Global Cauchy problem of 2D generalized magnetohydrodynamic equations

J. Math. Anal. Appl. 420 (2014) 1024–1032

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Global Cauchy problem of 2D generalized magnetohydrodynamic equations Jishan Fan a , Kun Zhao b,∗ a b

Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, PR China Department of Mathematics, Tulane University, New Orleans, LA 70118, USA

a r t i c l e

i n f o

Article history: Received 23 November 2013 Available online 19 June 2014 Submitted by J. Guermond Keywords: gMHD Global regularity Regularity criterion

a b s t r a c t In this paper we prove the global-in-time existence of smooth solutions of the 2D generalized magnetohydrodynamic equations with zero viscosity, when the power of the Laplacian operator −Δ in the dissipation term of the induction equation is between 1 and 2. We also show a regularity criterion on the direction of the magnetic b field b0 := |b| when β = 1. © 2014 Elsevier Inc. All rights reserved.

1. Introduction Recent development of mathematical fluid mechanics indicates that researchers have found it favorable to replace the Laplace operator −Δ by fractional powers of −Δ. Such a generalization of “normal” diffusion has been enforced to many fluid dynamic systems, including the Navier–Stokes [18,22], Boussinesq [11,10,12], magnetohydrodynamic (MHD) [4,21], and surface quasi-geostrophic (SQG) [2,8,7,6,9,14,15] equations. Mathematical study of these generalized models has inspired the invention of many new analytical methods, such as the recent breakthough in the study of the 2D SQG equations [2,8,14,15]. In this paper, we consider the following Cauchy problem of the 2D generalized MHD equations with zero viscosity: div u = div b = 0,   1 2 ut + u · ∇u + ∇ π + |b| = b · ∇b, 2 * Corresponding author. E-mail addresses: [email protected] (J. Fan), [email protected] (K. Zhao). http://dx.doi.org/10.1016/j.jmaa.2014.06.030 0022-247X/© 2014 Elsevier Inc. All rights reserved.

(1.1) (1.2)

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bt + u · ∇b + (−Δ)β b = b · ∇u,

(1.3)

in R .

(1.4)

(u, b)(·, 0) = (u0 , b0 )(·)

2

Here u is the fluid velocity field, π is the pressure and b is the magnetic field. β ≥ 0 is a constant. The problem of global-in-time well-posedness of (1.1)–(1.3) in three-dimensional space is highly challenging, due to the system contains the 3D Euler equations as a special case (obtained by setting b ≡ 0), for which the issue of global well-posedness has not been resolved. Even in two-dimensional space, quantitative behavior of system (1.1)–(1.4), such as global regularity and blow-up criteria, has not been fully understood. As a matter of fact, such properties have only been studied in limited detail by several researchers. For example, when β = 1 (the case of “normal” diffusion), the existence of a global classical solution was proved by Kozono [16] with small initial data b0 . Later on, Lei and Zhou [17] proved that, for arbitrarily large initial data, local smooth solutions to (1.1)–(1.4) when β = 1 can be extended beyond their local existing time if and only if   2  0 ω ∈ L1 0, T ; B˙ ∞,∞ R 0 where ω := curl u = ∂1 u2 − ∂2 u1 for u = (u1 , u2 )T and B˙ ∞,∞ (R2 ) denotes the homogeneous Besov space. Regarding the global regularity of large amplitude solutions to (1.1)–(1.4), one of the major breakthroughs is recently made by Tran, Yu and Zhai [20]. In that paper, the authors proved that (1.1)–(1.4) is globally regular for large initial data when β > 2. Their result indicates that, in order to guarantee the global regularity of (1.1)–(1.4), the dissipation in the induction equation must be “super strong” (super-dissipative) to compensate the effect of vanished viscosity and nonlinear coupling. The result is recently generalized to the case β = 2 by Yuan and Bai [24] and to the case β > 32 by Yamazaki [23] and Jiu and Zhao [13]. However, when 1 < β ≤ 32 , the question of global regularity of (1.1)–(1.4) is much more challenging than the aforementioned cases. Such a problem is of great importance since its resolution fills up the gap between the results of [13,20,23,24] and the limiting case β = 1. This in turn will serve as the foundation for researchers to tackle a long-standing open problem in the field, which is the global regularity of (1.1)–(1.4) when β = 1 (which is the case of “normal” diffusion). The first goal of this paper is to study the global regularity of (1.1)–(1.4) when 1 < β ≤ 32 . In addition, we provide a different version of proof for the case 3 2 < β ≤ 2, which is much simpler than those in [13,23,24]. The result is recorded in the following theorem.

