American Institute of Mathematical Sciences

Advanced
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 & 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.   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

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

[1]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259
[2]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348
[3]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241
[4]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110
[5]

Martin Kalousek, Joshua Kortum, Anja Schlömerkemper. Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 17-39. doi: 10.3934/dcdss.2020331
[6]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264
[7]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216
[8]

Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020352
[9]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081
[10]

Helmut Abels, Johannes Kampmann. Existence of weak solutions for a sharp interface model for phase separation on biological membranes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 331-351. doi: 10.3934/dcdss.2020325
[11]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272
[12]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, 2021, 20 (1) : 319-338. doi: 10.3934/cpaa.2020268
[13]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168
[14]

Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137
[15]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320
[16]

Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020443
[17]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318
[18]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456
[19]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440
[20]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (62)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences