American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Positive solutions of integral systems involving Bessel potentials
  • CPAA Home
  • This Issue
  • Next Article
    Elliptic equations involving linear and superlinear terms and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions
November  2013, 12(6): 2715-2719. doi: 10.3934/cpaa.2013.12.2715

Logarithmically improved criteria for Euler and Navier-Stokes equations

Yi Zhou 1, and  Zhen Lei 2,

1. 

Department and Institute of Mathematics, Fudan University, Shanghai 200433, China
2. 

Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433

Received  September 2012 Revised  February 2013 Published  May 2013

In this paper we prove the logarithmically improved Serrin's criteria to the three-dimensional incompressible Navier-Stokes equations.
Keywords: global regularity., Serrin's Criterion, Navier-Stokes equations.
Mathematics Subject Classification: Primary: 35Q35; Secondary: 76D0.
Citation: Yi Zhou, Zhen Lei. Logarithmically improved criteria for Euler and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2715-2719. doi: 10.3934/cpaa.2013.12.2715
References:
[1]

J. T. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the $3$-D Euler equations,, Comm. Math. Phys., 94 (1984), 61.   Google Scholar

[2]

J. M. Bony, Calcul symbolique et propagation des singularites pour les quations aux drivees partielles non lineaires,, Ann. Sci. Ecole Norm. Sup., 14 (1981), 209.   Google Scholar

[3]

C. H. Chan and A. Vasseur, Log improvement of the Prodi-Serrin criteria for Navier-Stokes equations,, available online at arXiv:0705.3659., ().   Google Scholar

[4]

J. Y. Chemin, "Perfect Incompressibe Fluids,", Oxford University Press, (1998).   Google Scholar

[5]

P. Constantin and C. Foias, "Navier-Stokes Equations,", Chicago University Press, (1988).   Google Scholar

[6]

P. Constantin and C. Fefferman, Direction of vorticity and the problem of global regularity for the Navier-Stokes equations,, Indiana Univ. Math. J., 42 (1993), 775.   Google Scholar

[7]

C. Fefferman, http://www.claymath.org/millennium/Navier-Stokes equations., preprint., ().   Google Scholar

[8]

L. Iskauriaza, G. A. Seregin and V. Shverak, $L_{3,\infty}$-solutions of Navier-Stokes equations and backward uniqueness,, (Russian) Uspekhi Mat. Nauk, 58 (2003), 3.   Google Scholar

[9]

H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations,, Math. Z., 235 (2000), 173.   Google Scholar

[10]

O. A. Ladyzhenskaya, "Mathematical Questions of the Dynamics of a Viscous Incompressible Fluid,", Nauka, (1970).   Google Scholar

[11]

A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow,", Cambridge Texts in Applied Mathematics, (2002).   Google Scholar

[12]

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

[13]

G. Seregin and V. Sverak, Navier-Stokes equations with lower bounds on the pressure,, Arch. Ration. Mech. Anal., 163 (2002), 65.   Google Scholar

[14]

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

[15]

J. Serrin, The initial value problem for the Navier-Stokes equations,, Nonlinear Problems, (1963), 69.   Google Scholar

[16]

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

[17]

T. Tao, Nonlinear dispersive equations. Local and global analysis,, CBMS Regional Conference Series in Mathematics, (2006).   Google Scholar

[18]

R. Temam, "Navier-Stokes Equations,", Second Edition, (2001).   Google Scholar

[19]

H. Triebel, "Theory of Function Spaces,", Birkauser Verlag, (1983).   Google Scholar

[20]

Y. Zhou and S. Gala, Logarithmically improved Serrin's criterion to the Navier-Stokes equations in multiplier spaces,, preprint, (2008).   Google Scholar

[21]

Y. Zhou and Z. Lei, Logarithmically improved criteria for Euler and Navier-Stokes equations,, avaliable on http://arxiv.org/abs/0805.2784v1, ().   Google Scholar

show all references

References:
[1]

J. T. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the $3$-D Euler equations,, Comm. Math. Phys., 94 (1984), 61.   Google Scholar

[2]

J. M. Bony, Calcul symbolique et propagation des singularites pour les quations aux drivees partielles non lineaires,, Ann. Sci. Ecole Norm. Sup., 14 (1981), 209.   Google Scholar

