Abstract
We obtain logarithmic improvements for conditions for regularity of the Navier-Stokes equation, similar to those of Prodi-Serrin or Beale-Kato-Majda. Some of the proofs make use of a stochastic approach involving Feynman-Kac-like inequalities. As part of our methods, we give a different approach to a priori estimates of Foias, Guillope and Temam.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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), 61–66.
B. Busnello, F. Flandoli, and M. Romito: A probabilistic representation for the vorticity of a 3D viscous fluid and for general systems of parabolic equations. Preprint, http://arxiv.org/abs/math/0306075.
M. Cannone: Wavelets, paraproducts and Navier-Stokes. Diderot Editeur, Paris, 1995. (In French.)
A. Chorin: Vorticity and Turbulence. Appl. Math. Sci., Vol. 103. Springer-Verlag, New York, 1994.
P. Constantin: An Eulerian-Lagrangian approach to the Navier-Stokes equations. Commun. Math. Phys. 216 (2001), 663–686.
P. Constantin, C. Foias: Navier-Stokes Equations. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, 1988.
C. R. Doering, J. D. Gibbon: Applied Analysis of the Navier-Stokes Equations. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 1995.
L. Escauriaza, G. Seregin, and V. Sverak: On L 3;∞-solutions to the Navier-Stokes equations and backward uniqueness. http://www.ima.umn.edu/preprints/dec2002/dec2002.html.
C. Foias, C. Guillope, and R. Temam: New a priori estimates for Navier-Stokes equations in dimension 3. Commun. Partial Differ. Equations 6 (1981), 329–359.
Z. Grujic, I. Kukavica: Space analyticity for the Navier-Stokes and related equations with initial data in L p. J. Funct. Anal. 152 (1998), 447–466.
I. Karatzas, S. E. Shreve: Brownian Motion and Stochastic Calculus, second edition. Graduate Texts in Mathematics Vol. 113. Springer-Verlag, New York, 1991.
H. Kozono, Y. Taniuchi: Bilinear estimates in BMO and the Navier-Stokes equations. Math. Z. 235 (2000), 173–194.
M. A. Krasnosel'skii, Ya. B. Rutitskii: Convex Functions and Orlicz Spaces. Translated from the first Russian edition. P. Noordhoff, Groningen, 1961.
P. G. Lemarie-Rieusset: Recent Developments in the Navier-Stokes Problem. Chapman and Hall/CRC, Boca Raton, 2002.
P. G. Lemarie-Rieusset: Further remarks on the analyticity of mild solutions for the Navier-Stokes equations in ℝ3. C. R. Math. Acad. Sci. Paris 338 (2004), 443–446. (In French.)
S. J. Montgomery-Smith, M. Pokorny: A counterexample to the smoothness of the solution to an equation arising in fluid mechanics. Comment. Math. Univ. Carolin. 43 (2002), 61–75.
G. Prodi: Un teorema di unicita per le equazioni di Navier-Stokes. Ann. Mat. Pura Appl. 48 (1959), 173–182. (In Italian.)
V. Scheffer: Turbulence and Hausdorff Dimension. Turbulence and Navier-Stokes Equations (Proc. Conf., Univ. Paris-Sud, Orsay, 1975). Lect. Notes Math. Vol. 565. Springer-Verlag, Berlin, 1976, pp. 174–183.
J. Serrin: On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Ration. Mech. Anal. 9 (1962), 187–195.
H. Sohr: Zur Regularitatstheorie der instationaren Gleichungen von Navier-Stokes. Math. Z. 184 (1983), 359–375.
R. Temam: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, second edition. Applied Mathematical Sciences Vol. 68. Springer-Verlag, New York, 1997.
Author information
Authors and Affiliations
Additional information
The author was partially supported by an NSF grant.
Rights and permissions
About this article
Cite this article
Montgomery-Smith, S. Conditions Implying Regularity of the Three Dimensional Navier-Stokes Equation. Appl Math 50, 451–464 (2005). https://doi.org/10.1007/s10492-005-0032-0
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s10492-005-0032-0