A new regularity criterion for the 3D incompressible MHD equations via partial derivatives

J. Math. Anal. Appl. 481 (2020) 123497

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A new regularity criterion for the 3D incompressible MHD equations via partial derivatives Sadek Gala a,b,∗ , Maria Alessandra Ragusa b,c a

Department of Sciences exactes, Ecole Normale Supérieure de Mostaganem, University of Mostaganem, Box 227, Mostaganem 27000, Algeria b Dipartimento di Matematica e Informatica, Università di Catania, Viale Andrea Doria, 6, 95125 Catania, Italy c RUDN University, 6 Miklukho - Maklay St, Moscow, 117198, Russia

a r t i c l e

i n f o

Article history: Received 5 April 2019 Available online 12 September 2019 Submitted by J. Guermond Keywords: MHD equations Regularity criterion A priori estimates

a b s t r a c t We consider the regularity criterion for the incompressible MHD equations. We show that the weak solution is regular, provided that    5r (∂3 w+ , ∂3 w− ) ∈ L 4r−6 0, T ; Lr R3

with

4 ≤ r ≤ ∞,

(0.1)

   11r (∂3 w+ , ∂3 w− ) ∈ L 7r−12 0, T ; Lr R3

with

5 ≤ r ≤ 4. 2

(0.2)

or

© 2019 Elsevier Inc. All rights reserved.

We are interested in the regularity of weak solutions to the viscous incompressible magnetohydrodynamics (MHD) equations in R3 ⎧ ⎪ ⎪ ∂t u + u · ∇u − (b · ∇) b − Δu + ∇π = 0, ⎪ ⎨ ∂t b + u · ∇b − b · ∇u − Δb = 0, ⎪ ∇ · u = ∇ · b = 0, ⎪ ⎪ ⎩ u(x, 0) = u0 (x), b(x, 0) = b0 (x),

(1.1)

where u = (u1 , u2 , u3 ) is the velocity field, b = (b1 , b2 , b3 ) is the magnetic field, and π is the scalar pressure, while u0 and b0 are the corresponding initial data satisfying ∇ · u0 = ∇ · b0 = 0 in the sense of distribution. * Corresponding author. E-mail addresses: [email protected] (S. Gala), [email protected] (M.A. Ragusa). https://doi.org/10.1016/j.jmaa.2019.123497 0022-247X/© 2019 Elsevier Inc. All rights reserved.

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S. Gala, M.A. Ragusa / J. Math. Anal. Appl. 481 (2020) 123497

As for the Navier-Stokes equations (b = 0 in (1.1)), it is well-known [7] that (1.1) possesses a global-in-time weak solution (u, b) for initial data of finite energy. But, whether this unique local solution can exist globally is an outstanding challenging open problem [19]. Guided by the regularity criteria for the Navier-Stokes equations, many authors were devoted to finding sufficient conditions to ensure a given weak solution (u, b) to be smooth. In [10,23], some fundamental Serrin’s-type regularity criteria only in terms of the velocity field were established u ∈ Lp (0, T ; Lq (R3 )) with

2 3 + = 1, 3 < q ≤ ∞, p q

or ∇u ∈ Lp (0, T ; Lq (R3 )) with

3 2 3 + = 2, < q ≤ ∞. p q 2

Some improvements and extensions to the weak Lp spaces, multiplier spaces, Morrey-Campanato spaces and Besov spaces were made (see for example [4,5,9,20,22,24,26]). The interested readers are referred to [2, 3,6,8,12,17,21,25,27] for more results on the regularity issue of related equations to (1.1) and the references cited therein. Recently, some regularity criteria in terms of partial velocity components and the derivative of the velocity field satisfy certain growth conditions were established in [1,11–16]. In [1], Cao and Wu showed an interesting result which considered the regularity criterion in terms of one directional derivative of the velocity. More precisely, they proved that if    3 2 + ≤ 1 and 3 ≤ r ≤ ∞, ∂3 u ∈ Lα 0, T ; Lr R3 with α r

(1.2)

then (u, b) is regular on the interval [0, T ]. Later on, condition (1.2) was partially improved by Jia and Zhou [13] as    3 3(r + 2) 2 + ≤ and 2 < r ≤ ∞. ∂3 u ∈ Lα 0, T ; Lr R3 with α r 4r

(1.3)