[3]

C. H. Chan and A. Vasseur, Log improvement of the Prodi-Serrin criteria for Navier-Stokes equations,, available online at arXiv:0705.3659., ().   Google Scholar

[4]

J. Y. Chemin, "Perfect Incompressibe Fluids,", Oxford University Press, (1998).   Google Scholar

[5]

P. Constantin and C. Foias, "Navier-Stokes Equations,", Chicago University Press, (1988).   Google Scholar

[6]

P. Constantin and C. Fefferman, Direction of vorticity and the problem of global regularity for the Navier-Stokes equations,, Indiana Univ. Math. J., 42 (1993), 775.   Google Scholar

[7]

C. Fefferman, http://www.claymath.org/millennium/Navier-Stokes equations., preprint., ().   Google Scholar

[8]

L. Iskauriaza, G. A. Seregin and V. Shverak, $L_{3,\infty}$-solutions of Navier-Stokes equations and backward uniqueness,, (Russian) Uspekhi Mat. Nauk, 58 (2003), 3.   Google Scholar

[9]

H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations,, Math. Z., 235 (2000), 173.   Google Scholar

[10]

O. A. Ladyzhenskaya, "Mathematical Questions of the Dynamics of a Viscous Incompressible Fluid,", Nauka, (1970).   Google Scholar

[11]

A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow,", Cambridge Texts in Applied Mathematics, (2002).   Google Scholar

[12]

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

[13]

G. Seregin and V. Sverak, Navier-Stokes equations with lower bounds on the pressure,, Arch. Ration. Mech. Anal., 163 (2002), 65.   Google Scholar

[14]

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

[15]

J. Serrin, The initial value problem for the Navier-Stokes equations,, Nonlinear Problems, (1963), 69.   Google Scholar

[16]

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

[17]

T. Tao, Nonlinear dispersive equations. Local and global analysis,, CBMS Regional Conference Series in Mathematics, (2006).   Google Scholar

[18]

R. Temam, "Navier-Stokes Equations,", Second Edition, (2001).   Google Scholar

[19]

H. Triebel, "Theory of Function Spaces,", Birkauser Verlag, (1983).   Google Scholar

[20]

Y. Zhou and S. Gala, Logarithmically improved Serrin's criterion to the Navier-Stokes equations in multiplier spaces,, preprint, (2008).   Google Scholar

[21]

Y. Zhou and Z. Lei, Logarithmically improved criteria for Euler and Navier-Stokes equations,, avaliable on http://arxiv.org/abs/0805.2784v1, ().   Google Scholar

[1]

Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure & Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675
[2]

Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637
[3]

Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617
[4]

Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137
[5]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056
[6]

Luigi C. Berselli, Jishan Fan. Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain. Communications on Pure & Applied Analysis, 2015, 14 (2) : 637-655. doi: 10.3934/cpaa.2015.14.637
[7]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271
[8]

Z. Reichstein and B. Youssin. Parusinski's "Key Lemma" via algebraic geometry. Electronic Research Announcements, 1999, 5: 136-145.

[9]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094
[10]

Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212
[11]

Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597
[12]

Philippe G. Lefloch, Cristinel Mardare, Sorin Mardare. Isometric immersions into the Minkowski spacetime for Lorentzian manifolds with limited regularity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 341-365. doi: 10.3934/dcds.2009.23.341
[13]

Ronald E. Mickens. Positivity preserving discrete model for the coupled ODE's modeling glycolysis. Conference Publications, 2003, 2003 (Special) : 623-629. doi: 10.3934/proc.2003.2003.623
[14]

Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397
[15]

A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044
[16]

Irena PawŃow, Wojciech M. Zajączkowski. Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1331-1372. doi: 10.3934/cpaa.2017065
[17]

Hakan Özadam, Ferruh Özbudak. A note on negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$. Advances in Mathematics of Communications, 2009, 3 (3) : 265-271. doi: 10.3934/amc.2009.3.265
[18]

Hyeong-Ohk Bae, Hyoungsuk So, Yeonghun Youn. Interior regularity to the steady incompressible shear thinning fluids with non-Standard growth. Networks & Heterogeneous Media, 2018, 13 (3) : 479-491. doi: 10.3934/nhm.2018021
[19]

Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827
[20]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (61)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences