- Email: [email protected]
A new regularity criterion of the 2D MHD equations
Computers and Mathematics with Applications (
)
–
Contents lists available at ScienceDirect
Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa
A new regularity criterion of the 2D MHD equations Zhihong Wen Department of Mathematics and Statistics, Jiangsu Normal University, 101 Shanghai Road, Xuzhou 221116, Jiangsu, PR China
article
info
a b s t r a c t
Article history: Received 9 April 2018 Received in revised form 9 June 2018 Accepted 17 June 2018 Available online xxxx
In this paper, we establish a new regularity criterion for the two-dimensional incompressible generalized magnetohydrodynamics equations. © 2018 Elsevier Ltd. All rights reserved.
Keywords: Regularity criterion Generalized MHD
1. Introduction and main results This note is concerned with the following two-dimensional (2D) generalized magnetohydrodynamics (GMHD) equations:
⎧ ∂t u + (u · ∇ )u + (−∆)α u + ∇ p = (b · ∇ )b, ⎪ ⎪ ⎪ ⎨ ∂t b + (u · ∇ )b + (−∆)β b = (b · ∇ )u, ⎪ ∇ · u = 0, ∇ · b = 0, ⎪ ⎪ ⎩ u(x, 0) = u0 (x), b(x, 0) = b0 (x),
x ∈ R2 , t > 0, (1.1)
where α ∈ [0, 2] and β ∈ [0, 2] are real parameters, u = u(x, t) = (u1 (x, t), u2 (x, t)), b = b(x, t) = (b1 (x, t), b2 (x, t)) and p = p(x, t) denote the velocity vector, the magnetic vector and pressure scalar fields, respectively. For simplicity, we denote γ γ f (ξ ) = |ξ |γ fˆ (ξ ). The GMHD equations play a fundamental ˆ Λγ := (−∆) 2 , which is defined via the Fourier transform as Λ role in geophysics, astrophysics, cosmology and engineering (see, e.g., [1]). We point out the convention that by α = 0 we mean that there is no dissipation in (1.1)1 , and similarly β = 0 represents that there is no diffusion in (1.1)2 . Many important contributions have been made on the well-posedness result for the 2D GMHD equations (1.1), and we list only some results relevant to our concerns (see [2–9] with no intention to be complete). It is worthwhile to point out that the latest global regularity results of the 2D GMHD equations (1.1) can be summarized as (1) α > 0, β = 1;
(2) α = 0, β > 1;
(3) α = 2, β = 0,
see [2,4,5] for details (one also refers to [10,8,9] for logarithmic type dissipation). To the best of our knowledge, apart from the above mentioned cases, the global regular result for the remainder cases is not known up to date. Therefore, it is interesting to consider regularity criteria (see [11–15,7,16–18]). The target of this paper is to establish a new scaling invariant regularity criterion for the system (1.1) with the special case α = β . More precisely, our main results read as follows:
E-mail address: [email protected]. https://doi.org/10.1016/j.camwa.2018.06.026 0898-1221/© 2018 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Z. Wen, A new regularity criterion of the 2D MHD equations, Computers and Mathematics with Applications (2018), https://doi.org/10.1016/j.camwa.2018.06.026.
2
Z. Wen / Computers and Mathematics with Applications (
)
–
Theorem 1.1. Consider the system (1.1) with 0 < α = β < 1 and assume (u0 , b0 ) ∈ H s (R2 ) × H s (R2 ) with s > 2. Let (u, b) be a local smooth solution to the system (1.1) with the initial data. If the following condition holds 1
∥T (∇ u, ∇ b)(t)∥B˙20
T
∫
∞,∞
(
ln e + ∥u(t)∥2H s + ∥b(t)∥2H s
0
) 43 dt < ∞,
(1.2)
then (u, b) can be extended beyond time T , where T (∇ u, ∇ b) is given by T (∇ u, ∇ b) = 2∂1 b1 (∂2 u1 + ∂1 u2 ) − 2∂1 u1 (∂2 b1 + ∂1 b2 ).
(1.3)
Remark 1.2. At present, we are not able to prove that Theorem 1.1 is true for more general α ̸ = β . The essential reason is that we need the nice structure of the system (1.1) with α = β , namely, the two quantities G1 and G2 in (2.3) introduced in [19]. Finally, whether Theorem 1.1 holds true for the completely inviscid case (α = β = 0) also remains unknown. However, it follows from (2.4) and (2.7) that for the system (1.1) with α = β = 0, the following condition T
∫
∥T (∇ u, ∇ b)(t)∥L∞ dt < ∞ 0
implies that (u, b) can be extended beyond time T . According to the scaling invariant argument, the following is actually a scaling invariant T
∫
1
∥T (∇ u, ∇ b)(t)∥L2∞ dt < ∞. 0
Unfortunately, it is not clear to prove it. 2. The proof of Theorem 1.1 Throughout the paper, C stands for some real positive constants which may be different in each occurrence and C (a) denotes the positive constant depending on a. The basic L2 -energy estimate shows that
∥u(t)∥2L2 + ∥b(t)∥2L2 + 2
t
∫ 0
(∥Λα u(τ )∥2L2 + ∥Λα b(τ )∥2L2 ) dτ ≤ ∥u0 ∥2L2 + ∥b0 ∥2L2 .
