Two regularity criteria for 3D Navier-Stokes equations in a bounded domain

Abstract

We prove two new regularity criteria for the 3D incompressible Navier-Stokes equations in a bounded domain. Our results also hold for the 3D Boussinesq system with zero heat conductivity.

Keywords

3D incompressible Navier-Stokes equations / Boussinesq system / regularity criterion

References

[1] Adams R A, Fournier J F. Sobolev Spaces. 2nd ed. Pure and Appl Math Ser, Vol 140. Amsterdam: Elsevier/Academic Press, 2003

[2] Azzam J, Bedrossian J. Bounded mean oscillation and the uniqueness of active scalar equations. arXiv: 1108.2735 v2 [math. AP]

[3] Beirão da Veiga H. A new regularity class for the Navier-Stokes equations in Rn. Chin Ann Math Ser B, 1995, 16: 407–412

[4] Beirão da Veiga H, Cripo F. Sharp inviscid limit results under Navier type boundary conditions. An L^p theory. J Math Fluid Mech, 2010, 12: 397–411

[5] Berselli L, Galdi G P. Regularity criteria involving the pressure for the weak solutions to the Navier-Stokes equations. Proc Amer Math Soc, 2002, 130: 3585–3595

[6] Cao C, Titi E S. Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor. Arch Ration Mech Anal, 2011, 202: 919–932

[7] Constantin P, Fefferman C. Direction of vorticity and the problem of global regularity for the Navier-Stokes equations. Indiana Univ Math J, 1993, 42: 775–789

[8] Fan J, Jiang S, Nakamura G, Zhou Y. Logarithmically improved regularity criteria for the Navier-Stokes and MHD equations. J Math Fluid Mech, 2011, 13: 557–571

[9] Giga Y. Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier-Stokes system. J Differential Equations, 1986, 62: 186–212

[10] Kang K, Lee J. On regularity criteria in conjunction with the pressure of the Navier-Stokes equations. Int Math Res Not IMRN, 2006, Artical ID 80762 (pp 1–25)

[11] Kang K, Lee J. Erratum: On regularity criteria in conjunction with the pressure of the Navier-Stokes equations. Int Math Res Not IMRN, 2010, (9): 1772–1774

[12] Kim H. A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations. SIAM J Math Anal, 2006, 37: 1417–1434

[13] Lunardi A. Interpolation Theory. 2nd ed. Lecture Notes Scuola Normale Superiore di Pisa (New Series).Pisa: Edizioni della Normale, 2009

[14] Ohyama T. Interior regularity of weak solutions of the time-dependent Navier-Stokes equation. Proc Japan Acad, 1960, 36: 273–277

[15] Pedlosky J. Geophysical Fluid Dynamics.New York: Springer-Verlag, 1987

[16] Serrin J. On the interior regularity of weak solutions of the Navier-Stokes equations. Arch Ration Mech Anal, 1962, 9: 187–195

[17] Triebel H. Theory of Function Spaces.Basel: Birkhäuser, 1983

[18] Vasseur A. Regularity criterion for 3D Navier-Stokes equations in terms of the direction of the velocity. Appl Math, 2009, 54(1): 47–52

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