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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

Vittorino Pata 1,

1. 

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

Received  November 2010 Revised  April 2011 Published  October 2011

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 and Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747
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), 381-391.

[2]

C. Bjorland and A. Vasseur, Weak in space, log in time improvement of the Ladyž zenskaja-Prodi-Serrin criteria, J. Math. Fluid Mech., in press.

[3]

C. H. Chan and A. Vasseur, Log improvement of the Prodi-Serrin criteria for Navier-Stokes equations, Methods Appl. Anal., 14 (2007), 197-212.

[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), 211-250.

[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), 329-359.

[6]

E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr., 4 (1950), 213-231.

[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), 186-212.

[8]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.

[9]

S. Montgomery-Smith, Conditions implying regularity of the three dimensional Navier-Stokes equation, Appl. Math., 50 (2005), 451-464.

[10]

J. Nečas, M. Ruzička and V. Šverák, On Leray's self-similar solutions of the Navier-Stokes equations, Acta Math., 176 (1996), 283-294.

[11]

G. Prodi, Un teorema di unicità per le equazioni di Navier-Stokes, Ann. Mat. Pura Appl., 48 (1959), 173-182.

[12]

J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 9 (1962), 187-195.

[13]

H. Sohr, A regularity class for the Navier-Stokes equations in Lorentz spaces, J. Evol. Equ., 1 (2001), 441-467.

[14]

H. Sohr, "The Navier-Stokes Euations," Birkhäuser Verlag, Basel, 2001.

[15]

M. Struwe, On partial regularity results for the Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 437-458.

[16]

S. Takahashi, On interior regularity criteria for weak solutions of the Navier-Stokes equations, Manuscripta Math., 69 (1990), 237-254.

[17]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372.

[18]

R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis," CBMS-NSF Regional conference Series in Applied Mathematics, Philadelphia, 1983.

[19]

R. Temam, "Navier-Stokes Equations," AMS Chelsea Publishing, Providence, 2001.

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), 381-391.

[2]

C. Bjorland and A. Vasseur, Weak in space, log in time improvement of the Ladyž zenskaja-Prodi-Serrin criteria, J. Math. Fluid Mech., in press.

[3]

C. H. Chan and A. Vasseur, Log improvement of the Prodi-Serrin criteria for Navier-Stokes equations, Methods Appl. Anal., 14 (2007), 197-212.

[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), 211-250.

[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), 329-359.

[6]

E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr., 4 (1950), 213-231.

[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), 186-212.

[8]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.

[9]

S. Montgomery-Smith, Conditions implying regularity of the three dimensional Navier-Stokes equation, Appl. Math., 50 (2005), 451-464.

[10]

J. Nečas, M. Ruzička and V. Šverák, On Leray's self-similar solutions of the Navier-Stokes equations, Acta Math., 176 (1996), 283-294.

[11]

G. Prodi, Un teorema di unicità per le equazioni di Navier-Stokes, Ann. Mat. Pura Appl., 48 (1959), 173-182.

[12]

J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 9 (1962), 187-195.

[13]

H. Sohr, A regularity class for the Navier-Stokes equations in Lorentz spaces, J. Evol. Equ., 1 (2001), 441-467.

[14]

H. Sohr, "The Navier-Stokes Euations," Birkhäuser Verlag, Basel, 2001.

[15]

M. Struwe, On partial regularity results for the Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 437-458.

[16]

S. Takahashi, On interior regularity criteria for weak solutions of the Navier-Stokes equations, Manuscripta Math., 69 (1990), 237-254.

[17]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372.

[18]

R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis," CBMS-NSF Regional conference Series in Applied Mathematics, Philadelphia, 1983.

[19]

R. Temam, "Navier-Stokes Equations," AMS Chelsea Publishing, Providence, 2001.

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