Taking curls on the GMHD equations (1.1), it follows that the vorticity ω := ∂x1 u2 − ∂x2 u1 and the current density j := ∂x1 b2 − ∂x2 b1 satisfy
∂t ω + (u · ∇ )ω + Λ2α ω − (b · ∇ )j = 0,
(2.1)
∂t j + (u · ∇ )j + Λ2α j − (b · ∇ )ω = T (∇ u, ∇ b),
(2.2)
where T (∇ u, ∇ b) is given by (1.3). Following [19], we introduce G1 := ω + j,
G2 := ω − j,
(2.3)
which obey
∂t G1 + (u · ∇ )G1 + Λ2α G1 − (b · ∇ )G1 = T (∇ u, ∇ b),
(2.4)
∂t G2 + (u · ∇ )G2 + Λ2α G2 + (b · ∇ )G2 = −T (∇ u, ∇ b).
(2.5)
Multiplying Eqs. (2.4) and (2.5) by |G1 |p−2 G1 and |G2 |p−2 G2 , respectively, and using the fact
∫
2α
R2
Λ G1 (|G1 |
p−2
∫
G1 ) dx ≥ 0,
R2
Λ2α G2 (|G2 |p−2 G2 ) dx ≥ 0,
we have 1 d p dt
∫
p
p
(∥G1 (t)∥Lp + ∥G2 (t)∥Lp )
|T (∇ u, ∇ b)| |G1 |
p−1
≤ R2
∫ dx + R2
|T (∇ u, ∇ b)| |G2 |p−1 dx
≤ C ∥T (∇ u, ∇ b)∥Lp (∥G1 ∥pLp−1 + ∥G2 ∥pLp−1 ) 1 1 = C |T (∇ u, ∇ b)| 2 |T (∇ u, ∇ b)| 2 (∥G1 ∥pLp−1 + ∥G2 ∥pLp−1 ) Lp
Please cite this article in press as: Z. Wen, A new regularity criterion of the 2D MHD equations, Computers and Mathematics with Applications (2018), https://doi.org/10.1016/j.camwa.2018.06.026.
Z. Wen / Computers and Mathematics with Applications (
)
–
3
1 1 ≤ C (p)|T (∇ u, ∇ b)| 2 Lp |T (∇ u, ∇ b)| 2 BMO (∥G1 ∥pLp−1 + ∥G2 ∥pLp−1 ) 1
1
1
p−1 p−1 2 (∥G1 ∥Lp + ∥G2 ∥Lp ) ≤ C (p)∥∇ u∥L2p ∥∇ b∥L2p ∥T (∇ u, ∇ b)∥BMO 1
1
1
p−1 p−1 2 ≤ C (p)∥ω∥L2p ∥j∥L2p ∥T (∇ u, ∇ b)∥BMO (∥G1 ∥Lp + ∥G2 ∥Lp ) 1
p−1 p−1 2 ≤ C (p)(∥G1 ∥Lp + ∥G2 ∥Lp )∥T (∇ u, ∇ b)∥BMO (∥G1 ∥Lp + ∥G2 ∥Lp ) 1
p p 2 ≤ C (p)∥T (∇ u, ∇ b)∥BMO (∥G1 ∥Lp + ∥G2 ∥Lp ),
(2.6)
where we have used the bilinear estimate
∥fg ∥Lr ≤ C (r)(∥f ∥Lr ∥g ∥BMO + ∥g ∥Lr ∥f ∥BMO ),
1
and the following fact (see [20, Exercises 3.1.3])
γ |f |
γ
BMO
≤ 2∥f ∥BMO ,
∀ 0 < γ ≤ 1.
Consequently, one has d
p
1
p
p
p
2 (∥G1 (t)∥Lp + ∥G2 (t)∥Lp ) ≤ C (p)∥T (∇ u, ∇ b)∥BMO (∥G1 ∥Lp + ∥G2 ∥Lp ).
(2.7)
dt Thanks to
1
∥T (∇ u, ∇ b)(τ )∥B˙20
T
∫
∞,∞
ln e + ∥u(τ )∥
(
0
2 Hs
+ ∥b(τ )∥2H s
) 43 dτ < ∞,
one may check that for sufficiently small number ϵ > 0, there exists T0 = T0 (ϵ ) close enough to T such that 1
∫
∥T (∇ u, ∇ b)(τ )∥B˙20
T
∞,∞
T0
ln e + ∥u(τ )∥
(
2 Hs
+ ∥b(τ )∥2H s
) 43 dτ ≤ ϵ.
We now denote Y (t) := max (∥u(τ )∥2H s + ∥b(τ )∥2H s ), for s > 2, τ ∈[T0 , t ]
then it is easy to check that Y (t) is a monotonically nondecreasing function. Let us state the following logarithmic-type Sobolev inequality (see [21, Theorem 2.1])
∥f ∥B˙ 0
∞,2
( ) √ ≤ C 1 + ∥f ∥B˙ 0∞, ∞ ln(e + ∥f ∥H˙ s1 + ∥f ∥H˙ s2 ) ,
(2.8)
where 0 < s1 < 1 < s2 . This implies
∥f ∥B˙ 0
∞,2
) ( √ s > 1. ≤ C 1 + ∥f ∥B˙ 0∞, ∞ ln(e + ∥f ∥H˜s ) , ˜
(2.9)
Now recalling the following fact
∥f ∥BMO ≤ ∥f ∥B˙ 0
∞,2
and applying (2.9) to (2.7), we deduce
∥G1 (t)∥pLp + ∥G2 (t)∥pLp
∫ t [ ] 1 2 ≤ (∥G1 (T0 )∥pLp + ∥G2 (T0 )∥pLp ) exp C (p) ∥T (∇ u, ∇ b)(τ )∥BMO dτ T0
t
[
∫
[
∫ 0t (
[
∫
∥T (∇ u, ∇ b)(τ )∥
≤ C exp C (p)
dτ
]
∞,2
T
≤ C exp C (p)
1 2 B˙ 0
1
1 + ∥T (∇ u, ∇ b)(τ )∥ ˙20
B∞,∞
T0 t
∥T (∇ u, ∇ b)(τ )∥
≤ C exp C (p) T0
1 2 ˙B0
∞,∞
(
ln(e + ∥T (∇ u, ∇ b)(τ )∥H˜s )
ln(e + ∥u(τ )∥2H s + ∥b(τ )∥2H s )
(
) 41
) 41 ) dτ
dτ
]
]
1
[
∫
t
∥T (∇ u, ∇ b)(τ )∥B˙20
∞,∞
= C exp C (p) T0
ln(e + ∥u(τ )∥2H s + ∥b(τ )∥2H s )
ln e + ∥u(τ )∥2H s + ∥b(τ )∥2H s dτ
( 3 4
)
]
Please cite this article in press as: Z. Wen, A new regularity criterion of the 2D MHD equations, Computers and Mathematics with Applications (2018), https://doi.org/10.1016/j.camwa.2018.06.026.
