The 2D Incompressible Magnetohydrodynamics Equations with only ...

THE 2D INCOMPRESSIBLE MAGNETOHYDRODYNAMICS EQUATIONS WITH ONLY MAGNETIC DIFFUSION

arXiv:1306.3629v1 [math.AP] 16 Jun 2013

CHONGSHENG CAO1 , JIAHONG WU2 , BAOQUAN YUAN3 Abstract. This paper examines the global (in time) regularity of classical solutions to the 2D incompressible magnetohydrodynamics (MHD) equations with only magnetic diffusion. Here the magnetic diffusion is given by the fractional Laplacian operator (−∆)β . We establish the global regularity for the case when β > 1. This result significantly improves previous work which requires β > 32 and brings us closer to the resolution of the well-known global regularity problem on the 2D MHD equations with standard Laplacian magnetic diffusion, namely the case when β = 1.

1. Introduction This paper focuses on the initial-value problem (IVP) for the two-dimensional (2D) incompressible magneto-hydrodynamics (MHD) equations with fractional Laplacian magnetic diffusion  ∂t u + u · ∇u = −∇p + b · ∇b, x ∈ R2 , t > 0,    ∂ b + u · ∇b + (−∆)β b = b · ∇u, x ∈ R2 , t > 0, t (1.1)  ∇ · u = 0, ∇ · b = 0, x ∈ R2 , t > 0,    u(x, 0) = u0 (x), b(x, 0) = b0 (x), x ∈ R2 , where the fractional Laplacian operator (−∆)β is defined through the Fourier transform \ β f (ξ) = |ξ|2β fb(ξ) (−∆)

with fb being the Fourier transform of f , namely Z b f (ξ) = e−ix·ξ f (x) dx. R2

When β = 1, (1.1) reduces to the MHD equations with Laplacian magnetic diffusion, which models many significant phenomena such as the magnetic reconnection in astrophysics and geomagnetic dynamo in geophysics (see, e.g., [20]). What we care about here is the global regularity problem, namely whether the solutions of (1.1) emanating from sufficiently smooth data (u0 , b0 ) remain regular for all time. The main result of this paper states that (1.1) with any β > 1 always possesses a unique global solution. More precisely, we have the following theorem. Theorem 1.1. Consider (1.1) with β > 1. Assume that (u0 , b0 ) ∈ H s (R2 ) with s > 2, ∇ · u0 = 0, ∇ · b0 = 0 and j0 = ∇ × b0 satisfying k∇j0 kL∞ < ∞.

2000 Mathematics Subject Classification. 35Q35, 35B35, 35B65, 76D03. Key words and phrases. MHD equations, partial dissipation, classical solutions, global regularity. 1

2

C. CAO, J. WU, B. YUAN

Then (1.1) has a unique global solution (u, b) satisfying, for any T > 0, (u, b) ∈ L∞ ([0, T ]; H s (R2 )),

where j = ∇ × b.

∇j ∈ L1 ([0, T ]; L∞ (R2 ))

