Conditions of Prodi-Serrin's type for local regularity of suitable weak solutions to the Navier-Stokes equations

Zdeněk Skalák

Commentationes Mathematicae Universitatis Carolinae (2002)

  • Volume: 43, Issue: 4, page 619-639
  • ISSN: 0010-2628

Abstract

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In the context of suitable weak solutions to the Navier-Stokes equations we present local conditions of Prodi-Serrin’s type on velocity 𝐯 and pressure p under which ( 𝐱 0 , t 0 ) ∈ Ω × ( 0 , T ) is a regular point of 𝐯 . The conditions are imposed exclusively on the outside of a sufficiently narrow space-time paraboloid with the vertex ( 𝐱 0 , t 0 ) and the axis parallel with the t -axis.

How to cite

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Skalák, Zdeněk. "Conditions of Prodi-Serrin's type for local regularity of suitable weak solutions to the Navier-Stokes equations." Commentationes Mathematicae Universitatis Carolinae 43.4 (2002): 619-639. <http://eudml.org/doc/248959>.

@article{Skalák2002,
abstract = {In the context of suitable weak solutions to the Navier-Stokes equations we present local conditions of Prodi-Serrin’s type on velocity $\{\mathbf \{v\}\}$ and pressure $p$ under which $(\{\mathbf \{x\}\}_0,t_0)\in \Omega \times (0,T)$ is a regular point of $\{\mathbf \{v\}\}$. The conditions are imposed exclusively on the outside of a sufficiently narrow space-time paraboloid with the vertex $(\{\mathbf \{x\}\}_0,t_0)$ and the axis parallel with the $t$-axis.},
author = {Skalák, Zdeněk},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Navier-Stokes equations; suitable weak solutions; local regularity; Navier-Stokes equations; suitable weak solution; local regularity},
language = {eng},
number = {4},
pages = {619-639},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Conditions of Prodi-Serrin's type for local regularity of suitable weak solutions to the Navier-Stokes equations},
url = {http://eudml.org/doc/248959},
volume = {43},
year = {2002},
}

TY - JOUR
AU - Skalák, Zdeněk
TI - Conditions of Prodi-Serrin's type for local regularity of suitable weak solutions to the Navier-Stokes equations
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2002
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 43
IS - 4
SP - 619
EP - 639
AB - In the context of suitable weak solutions to the Navier-Stokes equations we present local conditions of Prodi-Serrin’s type on velocity ${\mathbf {v}}$ and pressure $p$ under which $({\mathbf {x}}_0,t_0)\in \Omega \times (0,T)$ is a regular point of ${\mathbf {v}}$. The conditions are imposed exclusively on the outside of a sufficiently narrow space-time paraboloid with the vertex $({\mathbf {x}}_0,t_0)$ and the axis parallel with the $t$-axis.
LA - eng
KW - Navier-Stokes equations; suitable weak solutions; local regularity; Navier-Stokes equations; suitable weak solution; local regularity
UR - http://eudml.org/doc/248959
ER -

References

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  1. Caffarelli L., Kohn R., Nirenberg L., Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Apl. Math. 35 (1982), 771-831. (1982) Zbl0509.35067MR0673830
  2. Kučera P., Skalák Z., Smoothness of the velocity time derivative in the vicinity of regular points of the Navier-Stokes equations, Proceedings of the Seminar “Euler and Navier-Stokes Equations (Theory, Numerical Solution, Applications)”, Institute of Thermomechanics AS CR, Editors: K. Kozel, J. Příhoda, M. Feistauer, Prague, 2001, pp.83-86. 
  3. Ladyzhenskaya O.A., Seregin G.A., On partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations, J. Math. Fluid Mech. 1 (1999), 356-387. (1999) Zbl0954.35129MR1738171
  4. Lin F., A new proof of the Caffarelli-Kohn-Nirenberg Theorem, Comm. Pure Appl. Math. 51 (1998), 241-257. (1998) Zbl0958.35102MR1488514
  5. Neustupa J., Partial regularity of weak solutions to the Navier-Stokes equations in the class L ∞ ( 0 , T , L 3 ( Ø m e g a ) 3 ) , J. Math. Fluid Mech. 1 (1999), 309-325. (1999) MR1738173
  6. Neustupa J., A removable singularity of a suitable weak solution to the Navier-Stokes equations, preprint. 
  7. Nečas J., Neustupa J., New conditions for local regularity of a suitable weak solution to the Navier-Stokes equations, preprint. 
  8. Skalák Z., Removable Singularities of Weak Solutions of the Navier-Stokes Equations, Proceedings of the Seminar “Euler and Navier-Stokes Equations (Theory, Numerical Solution, Applications)”, Institute of Thermomechanics AS CR, Editors: K. Kozel, J. Příhoda, M. Feistauer, Prague, 2001, pp.121-124. 
  9. Temam R., Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland Publishing Company, Amsterdam, New York, Oxford, revised edition, 1979. Zbl0981.35001MR0603444

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