4
Z. Wen / Computers and Mathematics with Applications (
)
–
1
∥T (∇ u, ∇ b)(τ )∥B˙20
t
∫
[
ln e + Y (τ ) dτ
∞,∞
≤ C exp C (p)
(
ln e + ∥u(τ )∥2H s + ∥b(τ )∥2H s
(
T0
)
3 4
)
]
1
[
≤ C exp C (p)
∥T (∇ u, ∇ b)(τ )∥B˙20
t
(∫
) (
dτ ln e + Y (t)
∞,∞
T0
ln e + ∥u(τ )∥2H s + ∥b(τ )∥2H s
(
) 34
)]
≤ C (e + Y (t))C (p)ϵ ,
(2.10)
where we have used
∥T (∇ u, ∇ b)(τ )∥H˜s ≤ ≤ ≤ ≤
C (∥∇ u(τ )∥L∞ ∥∇ b(τ )∥H˜s + ∥∇ b(τ )∥L∞ ∥∇ u(τ )∥H˜s ) C (∥u(τ )∥H s ∥∇ b(τ )∥H˜s + ∥b(τ )∥H s ∥∇ u(τ )∥H˜s ) C ∥u(τ )∥H s ∥b(τ )∥H s C (∥u(τ )∥2H s + ∥b(τ )∥2H s )
with s > 2 and˜ s = s − 1 > 1. Therefore, it follows that for any T0 ≤ t < T
∥G1 (t)∥pLp + ∥G2 (t)∥pLp ≤ C (e + Y (t))C (p)ϵ . The above estimate is equal to
∥G1 (t)∥Lp + ∥G2 (t)∥Lp ≤ C (e + Y (t))C (p)ϵ , ˜
which together with (2.3) further leads to
∥ω(t)∥Lp + ∥j(t)∥Lp ≤ C (e + Y (t))C (p)ϵ .
(2.11)
˜
We apply J := (I + Λ) to the Eqs. (1.1)1 and (1.1)2 , take the L inner product of the obtained equations with J u and J s b, respectively, and add them up to deduce s
s
1 d 2 dt ∫
2
s
(∥J s u(t)∥2L2 + ∥J s b(t)∥2L2 ) + ∥J s+α u∥2L2 + ∥J s+α b∥2L2
= − 2 ∫R
[J , u · ∇]b · J b dx +
− R2
s
s
∫
[J s , u · ∇]u · J s u dx +
2 ∫R
[J s , b · ∇]u · J s b dx
R2
:= F1 + F2 + F3 + F4 .
[J s , b · ∇]b · J s u dx (2.12)
Making use of the following commutator estimates estimate (see [22])
∥[J s , f ]g ∥Lp ≤ C (∥∇ f ∥Lp1 ∥J s−1 g ∥Lp2 + ∥J s f ∥Lp3 ∥g ∥Lp4 ), with s > 0, p1 , p4 ∈ (1, ∞], p2 , p3 ∈ (1, ∞) satisfying
F1 , F2 ≤ C ∥[J , u · ∇]b∥ s
2p
s
∥ J b∥
L p+1
s
≤ C (∥∇ u∥Lp ∥J b∥
2p
L p−1
× ∥b∥ ≤
1
∥
s
u∥
2p
L p−1
s
)∥J b∥
1 p2
=
2p
1 p3
∥ J b∥
L p+1
we can show
2p
L p−1
2p
L p−1 α p−1 (s+α )p
+ ∥∇ b∥Lp ∥u∥L2
s
1 , p4
+
sp+1
∥J s+α u∥L(s2+α)p )
∥
(s+α )p
1
(s+α )p
8
α p−1
2p L p−1
+
∥J s+α u∥2L2 + ∥J s+α b∥2L2 + C ∥ω∥Lαp p−1 + C ∥j∥Lαp p−1 ,
8 where we have used
∥J s f ∥
∥
1 p1
+ C ∥[J , b · ∇]u∥
+ ∥∇ ∥ ∥
∥
=
s
b Lp J 2p L p−1 α p−1 sp+1 (s+α )p (s+α )p u Lp b L2 J s+α b 2 L α p−1 sp+1 (s+α )p (s+α )p s+α J b L2 L2
≤ C (∥∇ ∥ ∥ ∥
1 p
sp+1
≤ C ∥f ∥L(s2+α)p ∥J s+α f ∥L(s2+α)p ,
p>
1
α
.