We now review some recent work to put our result in a proper content. Due to their prominent roles in modeling many phenomena in astrophysics, geophysics and plasma physics, the MHD equations have been studied extensively mathematically. G. Duvaut and J.-L. Lions constructed a global Leray-Hopf weak solution and a local strong solution of the 3D incompressible MHD equations [10]. M. Sermange and R. Temam further examined the properties of these solutions [22]. More recent work on the MHD equations develops regularity criteria in terms of the velocity field and deals with the MHD equations with dissipation and magnetic diffusion given by general Fourier multiplier operators such as the fractional Laplacian operators (see, e.g., [7, 8, 11, 12, 14, 24, 26, 27, 28, 29, 31, 32, 33]). Another direction that has generated considerable interest recently is the global regularity problem on the MHD equations with partial dissipation, especially in the 2D case (see, e.g., [3, 4, 5, 13, 15, 16, 17, 30, 34]). Since the global regularity of the 2D MHD equations with both Laplacian dissipation and magnetic diffusion is easy to obtain while the regularity of the completely inviscid MHD equations appear to be out of reach, it is natural to examine the MHD equations with partial dissipation. One particular partial dissipation case is (1.1). When β ≥ 1, any solution of (1.1) with (u0 , b0 ) ∈ H 1 generates a global weak solution (u, b) that remains bound in H 1 for all time (see, e.g., [3, 15]). However, it is not clear if such weak solutions are unique or their regularity can be improved to be in H 2 if the initial data (u0, b0 ) is in H 2 . The result of this paper obtains the uniqueness and global regularity for the case when β > 1. Previous global regularity results require that β > 23 (see, e.g., [14, 24, 31, 32]). Our approach here does not appear to be able to extend to the borderline case when β = 1, which is currently being studied by a different method [19]. Progress has also been made on several other partial dissipation cases of the MHD equations. F. Lin, L. Xu and P. Zhang recently studied the MHD equations with the Laplacian dissipation in the velocity equation but without magnetic diffusion and were able to establish the global existence of small solutions after translating the magnetic field by a constant vector ([16, 17, 30]). C. Cao and J. Wu examined the anisotrophic 2D MHD equations with horizontal dissipation and vertical magnetic diffusion (or vertical dissipation and horizontal magnetic diffusion) and obtained the global regularity for this partial dissipation case [3]. The anisotrophic MHD equations with horizontal dissipation and horizontal magnetic diffusion were also investigated very recently and progress was also made (see [4]). We now explain our proof of Theorem 1.1. Since the local (in time) well-posedness can be established following a standard approach, our efforts are devoted to proving global a priori bounds for (u, b) in the initial functional setting H s with s > 2. (u, b) indeed admits a global H 1 -bound, but direct energy estimates do not appear to easily yield a global H 2 -bound. The difficulty comes from the nonlinear term in the velocity equation due to the lack of dissipation. To bypass this difficulty, we first make use of the magnetic diffusion (−∆)β b with β > 1 to show the global bound for kωkLq and kjkLq for 2 2 ≤ q ≤ 2−β (the range is modified to 2 ≤ q < ∞ when β = 2). The magnetic diffusion

MHD EQUATIONS WITH MAGNETIC DIFFUSION

3

s is further exploited to obtain a global bound for j in the space-time Besov space L1t Bq,1 , namely, for any T > 0 and t ≤ T 2 2 s kjkL1t Bq,1 ≤ C(T, u0 , b0 ) < ∞ with 2 ≤ q ≤ , < s < 2β − 1. (1.2) 2−β q

Roughly speaking, this global bound provides the time integrability of the Lq -norm of j up to (2β − 1)-derivative. This global bound is proven through Besov space techniques. Special consequences of this global bound include the time integrability of kjkL∞ and of k∇jkLr for r > q. To gain higher regularity, we go through an iterative process. The bound k∇jkL1t Lr < ∞ allows us to further prove a global bound for kωkLr and kjkLr with r > q, which can be employed to prove (1.2) with q replaced by r. Repeating this process leads to the global bound in (1.2) for any q ∈ [2, ∞), which especially implies, for any t > 0, Z t (1.3) k∇jkL∞ dτ < ∞. 0

(1.3) can then be further used to obtain a global bound for kωkL∞ . The time integrability are enough to prove a global bound for (u, b) of kjkL∞ and the boundedness of kωkL∞ t,x s in H . The rest of this paper is divided into three sections. The second section provides the definition of inhomogeneous Besov spaces and related facts such as Bernstein’s inequality. The third section proves the global Lq -bound for (ω, j) while the fourth section establishes the global bound in (1.2). The last section gains higher regularity through an iterative process and proves Theorem 1.1. 2. Functional spaces This section provides the definition of Besov spaces and related facts used in the subsequent sections. Materials presented here can be found in several books and many papers (see, e.g., [1, 2, 18, 21, 25]). We start with several notation. S denotes the usual Schwarz class and S ′ its dual, the space of tempered distributions. It is a simple fact in analysis that there exist two radially symmetric functions Ψ, Φ ∈ S such that 4 3 8 b ⊂ B 0, b ⊂ A 0, , suppΨ , suppΦ , 3 4 3 b Ψ(ξ) +

∞ X j=0

b j (ξ) = 1, Φ

ξ ∈ Rd ,

where B(0, r) denotes the ball centered at the origin with radius r, A(0, r1, r2 ) denotes the annulus centered at the origin with the inner radius r1 and the outer r2 , Φ0 = Φ and b j (ξ) = Φ b 0 (2−j ξ). Φj (x) = 2jd Φ0 (2j x) or Φ Then, for any ψ ∈ S,

Ψ∗ψ+

∞ X j=0

Φj ∗ ψ = ψ

4


and hence Ψ∗f +

∞ X j=0

Φj ∗ f = f

(2.1)

in S ′ for any f ∈ S ′ . To define the inhomogeneous Besov space, we set  if j ≤ −2,  0, Ψ ∗ f, if j = −1, ∆j f =  Φj ∗ f, if j = 0, 1, 2, · · · .

(2.2)

s Definition 2.1. The inhomogeneous Besov space Bp,q with 1 ≤ p, q ≤ ∞ and s ∈ R ′ consists of f ∈ S satisfying s kf kBp,q ≡ k2jsk∆j f kLp klq < ∞.

Besides the Fourier localization operators ∆j , the partial sum Sj is also a useful notation. For an integer j, j−1 X ∆k , Sj ≡ k=−1

where ∆k is given by (2.2). For any f ∈ S ′ , the Fourier transform of Sj f is supported on the ball of radius 2j . Bernstein’s inequalities are useful tools in dealing with Fourier localized functions and these inequalities trade integrability for derivatives. The following proposition provides Bernstein type inequalities for fractional derivatives. Proposition 2.2. Let α ≥ 0. Let 1 ≤ p ≤ q ≤ ∞. 1) If f satisfies

supp fb ⊂ {ξ ∈ Rd : |ξ| ≤ K2j },

for some integer j and a constant K > 0, then 1

1

k(−∆)α f kLq (Rd ) ≤ C1 22αj+jd( p − q ) kf kLp (Rd ) . 2) If f satisfies supp fb ⊂ {ξ ∈ Rd : K1 2j ≤ |ξ| ≤ K2 2j }

for some integer j and constants 0 < K1 ≤ K2 , then

1

1

C1 22αj kf kLq (Rd ) ≤ k(−∆)α f kLq (Rd ) ≤ C2 22αj+jd( p − q ) kf kLp (Rd ) , where C1 and C2 are constants depending on α, p and q only.

3. Global Lq -bound for (ω, j) This section proves a global a priori bound for ω and j in the Lebesgue space Lq (R2 ) for q in a suitable range. More precisely, we prove the following proposition.


5

Proposition 3.1. Assume (u0, b0 ) satisfies the conditions stated in Theorem 1.1. Let (u, b) be the corresponding solution of (1.1) with β > 1. Then, for any q satisfying 2≤q≤

2 2−β

(3.1)

(the range of q is modified to 2 ≤ q < ∞ when β = 2), and for any T > 0 and t ≤ T , there exists a constant C = C(T, u0 , b0 ) such that kj(t)kLq ≤ C.

kω(t)kLq ≤ C,

(3.2)

To prove Proposition 3.1, we first provide two simple bounds. A simple energy estimate yields the global L2 -bound of (1.1) with β ≥ 0. Lemma 3.2. Assume (u0 , b0 ) satisfies the conditions stated in Theorem 1.1. Let (u, b) be the corresponding solution of (1.1) with β ≥ 0. Then, for any t ≥ 0, Z t 2 2 ku(t)kL2 + kb(t)kL2 + 2 kΛβ b(τ )k2L2 dτ = ku0 k2L2 + kb0 k2L2 , 0

√ where Λ = −∆.

In addition, by resorting to the equations of ω and j,   ∂t ω + u · ∇ω = b · ∇j, ∂t j + u · ∇j + (−∆)β j = b · ∇ω + 2∂1 b1 (∂2 u1 + ∂1 u2 ) − 2∂1 u1(∂2 b1 + ∂1 b2 ),  ω(x, 0) = ω (x), ω(x, 0) = j (x), 0 0

(3.3)

the global H 1 -bound on (u, b) can be obtained in a similar fashion as in [3].

Lemma 3.3. Assume (u0 , b0 ) satisfies the conditions stated in Theorem 1.1. Let (u, b) be the corresponding solution of (1.1) with β ≥ 1. Then, for any T > 0 and t ≤ T , there exists a constant C = C(T, k(u0, b0 )kH 1 ) such that Z t 2 2 kω(t)kL2 + kj(t)kL2 + (3.4) kΛβ j(τ )k2L2 dτ ≤ C. 0

Proof of Proposition 3.1. Multiplying the first equation in (3.3) by ω|ω|q−2, integrating in space and applying Hölder’s inequality, we have Z 1d q kωkLq = b · ∇j ω|ω|q−2 ≤ kωkLq−1 q kbkL∞ k∇jkLq . q dt Recall the simple Sobolev inequalities, for β > 1, 1−

