On the regularity of solutions to the Navier-Stokes equations

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March 2012, 11(2): 747-761. doi: 10.3934/cpaa.2012.11.747

On the regularity of solutions to the Navier-Stokes equations

1.	Politecnico di Milano - Dipartimento di Matematica "F. Brioschi", Via Bonardi 9, 20133 Milano

Received November 2010 Revised April 2011 Published October 2011

Abstract
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This article is concerned with the incompressible Navier-Stokes equations in a three-dimensional domain. A criterion of Prodi-Serrin type up to the boundary for global existence of strong solutions is established.

Keywords: weak solutions, Navier-Stokes equations, regularity criteria., strong solutions, blow-up.

Mathematics Subject Classification: Primary: 35Q30; Secondary: 76D03, 76D0.

Citation: Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747

References:

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[6]	E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen,, Math. Nachr., 4 (1950), 213. Google Scholar
[7]	Y. Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier-Stokes equations,, J. Differential Equations, 62 (1986), 186. Google Scholar
[8]	J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace,, Acta Math., 63 (1934), 193. Google Scholar
[9]	S. Montgomery-Smith, Conditions implying regularity of the three dimensional Navier-Stokes equation,, Appl. Math., 50 (2005), 451. Google Scholar
[10]	J. Nečas, M. Ruzička and V. Šverák, On Leray's self-similar solutions of the Navier-Stokes equations,, Acta Math., 176 (1996), 283. Google Scholar
[11]	G. Prodi, Un teorema di unicità per le equazioni di Navier-Stokes,, Ann. Mat. Pura Appl., 48 (1959), 173. Google Scholar
[12]	J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations,, Arch. Ration. Mech. Anal., 9 (1962), 187. Google Scholar
[13]	H. Sohr, A regularity class for the Navier-Stokes equations in Lorentz spaces,, J. Evol. Equ., 1 (2001), 441. Google Scholar
[14]	H. Sohr, "The Navier-Stokes Euations,", Birkh\, (2001). Google Scholar
[15]	M. Struwe, On partial regularity results for the Navier-Stokes equations,, Comm. Pure Appl. Math., 41 (1988), 437. Google Scholar
[16]	S. Takahashi, On interior regularity criteria for weak solutions of the Navier-Stokes equations,, Manuscripta Math., 69 (1990), 237. Google Scholar
[17]	G. Talenti, Best constant in Sobolev inequality,, Ann. Mat. Pura Appl., 110 (1976), 353. Google Scholar
[18]	R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis,", CBMS-NSF Regional conference Series in Applied Mathematics, (1983). Google Scholar
[19]	R. Temam, "Navier-Stokes Equations,", AMS Chelsea Publishing, (2001). Google Scholar

show all references

References:

[1]	H. Beirão da Veiga, Remarks on the smoothness of the $L^\infty(0,T;L^3)$ solutions of the 3-D Navier-Stokes equations,, Portugal. Math., 54 (1997), 381. Google Scholar
[2]	C. Bjorland and A. Vasseur, Weak in space, log in time improvement of the Ladyž zenskaja-Prodi-Serrin criteria,, J. Math. Fluid Mech., (). Google Scholar
[3]	C. H. Chan and A. Vasseur, Log improvement of the Prodi-Serrin criteria for Navier-Stokes equations,, Methods Appl. Anal., 14 (2007), 197. Google Scholar
[4]	L. Escauriaza, G. Seregin and V. Šverák, $L_{3,\infty}$ -solutions of the Navier-Stokes equations and backward uniqueness,, Russian Math. Surveys, 58 (2003), 211. Google Scholar
[5]	C. Foias, C. Guillope and R. Temam, New a priori estimates for Navier-Stokes equations in dimension 3,, Comm. Partial Differential Equations, 6 (1981), 329. Google Scholar
[6]	E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen,, Math. Nachr., 4 (1950), 213. Google Scholar
[7]	Y. Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier-Stokes equations,, J. Differential Equations, 62 (1986), 186. Google Scholar
[8]	J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace,, Acta Math., 63 (1934), 193. Google Scholar
[9]	S. Montgomery-Smith, Conditions implying regularity of the three dimensional Navier-Stokes equation,, Appl. Math., 50 (2005), 451. Google Scholar
[10]	J. Nečas, M. Ruzička and V. Šverák, On Leray's self-similar solutions of the Navier-Stokes equations,, Acta Math., 176 (1996), 283. Google Scholar
[11]	G. Prodi, Un teorema di unicità per le equazioni di Navier-Stokes,, Ann. Mat. Pura Appl., 48 (1959), 173. Google Scholar
[12]	J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations,, Arch. Ration. Mech. Anal., 9 (1962), 187. Google Scholar
[13]	H. Sohr, A regularity class for the Navier-Stokes equations in Lorentz spaces,, J. Evol. Equ., 1 (2001), 441. Google Scholar
[14]	H. Sohr, "The Navier-Stokes Euations,", Birkh\, (2001). Google Scholar
[15]	M. Struwe, On partial regularity results for the Navier-Stokes equations,, Comm. Pure Appl. Math., 41 (1988), 437. Google Scholar
[16]	S. Takahashi, On interior regularity criteria for weak solutions of the Navier-Stokes equations,, Manuscripta Math., 69 (1990), 237. Google Scholar
[17]	G. Talenti, Best constant in Sobolev inequality,, Ann. Mat. Pura Appl., 110 (1976), 353. Google Scholar
[18]	R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis,", CBMS-NSF Regional conference Series in Applied Mathematics, (1983). Google Scholar
[19]	R. Temam, "Navier-Stokes Equations,", AMS Chelsea Publishing, (2001). Google Scholar

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