One of the most significant achievements in this direction is the celebrated Ni-Guo-Zhou criterion ([18], Theorem 1.3). More precisely, they showed that if the following properties hold: ⎧    ⎪ u3 ∈ Lα1 0, T ; Lr1 R3 ⎪ ⎪ ⎪    ⎪ ⎨ ∂3 u ∈ Lα2 0, T ; Lr2 R3    ⎪ b ∈ Lα3 0, T ; Lr3 R3 ⎪ ⎪ ⎪ ⎪ ⎩ ∂ b ∈ Lα4 0, T ; Lr4 R3  3

with with with with

2 3 α1 + r1 ≤ 1 and 3 < r1 ≤ ∞, 2 3 3 α2 + r2 ≤ 1 and 2 < r2 ≤ ∞, 2 3 α3 + r3 ≤ 1 and 3 < r3 ≤ ∞, 2 3 3 α4 + r4 ≤ 1 and 2 < r4 ≤ ∞,

(1.4)

then the corresponding solution (u, b) remains smooth on [0, T ]. Let w+ = u ± b,

w0± = u0 ± b0 .

We reformulate equation (1.1) as follows. Formally, if the first equation of MHD equations (1.1) plus and minus the second one, respectively, then MHD equations (1.1) can be re-written as: ⎧ ∂t w+ − Δw+ + (w− · ∇)w+ + ∇π = 0, ⎪ ⎪ ⎪ ⎪ ⎨ ∂t w− − Δw− + (w+ · ∇)w− + ∇π = 0, ⎪ div w+ = 0, div w− = 0, ⎪ ⎪ ⎪ ⎩ + w (x, 0) = w0+ (x), w− (x, 0) = w0− (x).

(1.5)

S. Gala, M.A. Ragusa / J. Math. Anal. Appl. 481 (2020) 123497

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The purpose of this paper is to give a new regularity criterion in terms of the partial derivatives of the combination of velocity and magnetic fields (∂3 w+ , ∂3 w− ). The advantage is that the equations become symmetric. Our result now reads as follows. Theorem 1.1. Suppose that (u0 , b0 ) ∈ L2 (R3 ) with ∇ · u0 = ∇ · b0 = 0 in the sense of distribution. Let (u, b) be a weak solution to the MHD equations on some interval [0, T ] with 0 < T ≤ ∞. If w+ and w− satisfy one of the following conditions    5r (∂3 w+ , ∂3 w− ) ∈ L 4r−6 0, T ; Lr R3

with

4 ≤ r ≤ ∞,

(1.6)

5 ≤ r ≤ 4, 2

(1.7)

or    11r (∂3 w+ , ∂3 w− ) ∈ L 7r−12 0, T ; Lr R3

with

then the weak (u, b) is smooth in R3 × (0, T ]. Before proving our result, we recall the following version of the three-dimensional Sobolev and Ladyzhenskaya inequalities in the whole space R3 (see e.g. [1]). There exists a constant C > 0 such that 1

1

1

2

1

f L3r ≤ Cr ∂1 f L3 2 ∂2 f L3 2 ∂3 f L3 r ≤ Cr ∇f L3 2 ∂3 f L3 r ,

(1.8)

    for any f satisfying ∂1 f, ∂2 f ∈ L2 R3 and ∂3 f ∈ Lr R3 , where r ∈ [1, ∞[. Also, we will use the following estimates for the pressure and the gradient of pressure in terms of w+ and − w (see [13]):      2

  2 πLq ≤ C w+ L2q w− L2q ≤ C w+ L2q + w− L2q ,   ∇πLq ≤ C w+ · ∇w− Lq ,   ∇πLq ≤ C w+− · ∇w+ Lq ,

(1.9) (1.10) (1.11)

with 1 < q < ∞. Proof. By using (1.4)1 , it suffices to show that u3 ∈ L3 (0, T ; L9 R3 )). First, taking the inner product of (1.5)1 with w+ and the inner product of (1.5)1 with w− , using the divergence free property and adding the resulting equations together, it follows that     + w (·, t)2 2 + w− (·, t)2 2 + 2 L L  2  2 ≤ w0+ L2 + w0− L2 ≤ C < 0,

t

 2  2 (∇w+ (·, s)L2 + ∇w− (·, s)L2 )ds

0

(1.12)

+ − where we have used the incompressibility condition + ∇ ·+w = ∇ · w = 0. Next, taking the inner product of (1.5)1 with w3 w3 , integration by parts and using the incompressible conditions, we have

S. Gala, M.A. Ragusa / J. Math. Anal. Appl. 481 (2020) 123497

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 1 d  w+ 3 3 + 4 3 L 3 dt 9

  + 32 2 ∇ w3 dx = − ∂3 π · w3+ w3+ dx. R3

(1.13)

R3

Similarly, taking the inner product of the second equation in (1.5) with w3− w3− , we obtain  1 d  w− 3 3 + 4 3 L 3 dt 9

  − 32 2 w dx = − ∂3 π · w3− w3− dx. ∇ 3 R3

(1.14)

R3

Adding (1.13) and (1.14) together yields   4 − 32 2 + 32 2 ∇ w3 dx + ∇ w3 dx 9

 3 3 1 d  4 (w3+ L3 + w3− L3 ) + 3 dt 9  =−

∂3 π · w3+ w3+ dx −

R3



≤C



R3

∂3 π · w3− w3− dx = I + J

R3



|π| ∂3 w+ w+ dx + C 3

3

R3

R3

|π| ∂3 w3− w3− dx.