1

1

1−

kbkL∞ ≤ CkbkL2 1+β kΛβ jkL1+β 2 , and note that (3.1) ensures that

Therefore,

2(q−1) βq

k∇jkLq ≤ C kjkL2

2(q−1) βq

2(q−1)

kΛβ jkL2βq

≤ 1. By Young’s inequality,

kbkL∞ k∇jkLq ≤ C (kbk2L2 + kjk2L2 + kΛβ jk2L2 ), kω(t)kLq ≤ kω0 kLq + C

Z

0

t

(kbk2L2 + kjk2L2 + kΛβ jk2L2 ) dt

(3.5)

6


and the bounds in Lemmas 3.2 and 3.3 yield the global bound for kω(t)kLq . To get a global bound for kjkLq , we first obtain from the equation of j in (3.3) Z 1d q kjkLq + j|j|q−2 (−∆)β j = K1 + K2 + K3 + K4 + K5 , (3.6) q dt where K1 , · · · , K5 are given by

Z

b · ∇ω j|j|q−2, Z 2 ∂1 b1 ∂2 u1 j|j|q−2, Z 2 ∂1 b1 ∂1 u2 j|j|q−2, Z 2 ∂1 u1 ∂2 b1 j|j|q−2, Z 2 ∂1 u1 ∂1 b2 j|j|q−2.

K1 = K2 = K3 = K4 = K5 =

According to [9], we have the lower bound, for C0 = C0 (β, q), Z Z q q−2 β j|j| (−∆) j ≥ C0 |Λβ (|j| 2 )|2 .

(3.7)

By integration by parts and Hölder’s inequality, Z q q 2(q − 1) −1 |K1 | = ω b · ∇(|j| 2 ) |j| 2 q q

−1

q

≤ C kbkL∞ kωkLq kjkL2 q k∇(|j| 2 )kL2 .

By the trivial embedding inequality, for β ≥ 1, q

q

q

q

q

q

k∇(|j| 2 )kL2 ≤ k|j| 2 kH β ≤ k|j| 2 kL2 + kΛβ (|j| 2 )kL2 = kjkL2 q + kΛβ (|j| 2 )kL2 , we obtain

Z q C0 q−2 2 2 |Λβ (|j| 2 )|2 + C kbkL∞ kωkLq kjkLq−1 |K1 | ≤ q + C kbkL∞ kωkLq kjkLq 2 Z q C0 ≤ |Λβ (|j| 2 )|2 + C (1 + kbkL∞ )kbkL∞ (kωkqLq + kjkqLq ). 2

K2 can be easily bounded. In fact, by the Sobolev inequality with β > 1, 1

1− 1

β β β kf kL∞ (R2 ) ≤ C kf kL2 (R 2 ) kΛ f kL2 (R2 ) ,

we have |K2 | ≤ 2k∂1 b1 kL∞ kωkLq kjkLq−1 q 1− 1

1

≤ C k∂1 b1 kL2 β kΛβ ∂1 b1 kLβ 2 (kωkqLq + kjkqLq ) 1− 1

1

≤ C kjkL2 β kΛβ jkLβ 2 (kωkqLq + kjkqLq ),

(3.8)


7

where the simple inequality k∂1 b1 kL2 ≤ k∇bkL2 = k∇ × bkL2 = kjkL2 . Clearly, K3 , K4 and K5 admits the same bound as in (3.8). Collecting the estimates, we have C0 1d kjkqLq + q dt 2

Z

q

|Λβ (|j| 2 )|2 1− 1

1

≤ C (kbkL∞ + kbk2L∞ + kjkL2 β kΛβ jkLβ 2 )(kωkqLq + kjkqLq ). 1− 1

1

Thanks to the time integrability of kbk2L∞ (see (3.5) for a bound) and kjkL2 β kΛβ jkLβ 2 on any finite time interval and the global bound for kωkLq , this differential inequality yields the global bound for kjkLq . This completes the proof of Proposition 3.1.

s 4. Global L1t Bq,1 -bound for j s This section establishes a global bound for j in the space-time Besov space L1t Bq,1 , where q satisfies (3.1) and 2q < s < 2β −1. For β > 1, this global bound provides a better integrability than the one given in (3.4) and will be exploited to gain higher regularity in the next section.