Similarly, the remainder two terms can be bounded by
F3 ≤ F4 ≤
1 8 1 8
(s+α )p
∥J s+α u∥2L2 + C ∥ω∥Lαp p−1 , 1
(s+α )p
∥J s+α u∥2L2 + ∥J s+α b∥2L2 + C ∥j∥Lαp p−1 . 8
Please cite this article in press as: Z. Wen, A new regularity criterion of the 2D MHD equations, Computers and Mathematics with Applications (2018), https://doi.org/10.1016/j.camwa.2018.06.026.
Z. Wen / Computers and Mathematics with Applications (
)
–
5
Summing up the above estimates, one obtains d
(s+α )p α p−1
(∥J s u(t)∥2L2 + ∥J s b(t)∥2L2 ) ≤ C ∥ω∥Lp
(s+α )p
+ C ∥j∥Lαp p−1 .
dt Now we integrate (2.13) over interval (T0 , t) and use the monotonicity of Y (t) as well as (2.11) to conclude
∫
t
∫
T0 t
e + Y (t) ≤ e + X (T0 ) + C
(s+α )p α p−1
(∥ω(τ )∥Lp
(
≤ e + G(T0 ) + C
e + Y (τ )
(2.13)
(s+α )p
+ ∥j(τ )∥Lαp p−1 ) dτ
)˜ C (p)ϵ ) (s+α α p−1
dτ .
(2.14)
T0
After fixing 0 < ϵ ≤
α p−1 , (s+α )˜ C (p)
∫
it follows that
t
(
e + Y (t) ≤ C + C
e + Y (τ ) dτ .
)
(2.15)
T0
The standard Gronwall type inequality implies that Y (t) remains bounded for any t ∈ [0, T ]. Therefore, we obtain sup (∥J s u(t)∥2L2 + ∥J s b(t)∥2L2 ) ≤ C .
0≤t ≤T
Thus, the proof of Theorem 1.1 is completed. Acknowledgments The author would like to express her deep thanks to the anonymous referees and the associated editor for their invaluable comments and suggestions. This work was supported by the Natural Science Foundation of Colleges of Jiangsu Province (No. 17KJD110002), The Foundation Project of Jiangsu Normal University (No. 16XLR033). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
P.A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, Cambridge, England, 2001. C. Cao, J. Wu, B. Yuan, The 2D incompressible magnetohydrodynamics equations with only magnetic diffusion, SIAM J. Math. Anal. 46 (2014) 588–602. C. Cao, J. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math. 226 (2011) 1803–1822. J. Fan, H. Malaikah, S. Monaquel, G. Nakamura, Y. Zhou, Global Cauchy problem of 2D generalized MHD equations, Monatsh. Math. 175 (2014) 127–131. Q. Jiu, J. Zhao, Global regularity of 2D generalized MHD equations with magnetic diffusion, Z. Angew. Math. Phys. 66 (2015) 677–687. C.V. Tran, X. Yu, Z. Zhai, On global regularity of 2D generalized magnetodydrodynamics equations, J. Differential Equations 254 (2013) 4194–4216. J. Wu, Generalized MHD equations, J. Differential Equations 195 (2003) 284–312. Z. Ye, Remark on the global regularity of 2D MHD equations with almost Laplacian magnetic diffusion, J. Evol. Equ. (2017). http://dx.doi.org/10.1007/ s00028-017-0421-3. B. Yuan, J. Zhao, Global regularity of 2D almost resistive MHD Equations, Nonlinear Anal. RWA 41 (2018) 53–65. L. Agelas, Global regularity for logarithmically critical 2D MHD equations with zero viscosity, Monatsh. Math. 181 (2016) 245–266. L. Agelas, Beyond the BKM criterion for the 2D resistive magnetohydrodynamic equations, Anal. PDE 11 (4) (2018) 899–918. J. Fan, X. Jia, G. Nakamura, Y. Zhou, On well-posedness and blowup criteria for the magnetohydrodynamics with the Hall and ion-slip effects, Z. Angew. Math. Phys. 66 (4) (2015) 1695–1706. X. Jia, Y. Zhou, Regularity criteria for the 3D MHD equations via partial derivatives, II, Kinet. Relat. Models 7 (2) (2014) 291–304. Z. Jiang, Y. Wang, Y. Zhou, On regularity criteria for the 2D generalized MHD system, J. Math. Fluid Mech. 18 (2) (2016) 331–341. Z. Lei, Y. Zhou, BKM’s criterion and global weak solutions for magnetohydrodynamics with zero viscosity, Discrete Contin. Dyn. Syst. 25 (2009) 575–583. Z. Ye, Two regularity criteria to the 2D generalized MHD equations with zero magnetic diffusivity, J. Math. Anal. Appl. 420 (2014) 954–971. Z. Ye, Remarks on regularity criteria for 2D generalized MHD equations, Rocky Mountain J. Math. 47 (2017) 1333–1353. Y. Zhou, J. Fan, A regularity criterion for the 2D MHD system wity zeao magnetic diffusivity, J. Math. Anal. Appl. 378 (2011) 169–172. Z. Ye, On regularity criteria of the 2D generalized MHD equations, J. Math. Anal. Appl. 463 (2) (2018) 989–1005. L. Grafakos, Modern Fourier Analysis, in: Graduate Texts in Mathematics, vol. 250, SpringerVerlag, New York, 2014. H. Kozono, T. Ogawa, Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Math. Z. 242 (2002) 251–278. T. Kato, G. Ponce, Commutator estimates and the Euler and Navier–Stokes equations, Comm. Pure Appl. Math. 41 (1988) 891–907.
Please cite this article in press as: Z. Wen, A new regularity criterion of the 2D MHD equations, Computers and Mathematics with Applications (2018), https://doi.org/10.1016/j.camwa.2018.06.026.