(1.15)

R3

Now we estimate (1.15). Case 1. Suppose that (1.6) holds. By Hölder’s inequality, (1.10), (1.8) and Young’s inequality, one has  2 I ≤ ∂3 πL3 w3+ L3    2 ≤ C w+ · ∂3 w− L3 w3+ L3      + 2 − 3r ∂3 w  r w  3 ≤ C w+  r−3 3 L L L

r+6    2(r−4)  +  3r−2    w  3r ∂3 w−  r w+ 2 3 ≤ C w+ L3r−2 2 3 L L L

  r+6   r+6   r+6     ∂2 w+  3(3r−2) ∂3 w+  3(3r−2) ∂3 w−  r w+ 2 3 ≤ C ∂1 w+ L3(3r−2) 2 2 3 L Lr L L  5r 5r    2  4r−6  4r−6 2 + + + − w  3 . ≤ C ∇w  2 + ∂3 w  r + ∂3 w  r L

L

L

3

L

(1.16)

Arguing similarly as the estimate of I, one has  2 J ≤ ∂3 πL3 w3− L3    2  5r  5r   −  4r−6  − 2  ∂ w 3 L3 . ≤ C ∇w− L2 + ∂3 w+ L4r−6 + w 3 r Lr

(1.17)

Substituting (1.16) and (1.17) into (1.15), we obtain  3 3 d  (w3+ L3 + w3− L3 ) dt  2 2  11r  11r  + 2     2 + ∂3 w− L7r−12 ≤ C ∇w+ L2 + ∇w− L2 + ∂3 w+ L7r−12 (w3 L3 + w3− L3 ). r r

(1.18)

 2  2 Dividing both sides by (w3+ L3 + w3− L3 ) and integrating with respect to t, together with the energy inequality (1.12) and the assumption (1.6) implies that  + w  3

L∞ (0,T ;L3 R3 ))

  + w3− L∞ (0,T ;L3 R3 )) ≤ C.

S. Gala, M.A. Ragusa / J. Math. Anal. Appl. 481 (2020) 123497

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Integrating (1.18) yields  + w  3

L3 (0,T ;L9 R3 ))

 3      + w3− L3 (0,T ;L9 R3 )) =  w3+ 2 

L2 (0,T ;L6 R3 ))

 3    ≤ ∇ w3+ 2 

 3    +  w3− 2 

L2 (0,T ;L2 R3 ))

L2 (0,T ;L6 R3 ))

 3    + ∇ w3− 2 

L2 (0,T ;L2 R3 ))

≤ C.

(1.19)

Then, it follows from the triangle inequality and (1.19) that 1 (u3 + b3 ) + (u3 − b3 )L3 (0,T ;L9 R3 )) 2

 −  1    w+  3 w ≤ + 3 L (0,T ;L9 R3 )) 3 L3 (0,T ;L9 R3 )) ≤ C, 2

u3 L3 (0,T ;L9 R3 )) =

and 1 (u3 + b3 ) − (u3 − b3 )L3 (0,T ;L9 R3 )) 2

 −  1  w+  3   ≤ 3 L (0,T ;L9 R3 )) + w3 L3 (0,T ;L9 R3 )) ≤ C. 2

b3 L3 (0,T ;L9 R3 )) =

Consequently, this concludes the proof of Case 1. Case 2. Suppose that (1.7) holds. By integration by parts, Hölder’s inequality, (1.8), (1.9) and Young’s inequality, it is easy to see that  R3

|π| ∂3 w3+ w3+ dx ≤ C π

3r L 2r−3

  ∂3 w+  3

 + w  3

Lr

L3

 2           ∂3 w+  r w+  3 + C w− 2 6r ∂3 w+  r w+  3 6r ≤ C w+ L 2r−3 3 L 3 L 3 3 L 2r−3 L L   2(2r−5)   2(r+3)     ∂3 w+  r w+  3 ≤ C w+ L23r−2 w+ L3r−2 3r 3 L 3 L 2(r+3)   −  2(2r−5)      ∂3 w+  r w+  +C w  23r−2 w−  3r−2 3r L