Proposition 4.1. Assume (u0, b0 ) satisfies the conditions stated in Theorem 1.1. Let (u, b) be the corresponding solution of (1.1) with β > 1. Then, for any q and s satisfying 2≤q≤

2 , 2−β

2 < s < 2β − 1 q

(4.1)

(the range of q is modified to 2 ≤ q < ∞ when β = 2), and for any T > 0 and t ≤ T , there exists a constant C = C(q, s, T, u0, b0 ) such that s ≤ C. kjkL1t Bq,1

(4.2)

We remark that the proof of this theorem makes use of the global Lq -bound for kωkLq and kjkLq and this explains why we need to restrict q to the range in (4.1). It can be seen from the proof that this theorem remains valid for any q ≥ 2 as long as kωkLq and kjkLq are bounded. We now prove Proposition 4.1. Proof of Proposition 4.1. Let k ≥ 0 be an integer. Applying ∆k to the equation of j in (3.3), multiplying by ∆k j |∆k j|q−2 and integrating in space, we obtain 1d k∆k jkqLq + q dt

Z

∆k j |∆k j|k−2 (−∆)β ∆k j = L1 + · · · + L6 ,

(4.3)

8


where L1 ,· · · , L6 are given by

Z

L1 = − ∆k j |∆k j|q−2 ∆k (u · ∇j), Z L2 = ∆k j |∆k j|q−2 ∆k (b · ∇ω), Z L3 = 2 ∆k j |∆k j|q−2 ∆k (∂1 b1 ∂2 u1 ), Z L4 = 2 ∆k j |∆k j|q−2 ∆k (∂1 b1 ∂1 u2 ), Z L5 = −2 ∆k j |∆k j|q−2 ∆k (∂1 u1 ∂2 b1 ), Z L6 = −2 ∆k j |∆k j|q−2 ∆k (∂1 u1 ∂1 b2 ). The term associated the magnetic diffusion admits the lower bound (see [6]) Z ∆k j |∆k j|q−2 (−∆)β ∆k j ≥ C1 22βk k∆k jkqLq for C1 = C1 (β, q). By Bony’s paraproducts decomposition, we write L1 = L11 + L12 + L13 + L14 + L15 , where L11 = −

Z

∆k j |∆k j|q−2

L12 = −

Z

∆k j |∆k j|q−2

L13 = −

Z

∆k j |∆k j|q−2 Sk u · ∇∆k j,

L14 = −

Z

∆k j |∆k j|q−2

L15 = −

Z

∆k j |∆k j|q−2

X

[∆k , Sl−1 u · ∇]∆l j,

X

(Sl−1 u − Sk u) · ∇∆k ∆l j,

|k−l|≤2

|k−l|≤2

X

|k−l|≤2

X

l≥k−1

∆k (∆l u · ∇Sl−1 j),

e l j) ∆k (∆l u · ∇∆

e l = ∆l−1 + ∆l + ∆l+1 . Thanks to ∇ · Sk u = 0, we have L13 = 0. By Hölder’s with ∆ inequality, a standard commutator estimate, and Bernstein’s inequality, X |L11 | ≤ k∆k jkLq−1 k[∆k , Sl−1 u · ∇]∆l jkLq q |k−l|≤2

≤ C

≤ C

k∆k jkLq−1 q q−1 k∆k jkLq

k∇Sk−1 ukLq k∆k jkL∞ 2

kωkLq 2k q k∆k jkLq .

(4.4)

Here we have also applied the simple fact that, for fixed k, the summation is for a finite number of l satisfying |k − l| ≤ 2 and the estimate for the term with the index l is only


9

a constant multiple of the bound for the term with the index k. In addition, the simple bound k∇Sk−1 ukLq ≤ k∇ukLq ≤ CkωkLq

is also used here. It is easily seen that L12 obeys the same bound as in (4.4). By Hölder’s inequality and and Bernstein’s inequality (both lower bound and upper bound parts), |L14 | ≤ C k∆k jkLq−1 q k∆k ukLq k∇Sk−1 jkL∞ X (1+ 2 )m −k 2 q k∆m jkLq ≤ C k∆k jkLq−1 k∇∆k ukLq q 2 ≤ C

kωkLq k∆k jkLq−1 q

X

2−k

m≤k−1

m≤k−1

2

2(1+ q )m k∆m jkLq .

By the divergence-free condition, ∇ · ∆l u = 0, X e l jkL∞ 2k k∆l ukLq k∆ |L15 | ≤ C k∆k jkLq−1 q l≥k−1

≤ C k∆k jkLq−1 q

X

l≥k−1

2 e l jkLq 2k−l k∇∆l ukLq 2l q k∆

X

2

kq ≤ C kωkLq k∆k jkLq−1 q 2

l≥k−1

2

2(k−l)(1− q ) k∆l jkLq .