)
–
Contents lists available at ScienceDirect
Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa
A new regularity criterion of the 2D MHD equations Zhihong Wen Department of Mathematics and Statistics, Jiangsu Normal University, 101 Shanghai Road, Xuzhou 221116, Jiangsu, PR China
article
info
a b s t r a c t
Article history: Received 9 April 2018 Received in revised form 9 June 2018 Accepted 17 June 2018 Available online xxxx
In this paper, we establish a new regularity criterion for the two-dimensional incompressible generalized magnetohydrodynamics equations. © 2018 Elsevier Ltd. All rights reserved.
Keywords: Regularity criterion Generalized MHD
1. Introduction and main results This note is concerned with the following two-dimensional (2D) generalized magnetohydrodynamics (GMHD) equations:
⎧ ∂t u + (u · ∇ )u + (−∆)α u + ∇ p = (b · ∇ )b, ⎪ ⎪ ⎪ ⎨ ∂t b + (u · ∇ )b + (−∆)β b = (b · ∇ )u, ⎪ ∇ · u = 0, ∇ · b = 0, ⎪ ⎪ ⎩ u(x, 0) = u0 (x), b(x, 0) = b0 (x),
x ∈ R2 , t > 0, (1.1)
where α ∈ [0, 2] and β ∈ [0, 2] are real parameters, u = u(x, t) = (u1 (x, t), u2 (x, t)), b = b(x, t) = (b1 (x, t), b2 (x, t)) and p = p(x, t) denote the velocity vector, the magnetic vector and pressure scalar fields, respectively. For simplicity, we denote γ γ f (ξ ) = |ξ |γ fˆ (ξ ). The GMHD equations play a fundamental ˆ Λγ := (−∆) 2 , which is defined via the Fourier transform as Λ role in geophysics, astrophysics, cosmology and engineering (see, e.g., [1]). We point out the convention that by α = 0 we mean that there is no dissipation in (1.1)1 , and similarly β = 0 represents that there is no diffusion in (1.1)2 . Many important contributions have been made on the well-posedness result for the 2D GMHD equations (1.1), and we list only some results relevant to our concerns (see [2–9] with no intention to be complete). It is worthwhile to point out that the latest global regularity results of the 2D GMHD equations (1.1) can be summarized as (1) α > 0, β = 1;
(2) α = 0, β > 1;
(3) α = 2, β = 0,
see [2,4,5] for details (one also refers to [10,8,9] for logarithmic type dissipation). To the best of our knowledge, apart from the above mentioned cases, the global regular result for the remainder cases is not known up to date. Therefore, it is interesting to consider regularity criteria (see [11–15,7,16–18]). The target of this paper is to establish a new scaling invariant regularity criterion for the system (1.1) with the special case α = β . More precisely, our main results read as follows:
E-mail address: [email protected]. https://doi.org/10.1016/j.camwa.2018.06.026 0898-1221/© 2018 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Z. Wen, A new regularity criterion of the 2D MHD equations, Computers and Mathematics with Applications (2018), https://doi.org/10.1016/j.camwa.2018.06.026.
2
Z. Wen / Computers and Mathematics with Applications (
)
–
Theorem 1.1. Consider the system (1.1) with 0 < α = β < 1 and assume (u0 , b0 ) ∈ H s (R2 ) × H s (R2 ) with s > 2. Let (u, b) be a local smooth solution to the system (1.1) with the initial data. If the following condition holds 1
∥T (∇ u, ∇ b)(t)∥B˙20
T
∫
∞,∞
(
ln e + ∥u(t)∥2H s + ∥b(t)∥2H s
0
) 43 dt < ∞,
(1.2)
then (u, b) can be extended beyond time T , where T (∇ u, ∇ b) is given by T (∇ u, ∇ b) = 2∂1 b1 (∂2 u1 + ∂1 u2 ) − 2∂1 u1 (∂2 b1 + ∂1 b2 ).
(1.3)
Remark 1.2. At present, we are not able to prove that Theorem 1.1 is true for more general α ̸ = β . The essential reason is that we need the nice structure of the system (1.1) with α = β , namely, the two quantities G1 and G2 in (2.3) introduced in [19]. Finally, whether Theorem 1.1 holds true for the completely inviscid case (α = β = 0) also remains unknown. However, it follows from (2.4) and (2.7) that for the system (1.1) with α = β = 0, the following condition T
∫
∥T (∇ u, ∇ b)(t)∥L∞ dt < ∞ 0
implies that (u, b) can be extended beyond time T . According to the scaling invariant argument, the following is actually a scaling invariant T
∫
1
∥T (∇ u, ∇ b)(t)∥L2∞ dt < ∞. 0
Unfortunately, it is not clear to prove it. 2. The proof of Theorem 1.1 Throughout the paper, C stands for some real positive constants which may be different in each occurrence and C (a) denotes the positive constant depending on a. The basic L2 -energy estimate shows that
∥u(t)∥2L2 + ∥b(t)∥2L2 + 2
t
∫ 0
(∥Λα u(τ )∥2L2 + ∥Λα b(τ )∥2L2 ) dτ ≤ ∥u0 ∥2L2 + ∥b0 ∥2L2 .
