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A generalization of some regularity criteria to the Navier-Stokes equations involving one velocity component. (English) Zbl 1337.35109
Amann, Herbert (ed.) et al., Recent developments of mathematical fluid mechanics. Proceedings of the international conference on mathematical fluid dynamics on the occasion of Yoshihiro Shibata’s 60th birthday, Nara, Japan, March, 5–9, 2013. Basel: Birkhäuser/Springer (ISBN 978-3-0348-0938-2/hbk; 978-3-0348-0939-9/ebook). Advances in Mathematical Fluid Mechanics, 79-97 (2016).
Summary: We present generalizations of results concerning conditional global regularity of weak Leray-Hopf solutions to incompressible Navier-Stokes equations presented by Zhou and Pokorný in articles ([M. Pokorný [Electron. J. Differ. Equ. 2003, Paper No. 11, 8 p. (2003; Zbl 1014.35073)], Y. Zhou [Methods Appl. Anal. 9, No. 4, 563–578 (2002; Zbl 1166.35359); J. Math. Pures Appl. (9) 84, No. 11, 1496–1514 (2005; Zbl 1092.35081)]) see also [J. Neustupa et al., in: Topics in mathematical fluid mechanics. Meeting on the occasion of Professor John G. Heywood sixtieth birthday, Capo Miseno, Italy, May 27–30, 2000. Rome: Aracne. 163–183 (2002; Zbl 1050.35073)]. We are able to replace the condition on one velocity component (or its gradient) by a corresponding condition imposed on a projection of the velocity (or its gradient) onto a more general vector field. Comparing to our other recent results from [Š. Axmann and M. Pokorný, “A note on regularity criteria for the solutions to Navier-Stokes equations involving one velocity component” (submitted)], the conditions imposed on the projection are more restrictive here, however due to the technique used in [loc. cit.], there appeared a specific additional restriction on geometrical properties of the reference field, which could be omitted here.
For the entire collection see [Zbl 1341.35001].
MSC:
35Q30 Navier-Stokes equations
35B65 Smoothness and regularity of solutions to PDEs
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