Applications of Mathematics, first online, pp. 1-10


Regularity criterion for a nonhomogeneous incompressible Ginzburg-Landau-Navier-Stokes system

Nana Pan, Jishan Fan, Yong Zhou

Received November 11, 2019.   Published online December 14, 2020.

Abstract:  We prove a regularity criterion for a nonhomogeneous incompressible Ginzburg-Landau-Navier-Stokes system with the Coulomb gauge in $\mathbb R^3$. It is proved that if the velocity field in the Besov space satisfies some integral property, then the solution keeps its smoothness.
Keywords:  Ginzburg-Landau; Navier-Stokes; regularity criterion
Classification MSC:  35Q30, 35Q56, 76D03, 82D55
DOI:  10.21136/AM.2020.0298-19

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References:
[1] T. Akiyama, H. Kasai, M. Tsutsumi: On the existence of the solution of the time dependent Ginzburg-Landau equations in $\mathbb R^3$. Funkc. Ekvacioj., Ser. Int. 43 (2000), 255-270. MR 1795973 | Zbl 1142.35561
[2] J. Chen, C. M. Elliott, T. Qi: Justification of a two-dimensional evolutionary Ginzburg-Landau superconductivity model. RAIRO, Modélisation Math. Anal. Numér. 32 (1998), 25-50. DOI 10.1051/m2an/1998320100251 | MR 1619592 | Zbl 0905.35084
[3] H. J. Choe, H. Kim: Strong solutions of the Navier-Stokes equations for nonhomogeneous incompressible fluids. Commun. Partial Differ. Equations 28 (2003), 1183-1201. DOI 10.1081/PDE-120021191 | MR 1986066 | Zbl 1024.76010
[4] J. Fan, H. Gao, B. Guo: Uniqueness of weak solutions to the 3D Ginzburg-Landau superconductivity model. Int. Math. Res. Not. 2015 (2015), 1239-1246. DOI 10.1093/imrn/rnt253 | MR 3340353 | Zbl 1317.35248
[5] J. Fan, S. Jiang: Global existence of weak solutions of a time-dependent 3-D Ginzburg-Landau model for superconductivity. Appl. Math. Lett. 16 (2003), 435-440. DOI 10.1016/S0893-9659(03)80069-1 | MR 1961437 | Zbl 1055.35109
[6] J. Fan, L. Jing, G. Nakamura, T. Tang: Regularity criteria for a density-dependent incompressible Ginzburg-Landau-Navier-Stokes system in a bounded domain. Ann. Pol. Math. 125 (2020), 47-57. DOI 10.4064/ap190616-15-4 | MR 4121379 | Zbl 07233404
[7] J. Fan, T. Ozawa: Regularity criteria for the 3D density-dependent Boussinesq equations. Nonlinearity 22 (2009), 553-568. DOI 10.1088/0951-7715/22/3/003 | MR 2480102 | Zbl 1168.35416
[8] J. Fan, T. Ozawa: Global well-posedness of weak solutions to the time-dependent Ginzburg-Landau model for superconductivity. Taiwanese J. Math. 22 (2018), 851-858. DOI 10.11650/tjm/180102 | MR 3830823 | Zbl 1404.35425
[9] J. Fan, B. Samet, Y. Zhou: Uniform regularity for a 3D time-dependent Ginzburg-Landau model in superconductivity. Comput. Math. Appl. 75 (2018), 3244-3248. DOI 10.1016/j.camwa.2018.01.044 | MR 3785556 | Zbl 1409.82022
[10] J. Fan, Z. Zhang, Y. Zhou: Regularity criteria for a Ginzburg-Landau-Navier-Stokes in a bounded domain. Bull. Malays. Math. Sci. Soc. (2) 43 (2020), 1009-1024. DOI 10.1007/s40840-019-00866-x | MR 4044915 | Zbl 07173803
[11] J. Fan, Y. Zhou: A note on the time-dependent Ginzburg-Landau model for superconductivity in $\mathbb{R}^n$. Appl. Math. Lett. 103 (2020), Article ID 106208, 7 pages. DOI 10.1016/j.aml.2020.106208 | MR 4050827 | Zbl 07208182
[12] J. Fan, Y. Zhou: A regularity criterion to the time-dependent Ginzburg-Landau model for superconductivity in $\mathbb{R}^n$. J. Math. Anal. Appl. 483 (2020), Article ID 123653, 7 pages. DOI 10.1016/j.jmaa.2019.123653 | MR 4037584 | Zbl 07153756
[13] Q. Hou, X. Xu, Z. Ye: Logarithmically improved blow-up criteria for the 3D nonhomogeneous incompressible Navier-Stokes equations with vacuum. Electron. J. Differ. Equ. 2016 (2016), Article ID 192, 12 pages. MR 3547381 | Zbl 1344.35080
[14] H. Kim: A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations. SIAM J. Math. Anal. 37 (2006), 1417-1434. DOI 10.1137/S0036141004442197 | MR 2215270 | Zbl 1141.35432
[15] H. Kozono, Y. Shimada: Bilinear estimates in homogeneous Triebel-Lizorkin spaces and the Navier-Stokes equations. Math. Nachr. 276 (2004), 63-74. DOI 10.1002/mana.200310213 | MR 2100048 | Zbl 1078.35087
[16] Q. Tang: On an evolutionary system of Ginzburg-Landau equations with fixed total magnetic flux. Commun. Partial Differ. Equations 20 (1995), 1-36. DOI 10.1080/03605309508821085 | MR 1312698 | Zbl 0833.35132
[17] Q. Tang, S. Wang: Time dependent Ginzburg-Landau equation of superconductivity. Physica D 88 (1995), 139-166. DOI 10.1016/0167-2789(95)00195-A | MR 1360881 | Zbl 0900.35371

Affiliations:   Nana Pan, Department of General Education, Anhui Xinhua University, Hefei 230088, Anhui, People's Republic of China, e-mail: pannanamath@126.com; Jishan Fan, Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, People's Republic of China, e-mail: fanjishan@njfu.edu.cn; Yong Zhou (corresponding author), School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai 519082, Guangdong and Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, People's Republic of China, e-mail: zhouyong3@mail.sysu.edu.cn


 
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