A new improved regularity criterion of solutions to Leray-𝛼-MHD model and Navier-Stokes equation

Wu, Jiayan; Zhang, Ting

doi:10.1090/proc/16010

A new improved regularity criterion of solutions to Leray-$\alpha$-MHD model and Navier-Stokes equation
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by Jiayan Wu and Ting Zhang PDF

Proc. Amer. Math. Soc. 150 (2022), 4819-4829 Request permission

Abstract:

In this paper, we obtain the $n$th-logarithmically improved regularity criterion of smooth solutions for the incompressible Leray-$\alpha$-MHD model in terms of the magnetic field $B$. Meanwhile, the new logarithmically improved regularity criterion for the 3D Navier-Stokes equation in terms of the pressure $\pi$ and gradient of velocity $\nabla u$ can also be established. Especially, we explore a new logarithmically improved Serrin’s criterion for the 3D Navier-Stokes equations, which improves the results of Lei and Zhou [Commn. Pure Appl. Anal. 12 (2013), pp. 2715–2719].

References

Hui Chen, Daoyuan Fang, and Ting Zhang, Regularity of 3D axisymmetric Navier-Stokes equations, Discrete Contin. Dyn. Syst. 37 (2017), no. 4, 1923–1939. MR 3640581, DOI 10.3934/dcds.2017081
Qionglei Chen, Changxing Miao, and Zhifei Zhang, On the regularity criterion of weak solution for the 3D viscous magneto-hydrodynamics equations, Comm. Math. Phys. 284 (2008), no. 3, 919–930. MR 2452599, DOI 10.1007/s00220-008-0545-y
Chi Hin Chan and Alexis Vasseur, Log improvement of the Prodi-Serrin criteria for Navier-Stokes equations, Methods Appl. Anal. 14 (2007), no. 2, 197–212. MR 2437103, DOI 10.4310/MAA.2007.v14.n2.a5
Boqing Dong, Gala Sadek, and Zhimin Chen, On the regularity criteria of the 3D Navier-Stokes equations in critical spaces, Acta Math. Sci. Ser. B (Engl. Ed.) 31 (2011), no. 2, 591–600. MR 2817117, DOI 10.1016/S0252-9602(11)60259-2
Jishan Fan, Song Jiang, Gen Nakamura, and Yong Zhou, Logarithmically improved regularity criteria for the Navier-Stokes and MHD equations, J. Math. Fluid Mech. 13 (2011), no. 4, 557–571. MR 2847281, DOI 10.1007/s00021-010-0039-5
Jishan Fan and Tohru Ozawa, Regularity criterion for weak solutions to the Navier-Stokes equations in terms of the gradient of the pressure, J. Inequal. Appl. , posted on (2008), Art. ID 412678, 6. MR 2460836, DOI 10.1155/2008/412678
Jishan Fan and Tohru Ozawa, Regularity criteria for the magnetohydrodynamic equations with partial viscous terms and the Leray-$\alpha$-MHD model, Kinet. Relat. Models 2 (2009), no. 2, 293–305. MR 2507450, DOI 10.3934/krm.2009.2.293
Sadek Gala, On the improved regularity criterion of the solutions to the Navier-Stokes equations, Commun. Korean Math. Soc. 35 (2020), no. 1, 339–345. MR 4060220, DOI 10.4134/CKMS.c190019
Z. Guo and S. Gala, Remarks on logarithmical regularity criteria for the Navier-Stokes equations, J. Math. Phys. 52 (2011), no. 6, 063503, 9. doi:10.1063/1.3569967.
Xiaowei He and Sadek Gala, Regularity criterion for weak solutions to the Navier-Stokes equations in terms of the pressure in the class $L^2(0,T;\dot {B}^{-1}_{\infty ,\infty }(\Bbb R^3))$, Nonlinear Anal. Real World Appl. 12 (2011), no. 6, 3602–3607. MR 2832995, DOI 10.1016/j.nonrwa.2011.06.018
C. A. Jones and P. H. Roberts, Turbulence models and plane layer dynamos, Fluid dynamics and dynamos in astrophysics and geophysics, Fluid Mech. Astrophys. Geophys., vol. 12, CRC Press, Boca Raton, FL, 2005, pp. 295–330. MR 2427610
T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), no. 7, 891–907. doi:10.1002/cpa.3160410704.
Qiao Liu, A logarithmical blow-up criterion for the 3D nematic liquid crystal flows, Bull. Malays. Math. Sci. Soc. 41 (2018), no. 1, 29–47. MR 3743825, DOI 10.1007/s40840-015-0217-y
Jasmine S. Linshiz and Edriss S. Titi, Analytical study of certain magnetohydrodynamic-$\alpha$ models, J. Math. Phys. 48 (2007), no. 6, 065504, 28. MR 2337020, DOI 10.1063/1.2360145
Wei Luo and Zhaoyang Yin, Global existence and well-posedness for the FENE dumbbell model of polymeric flows, Nonlinear Anal. Real World Appl. 37 (2017), 457–488. MR 3648391, DOI 10.1016/j.nonrwa.2017.03.006
P. D. Mininni, D. C. Montgomery, and A. G. Pouquet, A numerical study of the alpha model for two-dimensional magnetohydrodynamic turbulent flows, Phys. Fluids 17 (2005), no. 3, 035112, 17 doi:10.1063/1.1863260.
P. D. Mininni, D. C. Montgomery, and A. G. Pouquet, Numerical solutions of the three-dimensional magnetohydrodynamic alpha model, Phys. Rev. E 71 (2005), no. 4, 046304. doi:10.1103/PhysRevE.71.046304.
Ines Ben Omrane, Sadek Gala, Jae-Myoung Kim, and Maria Alessandra Ragusa, Logarithmically improved blow-up criterion for smooth solutions to the Leray-$\alpha$-magnetohydrodynamic equations, Arch. Math. (Brno) 55 (2019), no. 1, 55–68. MR 3939064, DOI 10.5817/AM2019-1-55
Michel Sermange and Roger Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math. 36 (1983), no. 5, 635–664. MR 716200, DOI 10.1002/cpa.3160360506
Fan Wu, Regularity criteria for the 3D tropical climate model in Morrey-Campanato space, Electron. J. Qual. Theory Differ. Equ. , posted on (2019), Paper No. 48, 11. MR 3991097, DOI 10.14232/ejqtde.2019.1.48
Yong Zhou and Jishan Fan, On the Cauchy problem for a Leray-$\alpha$-MHD model, Nonlinear Anal. Real World Appl. 12 (2011), no. 1, 648–657. MR 2729050, DOI 10.1016/j.nonrwa.2010.07.007
Yong Zhou and Sadek Gala, A new regularity criterion for weak solutions to the viscous MHD equations in terms of the vorticity field, Nonlinear Anal. 72 (2010), no. 9-10, 3643–3648. MR 2606809, DOI 10.1016/j.na.2009.12.045
Yi Zhou and Zhen Lei, Logarithmically improved criteria for Euler and Navier-Stokes equations, Commun. Pure Appl. Anal. 12 (2013), no. 6, 2715–2719. MR 3060904, DOI 10.3934/cpaa.2013.12.2715
Yong Zhou, Remarks on regularities for the 3D MHD equations, Discrete Contin. Dyn. Syst. 12 (2005), no. 5, 881–886. MR 2128731, DOI 10.3934/dcds.2005.12.881
Yong Zhou, Regularity criteria for the 3D MHD equations in terms of the pressure, Internat. J. Non-Linear Mech. 41 (2006), no. 10, 1174–1180. MR 2308660, DOI 10.1016/j.ijnonlinmec.2006.12.001