Gu et al. Boundary Value Problems (016 016:188 DOI 10.1186/s13661-016-0695-3 R E S E A R C H Open Access A regularity criterion for the generalized Hall-MHD system Weijiang Gu 1, Caochuan Ma,3* and Jianzhu Sun 4 * Correspondence: ccma@zjnu.edu.cn Department of Mathematics, Tianshui Normal University, Tianshui, P.R. China 3 Department of Mathematics, Zhejiang Normal University, Jinhua, P.R. China Full list of author information is available at the end of the article Abstract This paper proves a regularity criterion for the 3D generalized Hall-MHD system. MSC: 35Q35; 35B65; 76D05 Keywords: regularity criterion; generalized Hall-MHD; Gagliardo-Nirenberg inequality 1 Introduction In this paper, we consider the following 3D generalized Hall-MHD system: t u + u u + (π + 1 b +( α u = b b, (1.1 t b + u b b u +( β b + rot(rot b b=0, (1. div u = div b =0, (1.3 (u, b(,0=(u 0, b 0 inr 3. (1.4 Here u, π, andb denote the velocity, pressure, and magnetic field of the fluid, respectively. 0<α, β are two constants. The fractional Laplace operator ( α is defined through the Fourier transform, namely, ( α f (ξ= ξ α ˆf (ξ. The applications of the Hall-MHD system cover a very wide range of physical objects, for example, magnetic reconnection in space plasmas, star formation, neutron stars, and the geo-dynamo. When the Hall effect term rot(rot b b isneglected,thesystem(1.1-(1.4 reducesto the well-known generalized MHD system, which has received attention in many studies [1 6]. When α = β =1,thesystem(1.1-(1.4 reduces to the well-known Hall-MHD system, which has received many studies [7 11]. Reference [7] gave a derivation of the isentropic Hall-MHD system from a two-fluid Euler-Maxwell system. Chae-Degond-Liu [8]proved the local existence of smooth solutions. Chae and Schonbek [9] showed the time-decay and some regularity criteria were proved in [10 1]. Some other relevant results about Hall-MHD equations can be found in [13 17]. The local well-posedness is established in Wan and Zhou [18] when0<α 1and 1 < β 1. 016 Gu et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Gu et al. Boundary Value Problems (016 016:188 Page of 7 When 3 4 α < 5 4 and 1 β < 7,JiangandZhu[19] prove the following regularity criteria: 4 b L t( 0, T; L s with β t + 3 s β 1, 3 β 1 < s, and one of the following two conditions: or u L p( 0, T; L q with α p + 3 q α 1, 3 α 1 < q 6α α 1, α u L p( 0, T; L q with α p + 3 q 3α 1, 3 3α 1 < q 6α 3α 1. When 1 α < 5 4 and 1 β < 7,Ye[0] showed the following regularity criterion: 4 u L p( 0, T; L q and b L l( 0, T; L k, where p, q, l,andk satisfy the relation 3 q + α p α 1, 3 q + β p β 1, 3 k + β l β 1, and ( 3 max α 1, 3 < q, β 1 3 β 1 < k. The aim of this paper is to refine the results in [19, 0] as follows. Theorem 1.1 Let 3 4 α < 5 4 and 3 4 β < 7 4 and u 0, b 0 H with div u 0 = div b 0 =0in R 3. If uand bsatisfy u L α γ α ( 1 0, T; Ḃ γ 1, b L β ( β 1 γ 0, T; Ḃ γ with 0<γ 1 <α and 0<γ <β 1, (1.5 with 0<T <, then the solution (u, b can be extended beyond T. In the following proof, we will use the following bilinear products and commutator estimates due to Kato-Ponce [1]: s (fg L p C ( s f L p 1 g L q 1 + f L p s g L q, (1.6 s (fg f s g L p C ( f L p 1 s 1 g L q 1 + s f L p g L q, (1.7 with s >0, := ( 1,and 1 p = 1 p 1 + 1 q 1 = 1 p + 1 q. We will also use the improved Gagliardo-Nirenberg inequalities [ 4]: u 3 L 3 C u Ḃ γ 1 u γ 1, (1.8 b 3 L 3 C b Ḃ γ b γ, (1.9
Gu et al. Boundary Value Problems (016 016:188 Page 3 of 7 b L p 3 C b 1 θ 1 Ḃ γ b θ s, 1 (1.10 β b L q 3 C b θ 1 Ḃ γ b θ s, 1 (1.11 with θ 1 := γ s+γ, s := 1 + γ β, p 3 := θ 1 and q 3 := 1 θ 1 b L p 4 C b 1 θ Ḃ γ b θ s, (1.1 3 β b L q 4 C b θ Ḃ γ b 1 θ, (1.13 s with θ := γ 1+s+γ, p 4 := θ,andq 4 := 1 θ.wehave β b L p 5 C b 1 θ 3 Ḃ γ b θ 3 (1.14 b L q 5 C b θ 3 Ḃ γ b 1 θ 3, s (1.15 with θ 3 := s 1+s+γ, p 5 := θ 3,andq 5 := 1 θ 3. Proof of Theorem 1.1 This section is devoted to the proof of Theorem 1.1, we only need to prove aprioriestimates. First, testing (1.1byu and using (1.3, we see that dt u dx + α u dx = (b b udx. Testing (1.byb and using (1.3, we find that dt b dx + β b dx = (b u bdx. Summing up the above two equations, we get the well-known energy inequality 1 ( u + b dx + T 0 ( α u + β b dxdt 1 Testing (1.1by u and using (1.3, we infer that ( u0 + b 0 dx. (.1 u dx + 1+α u dx dt = u u udx b b udx = i,j j u i i u j udx+ i,j j b i i b j udx+ i,j b i i j b j udx C u 3 L 3 + C b 3 L 3 + i,j b i i j b j udx=: I 1 + I + I 3. (.