Theorem 1.1. Let 1 < β ≤ 2 and u0 , b0 ∈ H s (R2 ) with s ≥ 2 and div u0 = div b0 = 0 in R2 . Then the problem (1.1)–(1.4) has a unique global-in-time smooth solution (u, b) satisfying    u, b ∈ L∞ 0, T ; H s R2 ,

   b ∈ L2 0, T ; H s+β R2

(1.5)

for any T > 0. Remark 1.1. It should be mentioned that in [5], Cao, Wu and Yuan proved a similar result as the one obtained in Theorem 1.1. However, we notice that the proofs are in very different fashions. In [5], the global well-posedness result is proved through Besov space techniques, while the main result in this paper is obtained by utilizing the theory of nonautonomous evolution equations. The method used in this paper is much simpler than the one in [5]. In addition, our result is slightly stronger than the one obtained in [5] in the sense that our requirement on the initial data is weaker than that in [5]. Indeed, • in Theorem 1.1, the initial data are required to satisfy (u0 , b0 ) ∈ H s (R2 ), s ≥ 2, • in [5], the initial data are required to satisfy (u0 , b0 ) ∈ H s (R2 ), s > 2 and ∇(∇ × b0 ) ∈ L∞ (R2 ). This is mainly due to the difference between the methods of proof.

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Remark 1.2. Combining the above theorem with one of the main results in [20], we see that the 2D generalized MHD equations with zero viscosity are globally regular for all β > 1. Our result indicates that one does not need the dissipation in the induction equation to be “super strong” in order to ensure the global regularity of (1.1)–(1.4). This is a significant improvement of the result obtained in [13,20,23,24]. It also makes it feasible to attack the global regularity of (1.1)–(1.4) when β = 1. A possible approach would be to view β as somewhat a solution parameter, and then examine the behavior of the solution as β → 1. We leave the investigation for the future. Next, we observe that, instead of monitoring the velocity field (cf. [17]), the authors of [20] studied a b blow-up criterion on the direction of the magnetic field b0 := |b| . Indeed, they showed that, when β > 1, (1.1)–(1.4) is globally regular if    b0 ∈ L∞ 0, T ; W 2,∞ R2 .

(1.6)

Notice that Theorem 1.1 secures the global regularity of (1.1)–(1.4) when β > 1. However, it is not clear whether the system is globally regular when β = 1 (recall the result of [17]). Inspired by (1.6), in the following theorem, we prove a blow-up criterion on the direction of the magnetic field for (1.1)–(1.4) when β = 1. Our result provides an alternative way to examine the global regularity of (1.1)–(1.4) equipped with “normal diffusion”, compared with the criterion established in [17]. Theorem 1.2. Let β = 1 and u0 , b0 ∈ H s (R2 ) with s ≥ 2 and div u0 = div b0 = 0 in R2 . Let (u, b) be a local b smooth solution to the problem (1.1)–(1.4). If b0 := |b| satisfies    b0 ∈ L1 0, T ; W 2,∞ R2

(1.7)

with 0 < T < ∞, then the solution (u, b) can be extended beyond T > 0. We prove the above theorems by combining Lp -based energy method, theory of nonautonomous evolution equations [1], Gagliardo–Nirenberg type interpolation inequalities, and the Beale–Kato–Majda-type blow-up criterion for MHD equations [17]. Our approach is different from those used in [5,13,20,23,24]. In particular, the theory of nonautonomous equations plays an important role in our energy framework (see Step 3 of the proof of Theorem 1.1). For Lp -based energy method, our proof will frequently use the following well-known energy equality for MHD equations: T u 2L2 + b 2L2 + 2

 β 2 Λ b 2 dt = u0 2 2 + b0 2 2 , L L L

(1.8)

0

and the following estimate proved in [16,17,20]:

sup 0≤t≤T



∇u 2L2

+

∇b 2L2



T +

 1+β 2 Λ b 2 dt ≤ C L

(1.9)

0

1

where Λ := (−Δ) 2 . In Section 2 and Section 3, we prove Theorem 1.1 and Theorem 1.2, respectively. 2. Proof of Theorem 1.1 Since the local existence result for (1.1)–(1.4) is standard, here we only deal with the a priori estimates. We divide the proof into three steps.