Taking curls on the GMHD equations (1.1), it follows that the vorticity ω := ∂x1 u2 − ∂x2 u1 and the current density j := ∂x1 b2 − ∂x2 b1 satisfy
∂t ω + (u · ∇ )ω + Λ2α ω − (b · ∇ )j = 0,
(2.1)
∂t j + (u · ∇ )j + Λ2α j − (b · ∇ )ω = T (∇ u, ∇ b),
(2.2)
where T (∇ u, ∇ b) is given by (1.3). Following [19], we introduce G1 := ω + j,
G2 := ω − j,
(2.3)
which obey
∂t G1 + (u · ∇ )G1 + Λ2α G1 − (b · ∇ )G1 = T (∇ u, ∇ b),
(2.4)
∂t G2 + (u · ∇ )G2 + Λ2α G2 + (b · ∇ )G2 = −T (∇ u, ∇ b).
(2.5)
Multiplying Eqs. (2.4) and (2.5) by |G1 |p−2 G1 and |G2 |p−2 G2 , respectively, and using the fact
∫
2α
R2
Λ G1 (|G1 |
p−2
∫
G1 ) dx ≥ 0,
R2
Λ2α G2 (|G2 |p−2 G2 ) dx ≥ 0,
we have 1 d p dt
∫
p
p
(∥G1 (t)∥Lp + ∥G2 (t)∥Lp )
|T (∇ u, ∇ b)| |G1 |
p−1
≤ R2
∫ dx + R2
|T (∇ u, ∇ b)| |G2 |p−1 dx
≤ C ∥T (∇ u, ∇ b)∥Lp (∥G1 ∥pLp−1 + ∥G2 ∥pLp−1 ) 1 1 = C |T (∇ u, ∇ b)| 2 |T (∇ u, ∇ b)| 2 (∥G1 ∥pLp−1 + ∥G2 ∥pLp−1 ) Lp
Please cite this article in press as: Z. Wen, A new regularity criterion of the 2D MHD equations, Computers and Mathematics with Applications (2018), https://doi.org/10.1016/j.camwa.2018.06.026.
Z. Wen / Computers and Mathematics with Applications (
)
–
3
1 1 ≤ C (p)|T (∇ u, ∇ b)| 2 Lp |T (∇ u, ∇ b)| 2 BMO (∥G1 ∥pLp−1 + ∥G2 ∥pLp−1 ) 1
1
1
p−1 p−1 2 (∥G1 ∥Lp + ∥G2 ∥Lp ) ≤ C (p)∥∇ u∥L2p ∥∇ b∥L2p ∥T (∇ u, ∇ b)∥BMO 1
1
1
p−1 p−1 2 ≤ C (p)∥ω∥L2p ∥j∥L2p ∥T (∇ u, ∇ b)∥BMO (∥G1 ∥Lp + ∥G2 ∥Lp ) 1
p−1 p−1 2 ≤ C (p)(∥G1 ∥Lp + ∥G2 ∥Lp )∥T (∇ u, ∇ b)∥BMO (∥G1 ∥Lp + ∥G2 ∥Lp ) 1
p p 2 ≤ C (p)∥T (∇ u, ∇ b)∥BMO (∥G1 ∥Lp + ∥G2 ∥Lp ),
(2.6)
where we have used the bilinear estimate
∥fg ∥Lr ≤ C (r)(∥f ∥Lr ∥g ∥BMO + ∥g ∥Lr ∥f ∥BMO ),
1
and the following fact (see [20, Exercises 3.1.3])
γ |f |
γ
BMO
≤ 2∥f ∥BMO ,
∀ 0 < γ ≤ 1.
Consequently, one has d
p
1
p
p
p
2 (∥G1 (t)∥Lp + ∥G2 (t)∥Lp ) ≤ C (p)∥T (∇ u, ∇ b)∥BMO (∥G1 ∥Lp + ∥G2 ∥Lp ).
(2.7)
dt Thanks to
1
∥T (∇ u, ∇ b)(τ )∥B˙20
T
∫
∞,∞
ln e + ∥u(τ )∥
(
0
2 Hs
+ ∥b(τ )∥2H s
) 43 dτ < ∞,
one may check that for sufficiently small number ϵ > 0, there exists T0 = T0 (ϵ ) close enough to T such that 1
∫
∥T (∇ u, ∇ b)(τ )∥B˙20
T
∞,∞
T0
ln e + ∥u(τ )∥
(
2 Hs
+ ∥b(τ )∥2H s
) 43 dτ ≤ ϵ.
We now denote Y (t) := max (∥u(τ )∥2H s + ∥b(τ )∥2H s ), for s > 2, τ ∈[T0 , t ]
then it is easy to check that Y (t) is a monotonically nondecreasing function. Let us state the following logarithmic-type Sobolev inequality (see [21, Theorem 2.1])
∥f ∥B˙ 0
∞,2
( ) √ ≤ C 1 + ∥f ∥B˙ 0∞, ∞ ln(e + ∥f ∥H˙ s1 + ∥f ∥H˙ s2 ) ,
(2.8)
where 0 < s1 < 1 < s2 . This implies
∥f ∥B˙ 0
∞,2
) ( √ s > 1. ≤ C 1 + ∥f ∥B˙ 0∞, ∞ ln(e + ∥f ∥H˜s ) , ˜
(2.9)
Now recalling the following fact
∥f ∥BMO ≤ ∥f ∥B˙ 0
∞,2
and applying (2.9) to (2.7), we deduce
∥G1 (t)∥pLp + ∥G2 (t)∥pLp
∫ t [ ] 1 2 ≤ (∥G1 (T0 )∥pLp + ∥G2 (T0 )∥pLp ) exp C (p) ∥T (∇ u, ∇ b)(τ )∥BMO dτ T0
t
[
∫
[
∫ 0t (
[
∫
∥T (∇ u, ∇ b)(τ )∥
≤ C exp C (p)
dτ
]
∞,2
T
≤ C exp C (p)
1 2 B˙ 0
1
1 + ∥T (∇ u, ∇ b)(τ )∥ ˙20
B∞,∞
T0 t
∥T (∇ u, ∇ b)(τ )∥
≤ C exp C (p) T0
1 2 ˙B0
∞,∞
(
ln(e + ∥T (∇ u, ∇ b)(τ )∥H˜s )
ln(e + ∥u(τ )∥2H s + ∥b(τ )∥2H s )
(
) 41
) 41 ) dτ
dτ
]
]
1
[
∫
t
∥T (∇ u, ∇ b)(τ )∥B˙20
∞,∞
= C exp C (p) T0
ln(e + ∥u(τ )∥2H s + ∥b(τ )∥2H s )
ln e + ∥u(τ )∥2H s + ∥b(τ )∥2H s dτ
( 3 4
)
]
Please cite this article in press as: Z. Wen, A new regularity criterion of the 2D MHD equations, Computers and Mathematics with Applications (2018), https://doi.org/10.1016/j.camwa.2018.06.026.
