ∈ Wloc (Ω) be a suitable weak solution to the system (3.1). Then, Ω \ Σ(B) is an open set, on which B is . α-H¨older continuous with repect to any exponent 0 < α < λ−1 2
Proof Let x0 ∈ Ω \ Σ(B) (cf. (5.1)). Set d = dist(x0 , ∂Ω), and define K = Bd/2 (x0 ). Using Sobolev Poincar´e’s inequality, we have Z 2 Z 1 4 2 − |B − B r,x0 | dx ≤ c lim inf lim inf |∇B| dx = 0. r→0+ r→0+ r Br (x0 ) Br (x0 )
We set M := sup |B r,x0 | + 3 < +∞. 0 0 fulfills (5.4)
2τ −3 ε1 ≤ 1.
By the absolutely continuity of the Lebesgue integral there exists 0 < δ < for all y ∈ Bδ (x0 ) (5.5) (5.6)
E(y, R1 ) + 2R1α ≤ min{ε0 , ε1}, |B R1 ,y | ≤ sup |B r,x0 | + 1 = M − 2. 0 1 stands for an absolute constant. Next, we ssume that (7.11)
1 3τ −1 C1 M(R, x0 ) ≤ τ β . 2
We note here that (7.11) yields M(R, x0 ) ≤ 1 and thus (7.5) remains true. We make use of triangle inequality, then apply (7.3) (note that W is harmonic). This together with (7.10) and (7.6) gives 1/3 Z Z 1/3 3 3 −1 E(τ R, x0 ) ≤ − |W − W τ R,x0 | dx − |Z| dx + 3τ Bτ R (x0 )
BR/2 (x0 )
Z −1 − ≤ 2C0 τ E(R, x0 ) + 3τ
BR/2 (x0 )
1/3 3 |Z| dx
1 ≤ τ β E(R, x0 ) + 3τ −1 C1 M(R, x0 )E(R, x0 ) + 3τ −1 C1 γ0 Rα , 2 and observing (7.11), we therefore obtain (7.12)
E(τ R, x0 ) ≤ τ β E(R, x0 ) + C2 γ0 Rα ,
where C2 = 3τ −1 C1 . Next, we shall estimate |M(τ R, x0 )|. By using triangle inequality and SobolevPoincar´e inequality it follows that M(τ R, x0 ) ≤ |B τ R,x0 | + cτ −1/2 S(R/2, x0 ) (7.13)
≤ M(R, x0 ) + cτ −1/2 S(R/2, x0 ) + 2τ −1 E(R, x0 ) ≤ M(R, x0 ) + C3 (E(R, x0 ) + γ0 Rα )
with a constant C3 > 0 depending on τ only. Let 0 < M0 ≤ 1 such that (7.14)
1 3τ −1 C1 M0 = τ β . 2 19
Let 0 < R1 ≤ 1 chosen so that (7.15)
2C3 (2C2 τ −α + 1)γ0 R1α ≤
M0 . 2
Since B ∈ L6 , there exists 0 < ρ0 < +∞ such that (7.16)
M(R1 , x0 ) + 2C3 E(R1 , x0 ) ≤ (1 + 4C3 )M(R1 , x0 ) ≤
M0 2
∀ |x0 | ≥ ρ0 .
Let x0 ∈ Rn , |x0 | ≥ ρ0 . We claim that for every k ∈ N ∪ {0} (7.17) (7.18)
E(τ k R1 , x0 ) ≤ τ βk E(R1 , x0 ) + 2(1 − 2−k )τ α(k−1) C2 γ0 R1α , n o M(τ k R1 , x0 ) ≤ M(R1 , x0 ) + 2(1 − 2−k ) C3 E(R1 , x0 ) + C3 (2C2 τ −α + 1)γ0 R1α .
We prove the claim by using induction over k ∈ N ∪ {0}. Firstly, note that for k = 0 both (7.17) and (7.18) are trivially fulfilled. Assume (7.17) and (7.18) hold for k ∈ N ∪ {0}. Observing (7.15) and (7.16), the assumption (7.18) implies (7.19)
M(τ k R1 , x0 ) ≤ M0 .