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1

Step 1. Testing (1.3) by (−Δ)β b, and denoting Λ := (−Δ) 2 , we have    1 d Λβ b2 2 + Λ2β b2 2 = L L 2 dt

 (b · ∇u − u · ∇b) Λ2β b dx

   ≤ b L∞ ∇u L2 + u L4 ∇b L4 Λ2β bL2 ,

(2.1)

where the inequality is an easy consequence of Hölder’s inequality. In view of (1.8)–(1.9), and the Gagliardo– Nirenberg interpolation inequality in 2D: u 2L4 ≤ C ∇u L2 u L2 , we see that the right hand side of (2.1) can be estimated as 

     b L∞ ∇u L2 + u L4 ∇b L4 Λ2β bL2 ≤ C b L∞ + ∇b L4 Λ2β bL2 ,

(2.2)

for some absolute constant C. Again, by Gagliardo–Nirenberg type interpolation inequalities in 2D, we have 1 1− 1  b L∞ ≤ C b L2 2β Λ2β bL2β2 ,

(2.3)

 1 1− 1  ∇b L4 ≤ C ∇b L2 4β−2 Λ2β bL4β−2 . 2

(2.4)

and

Combining (2.1)–(2.4), we see that 1  1    1  1  2β    1 d Λ b 2 Λβ b2 2 + Λ2β b2 2 ≤ C b 1−2 2β Λ2β b 2β2 + ∇b 1−2 4β−2 Λ2β b 4β−2 L L L L2 L L L 2 dt 2 1 ≤ Λ2β bL2 + C, 2

(2.5)

where the Young inequality and (1.8)–(1.9) have been implemented in the last step of estimation. (2.5) then proves 2  sup Λβ bL2 +

0≤t≤T

T

 2β 2 Λ b 2 dt ≤ C. L

(2.6)

0

Step 2. We denote the vorticity ω := curl u and the current j := curl b. Then we have the well-known equations: ωt + u · ∇ω = b · ∇j,

(2.7)

jt + u · ∇j + (−Δ)β j = b · ∇ω + Q(∇u, ∇b),

(2.8)

with Q(∇u, ∇b) := 2∂1 b1 (∂1 u2 + ∂2 u1 ) + 2∂2 u2 (∂1 b2 + ∂2 b1 ). When β > 32 , it follows from (2.6) and Gagliardo–Nirenberg type interpolation inequality that   b ∈ L∞ 0, T ; L∞ ,

  ∇j ∈ L2 0, T ; L∞ .

(2.9)

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(2.7) and (2.9) then lead to T ω L∞ (0,T ;L∞ ) ≤ C +

b · ∇j L∞ dt ≤ C.

(2.10)

0

The desired estimate (1.5) is then a simple consequence of (2.10) and the Beale–Kato–Majda-type blow-up criterion established in [17,3]. Step 3. Now we assume 1 < β ≤ 32 . First we rewrite (1.3) as a nonautonomous evolution equation: bt + (−Δ)β b = f := b · ∇u − u · ∇b.

(2.11)

  (−Δ)β b 2 ≤ C f L2 (0,T ;Lp ) + C, L (0,T ;Lp )

(2.12)

Then it is well-known that [1]:

for any p ∈ (2, ∞). Since we still have (2.6), it follows that   b ∈ L∞ 0, T ; L∞ ,

  ∇b ∈ L2 0, T ; Lq ,

∀q ∈ (2, ∞].

(2.13)

Testing (2.7) by |ω|p−2 ω (2 < p < ∞) and using (1.1), we get 1 d ω pLp ≤ b L∞ ∇j Lp ω p−1 Lp , p dt which, together with (2.13), gives d ω 2Lp ≤ C ∇j Lp ω Lp ≤ C ∇j 2Lp + C ω 2Lp . dt Notice that, since β > 1, by Gagliardo–Nirenberg type interpolation inequality in 2D, ∇j Lp can be well controlled by b L∞ and (−Δ)β b Lp . Thus, integrating the above inequality in time and using (2.12)–(2.13) and Young inequality, we derive t ω 2Lp

≤C +C



 ∇j 2Lp + ω 2Lp dτ

0

t ≤C +C

   (−Δ)β b2 p + ω 2Lp dτ L

0

t ≤C +C



 b · ∇u − u · ∇b 2Lp + ω 2Lp dτ



 b 2L∞ ∇u 2Lp + u 2L∞ ∇b 2Lp + ω 2Lp dτ



 ω 2Lp + u 2W 1,p ∇b 2Lp + ω 2Lp dτ,

0

t ≤C +C 0

t ≤C +C 0

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where we also applied the well-known result: ∇u Lp ≤ C ω Lp , and the Sobolev embedding in 2D: W 1,p → L∞ (2 < p < ∞). Using (1.8)–(1.9) and the Sobolev embedding in 2D: H 1 → Lp (2 < p < ∞), we derive u 2W 1,p = u 2Lp + ∇u 2Lp ≤ C u 2H 1 + C ω 2Lp ≤ C + C ω 2Lp .