4
Z. Wen / Computers and Mathematics with Applications (
)
–
1
∥T (∇ u, ∇ b)(τ )∥B˙20
t
∫
[
ln e + Y (τ ) dτ
∞,∞
≤ C exp C (p)
(
ln e + ∥u(τ )∥2H s + ∥b(τ )∥2H s
(
T0
)
3 4
)
]
1
[
≤ C exp C (p)
∥T (∇ u, ∇ b)(τ )∥B˙20
t
(∫
) (
dτ ln e + Y (t)
∞,∞
T0
ln e + ∥u(τ )∥2H s + ∥b(τ )∥2H s
(
) 34
)]
≤ C (e + Y (t))C (p)ϵ ,
(2.10)
where we have used
∥T (∇ u, ∇ b)(τ )∥H˜s ≤ ≤ ≤ ≤
C (∥∇ u(τ )∥L∞ ∥∇ b(τ )∥H˜s + ∥∇ b(τ )∥L∞ ∥∇ u(τ )∥H˜s ) C (∥u(τ )∥H s ∥∇ b(τ )∥H˜s + ∥b(τ )∥H s ∥∇ u(τ )∥H˜s ) C ∥u(τ )∥H s ∥b(τ )∥H s C (∥u(τ )∥2H s + ∥b(τ )∥2H s )
with s > 2 and˜ s = s − 1 > 1. Therefore, it follows that for any T0 ≤ t < T
∥G1 (t)∥pLp + ∥G2 (t)∥pLp ≤ C (e + Y (t))C (p)ϵ . The above estimate is equal to
∥G1 (t)∥Lp + ∥G2 (t)∥Lp ≤ C (e + Y (t))C (p)ϵ , ˜
which together with (2.3) further leads to
∥ω(t)∥Lp + ∥j(t)∥Lp ≤ C (e + Y (t))C (p)ϵ .
(2.11)
˜
We apply J := (I + Λ) to the Eqs. (1.1)1 and (1.1)2 , take the L inner product of the obtained equations with J u and J s b, respectively, and add them up to deduce s
s
1 d 2 dt ∫
2
s
(∥J s u(t)∥2L2 + ∥J s b(t)∥2L2 ) + ∥J s+α u∥2L2 + ∥J s+α b∥2L2
= − 2 ∫R
[J , u · ∇]b · J b dx +
− R2
s
s
∫
[J s , u · ∇]u · J s u dx +
2 ∫R
[J s , b · ∇]u · J s b dx
R2
:= F1 + F2 + F3 + F4 .
[J s , b · ∇]b · J s u dx (2.12)
Making use of the following commutator estimates estimate (see [22])
∥[J s , f ]g ∥Lp ≤ C (∥∇ f ∥Lp1 ∥J s−1 g ∥Lp2 + ∥J s f ∥Lp3 ∥g ∥Lp4 ), with s > 0, p1 , p4 ∈ (1, ∞], p2 , p3 ∈ (1, ∞) satisfying
F1 , F2 ≤ C ∥[J , u · ∇]b∥ s
2p
s
∥ J b∥
L p+1
s
≤ C (∥∇ u∥Lp ∥J b∥
2p
L p−1
× ∥b∥ ≤
1
∥
s
u∥
2p
L p−1
s
)∥J b∥
1 p2
=
2p
1 p3
∥ J b∥
L p+1
we can show
2p
L p−1
2p
L p−1 α p−1 (s+α )p
+ ∥∇ b∥Lp ∥u∥L2
s
1 , p4
+
sp+1
∥J s+α u∥L(s2+α)p )
∥
(s+α )p
1
(s+α )p
8
α p−1
2p L p−1
+
∥J s+α u∥2L2 + ∥J s+α b∥2L2 + C ∥ω∥Lαp p−1 + C ∥j∥Lαp p−1 ,
8 where we have used
∥J s f ∥
∥
1 p1
+ C ∥[J , b · ∇]u∥
+ ∥∇ ∥ ∥
∥
=
s
b Lp J 2p L p−1 α p−1 sp+1 (s+α )p (s+α )p u Lp b L2 J s+α b 2 L α p−1 sp+1 (s+α )p (s+α )p s+α J b L2 L2
≤ C (∥∇ ∥ ∥ ∥
1 p
sp+1
≤ C ∥f ∥L(s2+α)p ∥J s+α f ∥L(s2+α)p ,
p>
1
α
.