We have thus completed the estimate for L1 . To bound L2 , we also decompose it into five terms, L2 = L21 + L22 + L23 + L24 + L25 , where L21 = −

Z

∆k j |∆k j|q−2

L22 = −

Z

∆k j |∆k j|q−2

L23 = −

Z

∆k j |∆k j|q−2 Sk b · ∇∆k ω,

L24 = −

Z

∆k j |∆k j|q−2

L25 = −

Z

∆k j |∆k j|q−2

X

[∆k , Sl−1 b · ∇]∆l ω,

X

(Sl−1 b − Sk b) · ∇∆k ∆l ω,

|k−l|≤2

|k−l|≤2

X

|k−l|≤2

X

l≥k−1

∆k (∆l b · ∇Sl−1 ω),

e l ω). ∆k (∆l b · ∇∆

The estimates of these terms are similar to those in L1 , but there are some differences, as can be seen from the bounds below. 2

q−1 k |L2 | ≤ C 2k q k∆k jkLq−1 q k∇bkLq kωkLq + C 2 k∆k jkLq kbkL∞ kωkLq X (k−l)(1− 2 ) 2 q k∇∆ bk q k∆ ωk q 2 +C 2k q k∆k jkLq−1 q l L l L l≥k−1

≤ C2

k 2q

q−1 k k∆k jkLq−1 q k∇bkLq kωkLq + C 2 k∆k jkLq kbkL∞ kωkLq .

10


To bound L3 , we decompose it into three terms as L3 = L31 + L32 + L33 , where

Z

∆k j |∆k j|q−2

L32 = 2

Z

∆k j |∆k j|q−2

L33 = 2

Z

∆k j |∆k j|q−2

L31 = 2

X

∆k (Sl−1 ∂1 b1 ∆l ∂2 u1 ),

X

∆k (∆l ∂1 b1 Sl−1 ∂2 u1 ),

|k−l|≤2

|k−l|≤2

X

l≥k−1

These terms can be bounded by

e l ∂2 u1 ). ∆k (∆l ∂1 b1 ∆

|L31 | ≤ C k∆k jkLq−1 q kSk−1 ∂1 b1 kL∞ k∆k ∂2 u1 kLq 2

kq ≤ C k∆k jkLq−1 kjkLq kωkLq q 2

where we have used the bound k∇ukLq ≤ C kωkLq and k∇bkLq ≤ C kjkLq . Clearly, L32 admits the same bound. L33 is bounded by X l2 2 q k∆l ∂1 b1 kLq k∆l ∂2 u1 kLq |L33 | ≤ C k∆k jkLq−1 q l≥k−1

≤ C kωkLq k∆k jkLq−1 q

X

2

l≥k−1

2l q k∆l jkLq .

Collecting all the estimates above, we obtain d k∆k jkLq + C1 22βk k∆k jkLq ≤ RHS(t), dt where RHS(t) denotes the bound 2

RHS(t) ≡ C kωkLq 2k q k∆k jkLq X (1+ 2 )m 2 q k∆m jkLq + C kωkLq 2−k m≤k−1

+C 2

k 2q

kωkLq

X

2

l≥k−1

2(k−l)(1− q ) k∆l jkLq

k 2q

kjkLq kωkLq + C 2k kbkL∞ kωkLq X l2 + C kωkLq 2 q k∆l jkLq . +C 2

(4.5)

l≥k−1

Integrating in time, we have k∆k j(t)k

Lq

−C1 22βk t

≤e

k∆k j(0)k

Lq

+

Z

t

2βk (t−τ )

e−C1 2

RHS(τ ) dτ.

0

For any fixed t > 0, we take the L1 -norm on [0, t] and apply Young’s inequality to obtain Z t Z t −2βk −2βk k∆k j(0)kLq + C 2 k∆k j(τ )kLq dτ ≤ C 2 RHS(τ ) dτ. 0

0


11

Multiplying by 2ks and summing over k = 0, 1, 2, · · · , we obtain Z t Z t ∞ X ks s k∆k j(τ )kLq dτ 2 kjkL1t Bq,1 = k∆−1 j(τ )kLq dτ + 0

≤ C(t) + C kj0 kBs−2β + q,1

k=0 ∞ X

0

2

k(s−2β)

Z

t

RHS(τ ) dτ,

(4.6)

0

k=0

where we have applied the global bound k∆−1 j(τ )kLq ≤ kjkLq . For the clarity of presentation, we write Z t ∞ X k(s−2β) 2 RHS(τ ) dτ ≡ M1 + M2 + M3 + M4 + M5 + M6 , k=0