Gu et al. Boundary Value Problems (016 016:188 Page 4 of 7 Testing (1.by b and using (1.3, we deduce that b dx + 1+β b dx dt = u b bdx b u bdx+ (rot b b rot bdx j b i i u j bdx = i,j j u i i b j bdx+ i,j (rot b i b i rot bdx + i,j b i i j u j bdx i C u 3 L 3 + C b 3 L 3 + i,j b i i j u j bdx i (rot b i b i rot bdx =: I 1 + I + I 4 + I 5. (.3 Summing up (.and(.3, using (1.6, (1.8, (1.9, (1.10, (1.11, and I 3 + I 4 =0,wederive dt ( u + b ( dx + 1+α u + 1+β b dx C u 3 L 3 + C b 3 L 3 i 1 β (rot b i b β 1 i rot bdx which gives C u 3 L 3 + C b 3 L 3 + C b L p 3 β b L q 3 1+β b L C u Ḃ γ 1 u γ 1 + C b Ḃ γ b γ + C b Ḃ γ b s 1+β b L γ 1 C u Ḃ γ 1 u (1 α 1+α L u γ 1 α L +C b Ḃ γ b (1 γ β 1+β L b γ β L s + C b Ḃ γ b 1 β 1+β L b 1+ s β L 1 1+α u L + 1 1+β b L + C u + C b β β γ Ḃ γ b L + C b β β 1 γ Ḃ γ α α γ 1 Ḃ γ 1 u L b L, (u, b L (0,T;H 1 + u L (0,T;H 1+α + b L (0,T;H 1+β C. (.4 In the following proofs, we will use the following Sobolev embedding theorem: u L 4 C u H 1+α, u L 4 C u H +α C +α u L + C, b L 4 C b H 1+β, b L 4 C +β b L + C. (.5 Taking to (1.1, testing by u and using (1.3, we have u dx + +α u dx dt ( (u ( (b = u u u udx+ b b b udx
Gu et al. Boundary Value Problems (016 016:188 Page 5 of 7 + b b udx =: I 6 + I 7 + I 8. (.6 Applying to (1., testing by b and using (1.3, we have b dx + +β b dx dt = ( (u b u b bdx+ ( (b u b u bdx + b u bdx (rot b b rot bdx =: I 9 + I 10 + I 11 + I 1. (.7 Note that I 8 + I 11 =0. Using (1.7, (.4, and (.5, we bound I 6 + I 7 + I 9 + I 10 as follows: I 6 + I 7 + I 9 + I 10 C u L 4 u L 4 u L + C b L 4 b L 4 u L + C ( b L 4 u L 4 + u L 4 b L 4 b L C u H 1+α u H +α u L + C b H 1+β b H +β u L + C ( b H 1+β u H +α + u H 1+α b H +β b L 1 8 u H +α + 1 8 b H +β + C ( u H 1+α + b H 1+β ( u L + b L. Using (1.6, (1.1, (1.13, (1.14, and (1.15, we bound I 1 as follows: I 1 = (rot b b rot bdx ( i rot b i b rot bdx i = 1 β (rot b b β 1 rot bdx 1 β ( i rot b i b β 1 rot bdx i C ( b L p 4 3 β b L q 4 + β b L p 5 b L q 5 +β b L C b Ḃ γ b s +β b L s C b Ḃ γ b 1 β +β L b 1+ s β L 1 +β b β β 1 γ 8 L + C b Ḃ γ b L. Inserting the above estimates into (.6and(.7, andsummingup theresultsandusing the Gronwall inequality, we arrive at (u, b L (0,T;H + u L (0,T;H +α + b L (0,T;H +β C. This completes the proof.