L

3

L

3

L3

  4(r+3)   2(r+3)     ∂3 w+  3(3r−2) ∂3 w+  r w+  3 ≤ C ∇w+ L3(3r−2) 2 3 L 3 L Lr   4(r+3)   2(r+3)     ∂3 w−  3(3r−2) ∂3 w+  r w+  3 +C ∇w− L3(3r−2) 2 3 L 3 L Lr   4(r+3)   2(r+3)   ∂3 w+ 1+r 3(3r−2) w+  3 ≤ C ∇w+ L3(3r−2) 2 3 L L   4(r+3)   2(r+3)     ∂3 w−  3(3r−2) ∂3 w+  r w+  3 +C ∇w− L3(3r−2) 2 3 L Lr L  11r      2 w+  3 ≤ C ∇w+ L2 + ∂3 w+ L7r−12 (1.20) r 3 L 

2  11r  11r  +    w  3 +C ∇w− L2 + ∂3 w− L7r−12 + ∂3 w+ L7r−12 r r 3 L  11r 11r          2 2 −  7r−12  +   ∂ w 3 L3 . ≤ C ∇w+ L2 + ∇w− L2 + ∂3 w+ L7r−12 + w 3 r Lr Using the similar way with the estimate (1.9), we get  R3

    2 2  11r  11r   −  7r−12  −   ∂ w 3 L3 . |π| ∂3 w3− w3− dx ≤ C ∇w+ L2 + ∇w− L2 + ∂3 w+ L7r−12 + w 3 r Lr

(1.21)

S. Gala, M.A. Ragusa / J. Math. Anal. Appl. 481 (2020) 123497

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Substituting (1.20) and (1.21) into (1.15), we have  3 3 d  (w3+ L3 + w3− L3 ) dt       2 2  11r  11r  +  + ∂3 w− L7r−12 ≤ C ∇w+ L2 + ∇w− L2 + ∂3 w+ L7r−12 (w3 L3 + w3− L3 ). r r     Dividing both sides by (w3+ L3 + w3− L3 ) and integrating with respect to t, together with the energy inequality (1.12) and the assumption (1.7) implies that  + w  3

L∞ (0,T ;L3 R3 ))

  + w3− L∞ (0,T ;L3 R3 )) ≤ C.

Integrating (1.18) yields  + w  3

L3 (0,T ;L9 R3 ))

  + w3− L3 (0,T ;L9 R3 )) ≤ C.

(1.22)

Then, it follows from the triangle inequality and (1.22) that 1 (u3 + b3 ) + (u3 − b3 )L3 (0,T ;L9 R3 )) 2

 −  1    w+  3 ≤ 3 L (0,T ;L9 R3 )) + w3 L3 (0,T ;L9 R3 )) ≤ C, 2

u3 L3 (0,T ;L9 R3 )) =

and 1 (u3 + b3 ) − (u3 − b3 )L3 (0,T ;L9 R3 )) 2

 −  1    w+  3 w ≤ + 3 L (0,T ;L9 R3 )) 3 L3 (0,T ;L9 R3 )) ≤ C. 2

b3 L3 (0,T ;L9 R3 )) =

Thus, according to the regularity results in [18], (u, b) is smooth on [0, T ]. This completes the proof of Theorem 1.1. 2 Acknowledgments This work was done while the first author was visiting the Catania University in Italy. He would like to thank the hospitality and support of the University, where this work was completed. This research is partially supported by Piano della Ricerca 2016-2018 - Linea di intervento 2: “Metodi variazionali ed equazioni differenziali”. The second author wishes to thank the support of “RUDN University Program 5-100”. The authors wish to express their thanks to the referees for their very careful reading of the paper, giving valuable comments and helpful suggestions. References [1] C. Cao, J. Wu, Two regularity criteria for the 3D MHD equations, J. Differential Equations 248 (2010) 2263–2274. [2] X. Chen, S. Gala, Z. Guo, A new regularity criterion in terms of the direction of the velocity for the MHD equations, Acta Appl. Math. 113 (2011) 207–213. [3] X. Chen, Z. Guo, M. Zhu, A new regularity criterion for the 3D MHD equations involving partial components, Acta Appl. Math. 134 (2014) 161–171. [4] Q.L. Chen, C.X. Miao, Z.F. Zhang, On the regularity criterion of weak solution for the 3D viscous magnetohydrodynamics equations, Comm. Math. Phys. 284 (2008) 919–930. [5] B.Q. Dong, Y. Jia, W. Zhang, An improved regularity criterion of three-dimensional magnetohydrodynamic equations, Nonlinear Anal. Real World Appl. 13 (2012) 1159–1169. [6] H. Duan, On regularity criteria in terms of pressure for the 3D viscous MHD equations, Appl. Anal. 91 (2012) 947–952.

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