0

where, according to (4.5), q M1 = C kωkL∞ t L

M2 = C kωk

q L∞ t L

q M3 = C kωkL∞ t L

∞ X

2

k(s+ q2 −2β)

X

2

k=0

Since

2 q

2

∞ X

2

(s+ q2 −2β)m

Z

0

X

2

q kbk 1 ∞ M5 = C kωkL∞ Lt L t L

M6 = C kωk

k∆k j(τ )kLq dτ,

(m−k)(1−s+2β)

k=0 m≤k−1 ∞ X k(1+s−2β)

∞ X

t

0

k=0 ∞ X

q kjkL∞ Lq M4 = C t kωkL∞ t t L

q L∞ t L

Z

2

Z

(−1+ q2 )l

l≥k−1

t

k∆m jkLq dτ,

t 0

k∆l j(τ )kLq dτ,

2

2k(s+ q −2β) ,

k=0

∞ X

2k(s+1−2β) ,

k=0

k(s−2β)

k=0

X

2

l q2

Z

t

0

l≥k−1

k∆l j(τ )kLq dτ.

− 2β < 0, we can choose an integer k0 > 0 such that 2

k0 ( q −2β) q 2 C kωkL∞ ≤ t L

1 . 16

We can split the sum in M1 into two parts, q M1 = C kωkL∞ t L

k0 X

2

k(s+ q2 −2β)

k=0

q + C kωkL∞ t L

= C(t, u0, b0 ) +

∞ X

2

Z

t 0

k∆k j(τ )kLq dτ

k(s+ q2 −2β)

k=k0 +1

1 s , kjkL1t Bq,1 16

Z

0

t

k∆k j(τ )kLq dτ

12


where we have used the bound Z t Z t k∆k j(τ )kLq dτ ≤ kj(τ )kLq dτ ≤ C. 0

0

To deal with M2 , we first realize that 1 − s + 2β > 0 and (m − k)(1 − s + 2β) < 0, we apply Young’s inequality for series convolution to obtain Z t ∞ X k(s+ q2 −2β) q k∆k j(τ )kLq dτ, M2 ≤ C kωkL∞ 2 t L 0

k=0

which obeys the same bound as M1 , namely 1 s . kjkL1t Bq,1 16 To bound M3 , we exchange the order of two sums to get Z t l+1 ∞ X X (−1+ q2 )l q q 2k(1+s−2β) . 2 dτ k∆ j(τ )k M3 = C kωkL∞ l L t L M2 ≤ C(t, u0, b0 ) +

0

l=−1

Since 1 + s − 2β < 0, we have

l+1 X k=0

k=0

2k(1+s−2β) ≤ C

and thus q M3 = C kωkL∞ t L

∞ X

2

(−1+ q2 −s)l

2

ls

Z

0

l=−1

t

k∆l j(τ )kLq dτ.

2 q

Noticing that −1 + − s < 0, we take a positive integer l0 such that 2

(−1+ q −s)l0 q 2 C kωkL∞ < t L

1 . 16

Then, M3 is bounded by M3 ≤ C(t, u0, b0 ) +

1 s . kjkL1t Bq,1 16

Since s + 1 − 2β < 0, we clearly have q kjkL∞ Lq + C kωkL∞ Lq kbkL1 L∞ . M4 + M5 ≤ C t kωkL∞ t t t L t

M6 can be bounded in a similar fashion as M3 and we have, for M6 ≤ C(t, u0, b0 ) +

2 q

< s,

1 s . kjkL1t Bq,1 16

Inserting the estimates above in (4.6), we have s ≤ C(t, u0 , b0 ) + C kj0 kBs−2β + kjkL1t Bq,1 q,1

This completes the proof of Proposition 4.1.

1 s . kjkL1t Bq,1 4


13

5. Higher regularity through an iterative process and proof of Theorem 1.1 This section explores some consequences of Proposition 4.1. In particular, we obtain ∞ global bounds for ∇j in L1t L∞ x and ω in Lt,x , which are sufficient for the proof of Theorem 1.1. We now state the proposition for high regularity. Proposition 5.1. Assume (u0, b0 ) satisfies the conditions stated in Theorem 1.1. Let (u, b) be the corresponding solution of (1.1) with β > 1. Then, for any T > 0 and t ≤ T , there exists a constant C = C(T, u0 , b0 ) such that Z t kω(t)kL∞ ≤ C. k∇j(τ )kL∞ dτ ≤ C, 0