(2.14)

Substituting (2.14) into (2.15) and using (2.13), we have t ω 2Lp

≤C +C



1+

∇b 2Lp



t ω 2Lp dτ

0

t ≤C +C

∇b 2Lp dτ

+C 0



 1 + ∇b 2Lp ω 2Lp dτ.

(2.15)

0

Applying the Gronwall inequality to (2.15) generates ω L∞ (0,T ;Lp ) ≤ C.

(2.16)

Furthermore, the combination of (2.12)–(2.13) and (2.16) gives   (−Δ)β b

L2 (0,T ;Lp )

≤C

(2.17)

for any 2 < p < ∞. From Gagliardo–Nirenberg type interpolation inequality we know that, by taking 1 p > β−1 ≥ 2 in (2.17), it holds that ∇j L2 (0,T ;L∞ ) ≤ C.

(2.18)

Finally, (2.7), (2.13) and (2.18) lead to (2.10). The desired estimate (1.5) is then a simple consequence of (2.10) and the Beale–Kato–Majda-type blow-up criterion established in [17,3]. This completes the proof. 2 3. Proof of Theorem 1.2 This section is devoted to the proof of Theorem 1.2. We still only establish the a priori estimates. First we observe that Eq. (1.3) can be rewritten as bt − Δb = div g

(3.1)

with gi := bi u − ui b (i = 1, 2). Notice that, according to (1.8)–(1.9) and Sobolev embedding, it holds that gi ∈ L∞ (0, T ; Lp ) for any 2 < p < ∞. By the well-known L∞ -estimate and Lq (0, T ; W 1,p )-estimate of the heat equation [1,19], we have b L∞ (0,T ;L∞ ) ≤ C,

(3.2)

∇b Lq (0,T ;Lp ) ≤ C

(3.3)

for any q, p ∈ (2, ∞). Testing (2.7) by |ω|p−2 ω (2 < p < ∞), using (1.1) and (3.2), we deduce that 1 d ω pLp = p dt

 b · ∇j · |ω|p−2 ωdx ≤ b L∞ ∇j Lp ω p−1 Lp ,

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which yields t ω pLp

≤C +C



 ∇j pLp + ω pLp dτ



 Δb pLp + ω pLp dτ



 b · ∇u − u · ∇b pLp + ω pLp dτ



 b pL∞ ∇u pLp + u pL∞ ∇b pLp + ω pLp dτ



 u pL∞ ∇b pLp + ω pLp dτ



 1 + ∇b pLp ω pLp dτ,

0

t ≤C +C 0

t ≤C +C 0

t ≤C +C 0

t ≤C +C 0

t ≤C +C 0

where in the last step of estimation we have used argument similar to that in deriving (2.15). The Gronwall inequality and (3.3) lead to ω L∞ (0,T ;Lp ) ≤ C,

∀p ∈ (2, ∞).

(3.4)

Notice that with the aid of (3.2)–(3.4) and the well-known L∞ -estimate and Lq (0, T ; W 1,p )-estimate of the heat equation [1,19], the regularity of b now can be improved as b Lq (0,T ;W 2,p ) ≤ C,

∀q, p ∈ (2, ∞),

(3.5)

∇b L2 (0,T ;L∞ ) ≤ C.

(3.6)

To establish higher order regularity of the velocity field, we use the idea in [20]. We write the vorticity equation as follows ωt + u · ∇ω = h := A|b|2 + B(b · curl b)

(3.7)

with   A := curl b0 · ∇b0 − b0 div b0 ,

B := b0 · ∇b0 − b0 div b0 ,

b0 =

b . |b|

(3.8)

Using the blow-up criterion (1.7) and the Gagliardo–Nirenberg type interpolation inequality:  0 2     ∇b  ∞ ≤ C b0  ∞ b0  2,∞ , L L W

(3.9)

we have   A ∈ L1 0, T ; L∞ ,

  B ∈ L2 0, T ; L∞ .

(3.10)

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Moreover, because of (3.2) and (3.6), we have   b · curl b ∈ L2 0, T ; L∞ ,

(3.11)

which, together with (3.10)–(3.11) and (3.2), implies   h ∈ L1 0, T ; L∞ .

