Logarithmically Improved Regularity Criteria for the Navier–Stokes and MHD Equations

Beirão da Veiga H.: A new regularity class for the Navier–Stokes equations in \({{\mathbb R}^n}\) . Chin. Ann. Math. Ser. B 16, 407–412 (1995) Chan C.H., Vasseur A.: Log Improvement of the Prodi–Serrin Criteria for Navier–Stokes Equations. Methods Appl. Anal. 14, 197–212 (2007) Chen Q., Miao C., Zhang Z.: On the regularity criterion of weak solution for the 3D viscous magneto-hydrodynamics equations. Commun. Math. Phys. 284, 919–930 (2008) Duoandikoetxea, J.: Fourier analysis. Translated and revised from the 1995 Spanish original by David Cruz-Uribe. Graduate Studies in Mathematics, vol. 29. American Mathematical Society, Providence (2001) Escauriaza L., Serëgin G.A., Sverak V.: L 3,∞-solutions of Navier–Stokes equations and backward uniqueness. Uspekhi Mat. Nauk 58, 3–44 (2003) Fan J., Jiang S., Ni G.: On regularity criteria for the n-dimensional Navier–Stokes equations in terms of the pressure. J. Differ. Equ. 244, 2936–2979 (2008) Fan, J., Jiang, S., Nakamura, G.: On logarithmically improved regularity criteria for the Navier–Stokes equations in \({{\mathbb R}^n}\) (2009, submitted) Fan, J., Ozawa, T.: Regularity criterion for weak solutions to the Navier–Stokes equations in terms of the gradient of the pressure. J. Inequal. Appl. Art.ID412678, 6pp (2008) Friedman A.: Partial differential equations. Holt, Rinehart and Winston Inc., New York-Montreal (1969) He C., Xin Z.: On the regularity of weak solutions to the magnetohydrodynamic equations. J. Differ. Equ. 213, 235–254 (2005) Hopf E.: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4, 213–231 (1951) Kato T., Ponce G.: Commutator estimates and the Euler and Navier–Stokes equations. Commun. Pure Appl. Math. 41, 891–907 (1988) Kozono H., Ogawa T., Taniuchi Y.: The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations. Math. Z. 242, 251–278 (2002) Kozono H., Taniuchi Y.: Bilinear estimates in BMO and the Navier–Stokes equations. Math. Z. 235, 173–194 (2000) Leray J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, 183–248 (1934) Montgomery-Smith S.: Conditions implying regularity of the three dimensional Navier–Stokes equation. Appl. Math. 50, 451–464 (2005) Ohyama T.: Interior regularity of weak solutions of the time-dependent Navier–Stokes equation. Proc. Jpn Acad. 36, 273–277 (1960) Prodi G.: Un teorema di unicità per le equazioni di Navier–Stokes. Ann. Mat. Pura Appl. 48, 173–182 (1959) Sermange M., Temam R.: Some mathematical questions related to the MHD equations. Commun. Pure Appl. Math. 36, 635–664 (1983) Serrin, J.: The initial value problem for the Navier–Stokes equations. In: Nonlinear Problems. Proc. Sympos., Madison, WI, pp. 69–98. University of Wisconsin Press, Madison, WI (1963) Struwe M.: On partial regularity results for the Navier–Stokes equations. Commun. Pure Appl. Math. 41, 437–458 (1988) Wu J.: Regularity criteria for the generalized MHD equations. Commun. Partial Differ. Equ. 33, 285–306 (2008) Zhou, Y., Lei, Z.: Logarithmically improved criteria for Euler and Navier–Stokes equations (2008, submitted) Zhou Y.: On regularity criteria in terms of pressure for the Navier–Stokes equations in \({\mathbb{R}^{3}}\) . Proc. Am. Math. Soc. 134, 149–156 (2005) Zhou Y.: On a regularity criterion in terms of the gradient of pressure for the Navier–Stokes equations in \({{\mathbb R}_N}\) . Z. Angew. Math. Phys. 57, 384–392 (2006) Zhou Y.: Remarks on regularities for the 3D MHD equations. Disc. Cont. Dyn. Syst. 12, 881–886 (2005) Zhou Y.: Regularity criteria for the 3D MHD equations in terms of the pressure. Int. J. Non-Linear Mech. 41, 1174–1180 (2006) Zhou Y.: Regularity criteria for the generalized viscous MHD equations. Ann. Inst. H. Poincar Anal. Non Linaire 24, 491–505 (2007) Zhou, Y., Fan, J.: Regularity criteria for the viscous Camassa–Holm equations. Int. Math. Res. Not. IMRN, 2508–2518 (2009) Zhou, Y., Fan, J.: On regularity criteria in terms of pressure for the 3D viscous MHD equations (2009, submitted) Zhou, Y., Fan, J.: Logarithmically improved regularity criteria for the 3-D viscous MHD equations (2009, submitted) Zhou Y., Gala S.: Logarithmically improved regularity criteria for the Navier–Stokes equations in multiplier spaces. J. Math. Anal. Appl. 356, 498–501 (2009) Zhou Y., Gala S.: Regularity criteria for the solutions to the 3D MHD equations in the multiplier space. Z. Angew. Math. Phys. 61, 193–199 (2010)