Gu et al. Boundary Value Problems (016 016:188 Page 6 of 7 3 Conclusions The applications of Hall-MHD system cover a very wide range of physical objects, such as magnetic reconnection in space plasmas, star formation, neutron stars, and the geodynamo. In this paper, we obtained a new regularity criterion that improves and extends some known regularity criteria of the 3D generalized Hall-MHD system. Competing interests The authors declare that they have no competing interests. Authors contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Author details 1 Center for Information, Nanjing Forestry University, Nanjing, 10037, P.R. China. Department of Mathematics, Tianshui Normal University, Tianshui, P.R. China. 3 Department of Mathematics, Zhejiang Normal University, Jinhua, P.R. China. 4 Department of Applied Mathematics, Nanjing Forestry University, Nanjing, 10037, P.R. China. Acknowledgements The authors would like to thank the anonymous referees and the editors for their valuable suggestions and comments, which improved the results and the quality of the paper. Ma is partially supported by NSFC (Grant No. 11661070, the Youth Science and Technology Support Program (Grant No. 1606RJYE37 and the Scientific Research Foundation of the Higher Education Institutions (Grant No. 015A-131 of Gansu Province. Received: 9 August 016 Accepted: 13 October 016 References 1. Fan, J, Gao, H, Nakamura, G: Regularity criteria for the generalized magnetohydrodynamic equations and the quasi-geostrophic equations. Taiwan. J. Math. 15, 1059-1073 (011. Wu, J: Generalized MHD equations. J. Differ. Equ. 195, 84-31 (003 3. Wu, J: Regularity criteria for the generalized MHD equatins. Commun. Partial Differ. Equ. 33, 85-306 (008 4. Zhou, Y: Regularity criteria for the generalized viscous MHD equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 4, 491-505 (007 5. Fan, J, Malaikah, H, Monaquel, S, Nakamura, G, Zhou, Y: Global Cauchy problem of D generalized MHD equations. Monatshefte Math. 175(1,17-131 (014 6. Jiang, Z, Wang, Y, Zhou, Y: On regularity criteria for the D generalized MHD system. J. Math. Fluid Mech. 18(, 331-341 (016 7. Acheritogaray, M, Degond, P, Frouvelle, A, Liu, J: Kinetic formulation and global existence for the Hall-magnetohydrodynamics system. Kinet. Relat. Models 4, 901-918 (011 8. Chae, D, Degond, P, Liu, J: Well-posedness for Hall-magnetohydrodynamics. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 31(3, 555-565 (014 9. Chae, D, Schonbek, M: On the temporal decay for the Hall-magnetohydrodynamic equations. J. Differ. Equ. 55(11, 3971-398 (013 10. Wan, R, Zhou, Y: On the global existence, energy decay and blow up criterions for the Hall-MHD system. J. Differ. Equ. 59, 598-6008 (015 11. Zhang, Z: A remark on the blow-up criterion for the 3D Hall-MHD system in Besov spaces. Preprint (015 1. Pan, N, Ma, C, Zhu, M: Global regularity for the 3D generalized Hall-MHD system. Appl. Math. Lett. 61, 6-66 (016 13. Fan, J, Alsaedi, A, Hayat, T, Nakamura, G, Zhou, Y: On strong solutions to the compressible Hall-magnetohydrodynamic system. Nonlinear Anal., Real World Appl., 43-434 (015 14. Fan, J, Jia, X, Nakamura, G, Zhou, Y: On well-posedness and blowup criteria for the magnetohydrodynamics with the Hall and ion-slip effects. Z. Angew. Math. Phys. 66(4, 1695-1706 (015 15. Fan, J, Ahmad, B, Hayat, T, Zhou, Y: On blow-up criteria for a new Hall-MHD system. Appl. Math. Comput. 74, 0-4 (016 16. Fan, J, Ahmad, B, Hayat, T, Zhou, Y: On well-posedness and blow-up for the full compressible Hall-MHD system. Nonlinear Anal., Real World Appl. 31,569-579 (016 17. He, F, Ahmad, B, Hayat, T, Zhou, Y: On regularity criteria for the 3D Hall-MHD equations in terms of the velocity. Nonlinear Anal., Real World Appl. 3,35-51 (016 18. Wan, R, Zhou, Y: Low regularity well-posedness for the 3D generalized Hall-MHD system. Acta Appl. Math. (016. doi:10.1007/s10440-016-0070-5 19. Jiang, Z, Zhu, M: Regularity criteria for the 3D generalized MHD and Hall-MHD systems. Bull. Malays. Math. Soc. (015. doi:10.1007/s40840-015-043-9 0. Ye, Z: Regularity criteria and small data global existence to the generalized viscous Hall-magnetohydrodynamics. Comput. Math. Appl. 70, 137-154 (015 1. Kato, T, Ponce, G: Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure Appl. Math. 41, 891-907 (1988. Hajaiej, H, Molinet, L, Ozawa, T, Wang, B: Necessary and sufficient conditions for the fractional Gagliardo-Nirenberg inequalities and applications to Navier-Stokes and generalized Boson equations. RIMS Kôkyûroku Bessatsu 6, 159-175 (011 3. Machihara, S, Ozawa, T: Interpolation inequalities in Besov spaces. Proc. Am. Math. Soc. 131, 1553-1556 (00
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