(3.12)

Finally, the combination of (3.7) and (3.12) gives   ω ∈ L∞ 0, T ; L∞ ,

(3.13)

and the proof is complete due to (3.13) and the Beale–Kato–Majda-type blow-up criterion established in [17,3]. 2 Acknowledgments The authors would like to thank the anonymous referees for their valuable comments and suggestions on improving the current paper. J. Fan was supported by NSFC (No. 11171154). K. Zhao was partially supported by the Louisiana EPSCoR Pilot Fund No. LEQSF-EPS(2014)-PFUND-379. K. Zhao also gratefully acknowledges a start up funding from the Department of Mathematics at Tulane University, and a CoR Research Fellowship from the Office of Provost at Tulane University. References [1] H. Amann, Maximal regularity for nonautonomous evolution equations, Adv. Nonlinear Stud. 4 (2004) 417–430. [2] L.A. Caffarelli, A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. (2) 171 (2010) 1903–1930. [3] R.E. Caflisch, I. Klapper, G. Steele, Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD, Comm. Math. Phys. 184 (1997) 443–455. [4] C. Cao, J. Wu, Global regularity for the 2d MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math. 226 (2011) 1803–1822. [5] C. Cao, J. Wu, B. Yuan, The 2D incompressible magnetohydrodynamics equations with only magnetic diffusion, SIAM J. Math. Anal. 46 (2014) 588–602. [6] D. Chae, P. Constantin, D. Córdoba, F. Gancedo, J. Wu, Generalized surface quasi-geostrophic equations with singular velocities, Comm. Pure Appl. Math. 65 (2012) 1037–1066. [7] D. Chae, P. Constantin, J. Wu, Inviscid models generalizing the two-dimensional Euler and the surface quasi-geostrophic equations, Arch. Ration. Mech. Anal. 202 (2011) 35–62. [8] P. Constantin, V. Vicol, Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geom. Funct. Anal. 22 (2012) 1289–1321. [9] H. Dong, D. Li, On the 2d critical and supercritical dissipative quasi-geostrophic equation in Besov spaces, J. Differential Equations 248 (2010) 2684–2702. [10] T. Hmidi, S. Keraani, F. Rousset, Global well-posedness for a Boussinesq–Navier–Stokes system with critical dissipation, J. Differential Equations 249 (2010) 2147–2174. [11] T. Hmidi, S. Keraani, F. Rousset, Global well-posedness for Euler–Boussinesq system with critical dissipation, Comm. Partial Differential Equations 36 (2011) 420–445. [12] T. Hmidi, M. Zerguine, On the global well-posedness of the Euler–Boussinesq system with fractional dissipation, Phys. D 239 (2010) 1387–1401. [13] Q. Jiu, J. Zhao, A remark on global regularity of 2D generalized magnetohydrodynamic equations, J. Math. Anal. Appl. 412 (2014) 478–484. [14] A. Kiselev, F. Nazarov, A variation on a theme of Caffarelli and Vasseur, J. Math. Sci. 166 (2010) 31–39. [15] A. Kiselev, F. Nazarov, A. Volberg, Global well-posedness for the critical 2d dissipative quasi-geostrophic equation, Invent. Math. 167 (2007) 445–453. [16] H. Kozono, Weak and classical solutions of the 2-D MHD equations, Tohoku Math. J. 41 (1989) 471–488. [17] Z. Lei, Y. Zhou, BKM’s criterion and global weak solutions for magnetohydrodynamics with zero viscosity, Discrete Contin. Dyn. Syst. 25 (2009) 575–583. [18] P. Li, Z. Zhai, Well-posedness and regularity of generalized Navier–Stokes equations in some critical q-spaces, J. Funct. Anal. 259 (2010) 2457–2519.

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J. Fan, K. Zhao / J. Math. Anal. Appl. 420 (2014) 1024–1032

[19] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinues, Ann. Inst. Fourier (Grenoble) 15 (1965) 189–258. [20] C.V. Tran, X. Yu, Z. Zhai, On global regularity of 2D generalized magnetohydrodynamic equations, J. Differential Equations 254 (2013) 4194–4216. [21] J. Wu, Generalized MHD equations, J. Differential Equations 195 (2003) 284–312. [22] J. Wu, The generalized incompressible Navier–Stokes equations in Besov spaces, Dyn. Partial Differ. Equ. 1 (2004) 381–400. [23] K. Yamazaki, Remarks on the global regularity of the two-dimensional magnetohydrodynamics system with zero dissipation, Nonlinear Anal. 94 (2014) 194–205. [24] B. Yuan, L. Bai, Remarks on global regularity of 2D generalized MHD equations, J. Math. Anal. Appl. 413 (2014) 633–640.