Geodesic
Necessary conditions of potential blow up for Navier–Stokes equations
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 41, Tome 385 (2010), pp. 187-199 
G. A. Seregin. Necessary conditions of potential blow up for Navier–Stokes equations. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 41, Tome 385 (2010), pp. 187-199. http://geodesic.mathdoc.fr/item/ZNSL_2010_385_a7/
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     title = {Necessary conditions of potential blow up for {Navier{\textendash}Stokes} equations},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2010_385_a7/}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Assuming that $T$ is a potential blow up time, we show that $H^\frac12$-norm of the velocity field goes to $\infty$ as time $t$ approaches $T$. Bibl. 9 titles.

[1] L. Escauriaza, G. Seregin, V. Šverák, “$L_{3,\infty}$-solutions to the Navier–Stokes equations and backward uniqueness”, Russ. Math. Surv., 58:2 (2003), 211–250 | DOI | MR | Zbl

[2] G. Koch, N. Nadirashvili, G. Seregin, V. Sverak, “Liouville theorems for the Navier–Stokes equations and applications”, Acta Mathematica, 203 (2009), 83–105 | DOI | MR | Zbl

[3] P. G. Lemarie-Rieusset, Recent developmnets in the Navier–Stokes problem, Research notes in mathematics series, 431, Chapman Hall/CRC, 2002 | DOI | MR | Zbl

[4] J. Leray, “Sur le mouvement dún liquide visqueux emplissant léspace”, Acta Math., 63 (1934), 193–248 | DOI | MR | Zbl

[5] N. Kikuchi, G. Seregin, “Weak solutions to the Cauchy problem for the Navier–Stokes equations satisfying the local energy inequality”, Amer. Math. Soc. Ser. 2, 220 (2007), 141–164 | MR | Zbl

[6] W. Rusin, V. Sverak, Miminimal initial data for potential Navier–Stokes singularities, arXiv: 0911.0500

[7] G. A. Seregin, “Navier–Stokes equations: almost $L_{3,\infty}$-cases”, J. Math. Fluid Mech., 9 (2007), 34–43 | DOI | MR | Zbl

[8] G. A. Seregin, A note on necessary conditions for blow-up of energy solutions to the Navier–Stokes equations, arXiv: 0909.3897

[9] G. Seregin, V. Šverák, “On Type I singularities of the local axi-symmetric solutions of the Navier–Stokes equations”, Commun. PDE's, 34 (2009), 171–201 | DOI | MR | Zbl