Abstract
This note is devoted to the study of the smoothness of weak solutions to the Cauchy problem for three-dimensional magneto-hydrodynamic system in terms of the pressure. It is proved that if the pressure π belongs to L 2(0, T, Ḃ −1∞,∞ (ℝ3)) or the gradient field of pressure ∇π belongs to L 2/3(0, T, BMO(ℝ3)), then the corresponding weak solution (u, b) remains smooth on [0, T].
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References
M. Sermange and R. Temam, “Some mathematical questions related to the MHD equations,” Comm. Pure Appl. Math. 36 (5), 635–664 (1983).
R. E. Caflish, I. Klapper, and G. Steel, “Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD,” Comm. Math. Phys. 184 (2), 443–455 (1997).
Q. Chen, C. Miao, and Z. Zhang, “On the regularity criterion of weak solution for the 3D viscous magneto-hydrodynamics equations,” Comm. Math. Phys. 284 (3), 919–930 (2008).
C. Cao and J. Wu, “Two regularity criteria for the 3D MHD equations,” J. Differential Equations 248 (9), 2263–2274 (2010).
Z. Lei and Y. Zhou, “BKM’s criterion and global weak solutions for magnetohydrodynamics with zero viscosity,” Discrete Contin. Dyn. Syst. 25 (2), 575–583 (2009).
Y. Zhou and S. Gala, “Regularity criteria in terms of the pressure for the Navier–Stokes equations in the critical Morrey–Campanato space,” Z. Anal. Anwend. 30, 83–93 (2011).
H. Duan, “On regularity criteria in terms of pressure for the 3D viscousMHDequations,” Appl. Anal. 91 (5), 947–952 (2012).
J. Fan and T. Ozawa, “Regularity criterion for weak solutions to the Navier–Stokes equations in terms of the gradient of the pressure,” J. Inequal. Appl., No. 412678 (2008).
X. Zhang, Y. Jia, and B.-Q. Dong, “On the pressure regularity criterion of the 3D Navier–Stokes equations,” J. Math. Anal. Appl. 393 (2), 413–420 (2012).
H. Triebel, Theory of Function Spaces (Birkhäuser Verlag, Basel, 1983).
Y. Zhou, “On regularity criteria in terms of pressure for the Navier–Stokes equations in R3,” Proc. Amer. Math. Soc. 134 (1), 149–156 (2006).
P. G. Lemarié-Rieusset, “The Navier–Stokes equations in the criticalMorrey–Campanato space,” Rev. Mat. Iberoam. 23 (3), 897–930 (2007).
P. Gerard, Y. Meyer, and F. Oru, “Inégalités de Sobolev précisées,” in Séminaire sur les équations aux dérivées partielles, Exp. No. IV (École Polytech., Palaiseau, 1997).
H. Kozono and Y. Taniuchi, “Bilinear estimates in BMO and the Navier–Stokes equations,” Math. Z. 235 (1), 173–194 (2000).
S. Gala, “Extension criterion on regularity for weak solutions to the 3D MHD equations,” Math. Methods Appl. Sci. 33 (12), 1496–1503 (2010).
C. He and Y. Wang, “Remark on the regularity for weak solutions to the magnetohydrodynamic equations,” Math. Methods Appl. Sci. 31 (14), 1667–1684 (2008).
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Original Russian Text © S. Gala, M. A. Ragusa, 2017, published in Matematicheskie Zametki, 2017, Vol. 102, No. 4, pp. 526–531.
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Gala, S., Ragusa, M.A. A note on regularity criteria in terms of pressure for the 3D viscous MHD equations. Math Notes 102, 475–479 (2017). https://doi.org/10.1134/S000143461709019X
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DOI: https://doi.org/10.1134/S000143461709019X