Abstract
We show that if (v; p) is a suitable weak solution to the Navier-Stokes equations (in the sense of L. Caffarelli, R. Kohn & L. Nirenberg - see [1]) such that v3 (the third component of v) is essentially bounded in a subdomain D of a time-space cylinder Q T then v has no singular points in D.
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References
Caffareili, L., Kohn, R. and Nirenberg, L. (1982). Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Apl. Math., 35:771–831.
Foias, C. and Temam, R. (1979). Some analytic and geometric properties of the solutions of the evolution Navier-Stokes equations. J. Math. Pures Appl., 58:339–368.
Galdi, G.P. (1994). An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol I: Linearized Steady Problems, Vol II: Nonlinear Steady Problems, Springer Tracts in Natural Philosophy, Vol. 38, 39, Springer-Verlag, New York-Berlin-Heidelberg.
Galdi, G.P. An Introduction to the Navier-Stokes Initial-Boundary Value Problem. To be published.
Giga, Y. (1986). Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier-Stokes equations. J. Differential Equations, 61:186–212.
Heywood, J.G. (1980). The Navier-Stokes equations: On the existence, uniqueness and decay of solutions. Indiana Univ. Math. J., 29:639–681.
Hopf, E. (1950). Uber die Anfangwertaufgabe fur die Hydrodynamirmhen Grundgleichungen. Math. Nachr., 4:213–231.
Kaniel, S. and Shinbrot, M. (1967). Smoothness of weak solutions of the Navier-Stokes equations. Arch. Rational Mech. Anal., 24:302–324.
Kiselev, K.K. and Ladyzhenskaya, O.A. (1957). On existence and uniqueness of the solutions of the nonstationary problem for a viscous incompressible fluid (in Russian). Izv. Akad. Nauk SSSR, 21:655–680.
Kozono, H. and Sohr, H. (1996). Remark on uniqueness of weak solutions to the Navier-Stokes equations. Analysis, 16:255–271.
Kozono, H. (1998). Uniqueness and regularity of weak solutions to the Navier-Stokes equations. Lecture Notes in Num. and Appl. Anal., 16:161–208.
Ladyzhenskaya, O.A. (1967). Uniqueness and smoothness of generalized solutions of the Navier-Stokes equations (in Russian). Zap. Nauch. Sem. LOMI, 5:169–185.
Leray, J. (1934). Sur le mouvements ďun liquide visqueux emplissant ľespace. Ada Math., 63:193–248.
Nečas, J., Růžička, M. and Sverak V. (1996). On Leray’s self-similar solutions of the Navier-Stokes equations. Ada Math., 176:283–294.
Prodi, G. (1959). Un teorema di unicita per el equazioni di Navier-Stokes. Ann. Mat. Pura Appl., 48:173–182.
Serrin, J. On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Rational Mech. Anal., 9:187–195.
Sohr, H. and von Wahl, W. (1984). On the singular set and the uniqueness of weak solutions of the Navier-Stokes equations. Manuscripta Math., 49:27–59.
Solonnikov, V.A. (1973). Estimates of solutions of a non-stationary Navier-Stokes system (in Russian). Zap. Nauch. Sem. LOMI, 38:153–231.
Temam, R. (1977). Navier-Stokes Equations. North-Holland, Amsterdam-New York-Oxford.
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Neustupa, J., Penel, P. (2002). Regularity of a Suitable Weak Solution to the Navier-Stokes Equations as a Consequence of Regularity of One Velocity Component. In: Sequeira, A., da Veiga, H.B., Videman, J.H. (eds) Applied Nonlinear Analysis. Springer, Boston, MA. https://doi.org/10.1007/0-306-47096-9_26
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DOI: https://doi.org/10.1007/0-306-47096-9_26
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-306-46303-7
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