Abstract
This paper proves a logarithmic regularity criterion for 3D Navier-Stokes system in a bounded domain with the Navier-type boundary condition.
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R. A. Adams, J. J. F. Fournier. Sobolev Spaces. Pure and Applied Mathematics 140, Academic Press, New York, 2003.
J. Azzam, J. Bedrossian: Bounded mean oscillation and the uniqueness of active scalar equations. Trans. Am. Math. Soc. 567 2015), 3095–3118.
H. Beirao da Veiga, L. C. Berselli: Navier-Stokes equations: Green’s matrices, vorticity direction, and regularity up to the boundary. J. Differ. Equation. 37 2009), 597–628.
H. Beirao da Veiga, F. Crispo: Sharp inviscid limit results under Navier type boundary conditions. An V theory. J. Math. Fluid Mech. 37 2010), 397–411.
A. Bendali, J. M. Dominguez, S. Gallic: A variational approach for the vector potential formulation of the Stokes and Navier-Stokes problems in three-dimensional domains. J. Math. Anal. Appl. 107 (1985), 537–560.
L. C. Berselli: On a regularity criterion for the solutions to the 3D Navier-Stokes equations. Differ. Integral Equ. 37 2002), 1129–1137.
V. Georgescu: Some boundary value problems for differential forms on compact Riemannian manifolds. Ann. Mat. Pura Appl. (4) 122. 1979), 159–198.
Y. Giga: Solutions for semilinear parabolic equations in LP and regularity of weak solutions of the Navier-Stokes system. J. Differ. Equation. 37 1986), 186–212.
F. He, C. Ma, Y. Wang: On regularity for the Boussinesq system in a bounded domain. Appl. Math. Comput. 37 2016), 148–151.
E. Hopf: Über die Anfangswertaufgabe fur die hydrodynamischen Grundgleichungen. Math. Nachr. 37 1951), 213–231. (In German.)
T. Huang, C. Wang, H. Wen: Strong solutions of the compressible nematic liquid crystal flow. J. Differ. Equation. 37 2012), 2222–2265.
H. Kim: A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations. SIAM J. Math. Anal. 37 2006), 1417–1434.
J. Leray: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 37 1934), 193–248. (In French.)
P.-L. Lions: Mathematical Topics in Fluid Mechanics. Vol. 2: Compressible Models. Oxford Lecture Series in Mathematics and Its Applications 10, Clarendon Press, Oxford, 1998.
A. Lunardi: Interpolation Theory. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) 9, Edizioni della Normale, Pisa, 2009.
K. Nakao, Y. Taniuchi: An alternative proof of logarithmically improved Beale-Kato-Majda type extension criteria for smooth solutions to the Navier-Stokes equations. Nonlinear Anal, Theory Methods Appl, Ser. A, Theory Method. 37 2018), 48–55.
K. Nakao, Y. Taniuchi: Brezis-Gallouet-Wainger type inequalities and blow-up criteria for Navier-Stokes equations in unbounded domains. Commun. Math. Phys. 37 2018), 951–973.
L. Nirenberg: On elliptic partial differential equations. Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 37 1959), 115–162.
W. von Wahl: Estimating ∇u by div u and curl u. Math. Methods Appl. Sci. 37 1992), 123–143.
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The authors are indebted the two referees for helpful comments and suggestions.
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This work is partially supported by the National Natural Science Foundation of China (Grants No. 11171154 and 11801585).
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Fan, J., Jia, X. & Zhou, Y. A logarithmic regularity criterion for 3D Navier-Stokes system in a bounded domain. Appl Math 64, 397–407 (2019). https://doi.org/10.21136/AM.2019.0246-18
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DOI: https://doi.org/10.21136/AM.2019.0246-18