By the choice of M0 (7.19) yields (7.11) for R = τ k R1 . Hence from (7.12) with R = τ k R1 we infer E(τ k+1 R1 , x0 ) ≤ τ β E(τ k R1 , x0 ) + C2 γ0 τ αk R1α . Now, estimating the first term by the assumption (7.17) taking into account (7.6), we arrive at E(τ k+1 R1 , x0 ) ≤ τ β(k+1) E(R1 , x0 ) + 2τ β−α (1 − 2−k )τ αk C2 γ0 R1α + τ αk C2 γ0 R1α ≤ τ β(k+1) E(R1 , x0 ) + (2 − 2−k )τ αk C2 γ0 R1α which results in (7.17) for k + 1. It remains to verify (7.18) for k + 1. In fact, by means of (7.13) with R = τ k R1 together with the assumption (7.17) and (7.18) we estimate M(τ k+1 R1 , x0 ) ≤ M(τ k R1 , x0 ) + C3 (E(τ k R1 , x0 ) + γ0 τ αk R1α ) o n ≤ M(R1 , x0 ) + 2(1 − 2−k ) C3 E(R1 , x0 ) + C3 (2C2 τ −α + 1)γ0 R1α
+ C3 (τ βk E(R1 , x0 ) + 2τ α(k−1) C2 γ0 R1α + γ0 τ αk R1α ) n o ≤ M(R1 , x0 ) + 2(1 − 2−k ) C3 E(R1 , x0 ) + C3 (2C2 τ −α + 1)γ0 R1α
+ 2−k (C3 E(R1 , x0 ) + C3 (2C2 τ −α + 1)γ0R1α ) n o = M(R1 , x0 ) + 2(1 − 2−k−1) C3 E(R1 , x0 ) + C3 (2C2 τ −α + 1)γ0R1α . Whence, (7.18) for k + 1. This implies the following 20
˙ 1, 2 be a suitable weak solution Theorem 7.3. Let F ∈ M2,λ , 1 < λ < 3. Let B ∈ W to (3.1). Then there exists ρ0 > 0 such that Σ(B) ⊂ Bρ0 . As a consequence of Theorem 7.3 we get Corollary 7.4. Let f ∈ L6/5 ∩L2 and g ∈ L2 ∩Lq for some 3 < q < +∞. Let (u, p, B) be a suitable weak solution to the steady Hall-MHD system. Then there exists ρ0 > 0 such that B is H¨older contiuous in {x : |x| > ρ0 }. In particular, Σ(B) is a compact set of Hausdorff dimension at most one. Proof To prove the corollary we only need to verify that −B × u + g ∈ M2,λ for some 1 < λ < 3. Then the assertion follows immediately from Theorem 7.3 with F = −B × u + g ∈ M2,λ . First, using H¨older’s inequality, we find kgk22,BR ≤ cR3(q−2)/q kgkq for every ball < 3. BR ⊂ R3 , which implies g ∈ M2,3(q−2)/q . Owing to 3 < q < +∞ we have 1 < 3(q−2) q 6 2 6 3/2 3 Next, as V = B+ω ∈ L +L and u ∈ L , we see that −V ×u+(f +g) ∈ L +L +L2 . By Calder´on-Zygmund’s inequality it follows that ∇V ∈ W 1, 3/2 + W 1, 3 + W 1, 2 . By means of Sobolev’s embedding theorem we get kωk3,B2 (x0 ) ≤ kV k3,B2 (x0 ) + kBk3,B2 (x0 ) ≤ c(kωk2 kuk6 + kBk6 kuk6 + kf + gk2 ) for all x0 ∈ R3 , with an absolute constant c > 0. As ∇ · u = 0, we obtain kuk10,B1 (x0 ) ≤ c(kuk6 (1 + kωk2 + kBk6 ) + ckf + gk2
∀ x0 ∈ R3 .
Accordingly, B × u ∈ M2,7/5 . This completes the proof of the corollary. Acknowledgements Chae was partially supported by NRF grants 2006-0093854 and 2009-0083521, while Wolf has been supported by the Brain Pool Project of the Korea Federation of Science and Technology Societies (141S-1-3-0022).
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