Similarly, the remainder two terms can be bounded by
F3 ≤ F4 ≤
1 8 1 8
(s+α )p
∥J s+α u∥2L2 + C ∥ω∥Lαp p−1 , 1
(s+α )p
∥J s+α u∥2L2 + ∥J s+α b∥2L2 + C ∥j∥Lαp p−1 . 8
Please cite this article in press as: Z. Wen, A new regularity criterion of the 2D MHD equations, Computers and Mathematics with Applications (2018), https://doi.org/10.1016/j.camwa.2018.06.026.
Z. Wen / Computers and Mathematics with Applications (
)
–
5
Summing up the above estimates, one obtains d
(s+α )p α p−1
(∥J s u(t)∥2L2 + ∥J s b(t)∥2L2 ) ≤ C ∥ω∥Lp
(s+α )p
+ C ∥j∥Lαp p−1 .
dt Now we integrate (2.13) over interval (T0 , t) and use the monotonicity of Y (t) as well as (2.11) to conclude
∫
t
∫
T0 t
e + Y (t) ≤ e + X (T0 ) + C
(s+α )p α p−1
(∥ω(τ )∥Lp
(
≤ e + G(T0 ) + C
e + Y (τ )
(2.13)
(s+α )p
+ ∥j(τ )∥Lαp p−1 ) dτ
)˜ C (p)ϵ ) (s+α α p−1
dτ .
(2.14)
T0
After fixing 0 < ϵ ≤
α p−1 , (s+α )˜ C (p)
∫
it follows that
t
(
e + Y (t) ≤ C + C
e + Y (τ ) dτ .
)
(2.15)
T0
The standard Gronwall type inequality implies that Y (t) remains bounded for any t ∈ [0, T ]. Therefore, we obtain sup (∥J s u(t)∥2L2 + ∥J s b(t)∥2L2 ) ≤ C .
0≤t ≤T
Thus, the proof of Theorem 1.1 is completed. Acknowledgments The author would like to express her deep thanks to the anonymous referees and the associated editor for their invaluable comments and suggestions. This work was supported by the Natural Science Foundation of Colleges of Jiangsu Province (No. 17KJD110002), The Foundation Project of Jiangsu Normal University (No. 16XLR033). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
P.A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, Cambridge, England, 2001. C. Cao, J. Wu, B. Yuan, The 2D incompressible magnetohydrodynamics equations with only magnetic diffusion, SIAM J. Math. Anal. 46 (2014) 588–602. C. Cao, J. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math. 226 (2011) 1803–1822. J. Fan, H. Malaikah, S. Monaquel, G. Nakamura, Y. Zhou, Global Cauchy problem of 2D generalized MHD equations, Monatsh. Math. 175 (2014) 127–131. Q. Jiu, J. Zhao, Global regularity of 2D generalized MHD equations with magnetic diffusion, Z. Angew. Math. Phys. 66 (2015) 677–687. C.V. Tran, X. Yu, Z. Zhai, On global regularity of 2D generalized magnetodydrodynamics equations, J. Differential Equations 254 (2013) 4194–4216. J. Wu, Generalized MHD equations, J. Differential Equations 195 (2003) 284–312. Z. Ye, Remark on the global regularity of 2D MHD equations with almost Laplacian magnetic diffusion, J. Evol. Equ. (2017). http://dx.doi.org/10.1007/ s00028-017-0421-3. B. Yuan, J. Zhao, Global regularity of 2D almost resistive MHD Equations, Nonlinear Anal. RWA 41 (2018) 53–65. L. Agelas, Global regularity for logarithmically critical 2D MHD equations with zero viscosity, Monatsh. Math. 181 (2016) 245–266. L. Agelas, Beyond the BKM criterion for the 2D resistive magnetohydrodynamic equations, Anal. PDE 11 (4) (2018) 899–918. J. Fan, X. Jia, G. Nakamura, Y. Zhou, On well-posedness and blowup criteria for the magnetohydrodynamics with the Hall and ion-slip effects, Z. Angew. Math. Phys. 66 (4) (2015) 1695–1706. X. Jia, Y. Zhou, Regularity criteria for the 3D MHD equations via partial derivatives, II, Kinet. Relat. Models 7 (2) (2014) 291–304. Z. Jiang, Y. Wang, Y. Zhou, On regularity criteria for the 2D generalized MHD system, J. Math. Fluid Mech. 18 (2) (2016) 331–341. Z. Lei, Y. Zhou, BKM’s criterion and global weak solutions for magnetohydrodynamics with zero viscosity, Discrete Contin. Dyn. Syst. 25 (2009) 575–583. Z. Ye, Two regularity criteria to the 2D generalized MHD equations with zero magnetic diffusivity, J. Math. Anal. Appl. 420 (2014) 954–971. Z. Ye, Remarks on regularity criteria for 2D generalized MHD equations, Rocky Mountain J. Math. 47 (2017) 1333–1353. Y. Zhou, J. Fan, A regularity criterion for the 2D MHD system wity zeao magnetic diffusivity, J. Math. Anal. Appl. 378 (2011) 169–172. Z. Ye, On regularity criteria of the 2D generalized MHD equations, J. Math. Anal. Appl. 463 (2) (2018) 989–1005. L. Grafakos, Modern Fourier Analysis, in: Graduate Texts in Mathematics, vol. 250, SpringerVerlag, New York, 2014. H. Kozono, T. Ogawa, Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Math. Z. 242 (2002) 251–278. T. Kato, G. Ponce, Commutator estimates and the Euler and Navier–Stokes equations, Comm. Pure Appl. Math. 41 (1988) 891–907.
Please cite this article in press as: Z. Wen, A new regularity criterion of the 2D MHD equations, Computers and Mathematics with Applications (2018), https://doi.org/10.1016/j.camwa.2018.06.026.