The first step is the following integrability result, as a special consequence of Proposition 4.1. Proposition 5.2. Assume (u0, b0 ) satisfies the conditions stated in Theorem 1.1. Let (u, b) be the corresponding solution of (1.1) with β > 1. Then, for any r satisfying 2 (5.1) 2 ≤ r ≤ ∞ if β > 34 , and 2 ≤ r < if β ≤ 43 , 4 − 3β

and, for any T > 0 and t ≤ T , there exists a constant C = C(T, u0 , b0 ) such that Z t k∇j(τ )kLr dτ ≤ C. 0

We remark that the range for r, namely (5.1) is bigger than the one for q in (3.1). An immediate consequence is the global bound for kωkLr and kjkLr , as explained in the proof of Proposition 5.1. Proof of Proposition 5.2. By Bernstein’s inequality, ∞ X k∆k ∇jkLr k∇jkLr ≤ k=−1 ∞ X

≤ In the case when β > such that

4 , 3

k=−1

2

2

2k(1+ q − r −s) 2ks k∆k jkLq .

we can choose q and s satisfying (4.1), say q = 3 and s = 35 , 1+

2 − s ≤ 0, q

and consequently, for any r ∈ [2, ∞], k∇jkLr ≤ In the case when β ≤ such that

4 , 3

∞ X

k=−1

s . 2ks k∆k jkLq ≡ kjkBq,1

we can choose q and s satisfying (4.1) and r satisfying (5.1) 1+

2 2 − −s ≤0 q r

14


and again s . k∇jkLr ≤ kjkBq,1

Proposition 5.2 then follows from Proposition 4.1.

We now prove Proposition 5.1. Proof of Proposition 5.1. Proposition 5.2 allows us to obtain a global bound for kωkLr . In fact, it follows from the vorticity equation that 1d kωkrLr ≤ kbkL∞ k∇jkLr kωkLr−1 r . r dt Due to the Sobolev inequality, for q > 2, q−2

q

kjkL2q−2 kbkL∞ ≤ CkbkL2(q−1) q 2

and the time integrability of k∇jkLr from Proposition 5.2, we obtain the global bound Z t ∞ ∞ r r kω(t)kL ≤ kω0 kL + kbkLt Lx k∇jkLr dτ < ∞. 0

By going through the proof of Proposition 3.1 with q replaced by r, we can show that kj(t)kLr ≤ C.

As a consequence of the global bounds for kω(t)kLr and kj(t)kLr , we can prove Proposition 4.1 again with q replaced by r. This iterative process allows us to establish the global bound s kjkL1t Br,1 0, Z t (5.2) k∇jkL∞ dτ < ∞. 0

In fact, k∇jkL∞ ≤ =

∞ X

k=−1 ∞ X

k∆k ∇jkL∞ ≤ 2

k(1+ 2r −s)

k=−1

∞ X

2

k=−1

2k(1+ r ) k∆k jkLr

ks

2 k∆k jkLr =

∞ X

k=−1

s , 2ks k∆k jkLr ≡ kjkBr,1

where we have choose r large and s < 2β − 1 such that 2 1 + − s ≤ 0. r The global bound in (5.2) further allows us to show that which follows from the inequality

kωkL∞ < ∞,

kω(t)kL∞ ≤ kω0 kL∞ + kbk This completes the proof of Proposition 5.1.

∞ L∞ t L

(5.3) Z

0

t

k∇jkL∞ dτ.


15

Finally we prove Theorem 1.1. Proof of Theorem 1.1. The proof of Theorem 1.1 is divided into two main steps. The first step is to construct a local (in time) solution while the second step extends the local solution into a global one by making use of the global a priori bounds obtained in Proposition 5.1. The construction of a local solution is quite standard and is thus omitted here. The global bounds in Proposition 5.1 are sufficient in proving the global bound k(u, b)kH s < ∞. This completes the proof of Theorem 1.1.

Acknowledgements Cao was partially supported by NSF grant DMS-1109022. Wu was partially supported by NSF grant DMS-1209153. Wu also acknowledges the support of Henan Polytechnic University. Yuan was partially supported by the National Natural Science Foundation of China (No. 11071057) and by Innovation Scientists and Technicians Troop Construction Projects of Henan Province (No. 104100510015).

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Department of Mathematics, Florida International University, Miami, FL 33199, USA E-mail address: [email protected] 2

Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK 74078, USA E-mail address: [email protected] 3

School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo City, Henan 454000, P. R. China E-mail address: [email protected]