×

zbMATH — the first resource for mathematics

Conditions implying regularity of the three dimensional Navier-Stokes equation. (English) Zbl 1099.35086
Summary: 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 Foiaş, Guillopé and Temam.

MSC:
35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
60H30 Applications of stochastic analysis (to PDEs, etc.)
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
PDF BibTeX XML Cite
Full Text: DOI EuDML arXiv
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), 61–66. · Zbl 0573.76029 · doi:10.1007/BF01212349
[2] 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. · Zbl 1075.76019
[3] M. Cannone: Wavelets, paraproducts and Navier-Stokes. Diderot Editeur, Paris, 1995. (In French.)
[4] A. Chorin: Vorticity and Turbulence. Appl. Math. Sci., Vol. 103. Springer-Verlag, New York, 1994. · Zbl 0795.76002
[5] P. Constantin: An Eulerian-Lagrangian approach to the Navier-Stokes equations. Commun. Math. Phys. 216 (2001), 663–686. · Zbl 0988.76020 · doi:10.1007/s002200000349
[6] P. Constantin, C. Foias: Navier-Stokes Equations. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, 1988.
[7] C. R. Doering, J. D. Gibbon: Applied Analysis of the Navier-Stokes Equations. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 1995. · Zbl 0838.76016
[8] 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.
[9] 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. · Zbl 0472.35070 · doi:10.1080/03605308108820180
[10] 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. · Zbl 0896.35105 · doi:10.1006/jfan.1997.3167
[11] I. Karatzas, S. E. Shreve: Brownian Motion and Stochastic Calculus, second edition. Graduate Texts in Mathematics Vol. 113. Springer-Verlag, New York, 1991. · Zbl 0734.60060
[12] H. Kozono, Y. Taniuchi: Bilinear estimates in BMO and the Navier-Stokes equations. Math. Z. 235 (2000), 173–194. · Zbl 0970.35099 · doi:10.1007/s002090000130
[13] M. A. Krasnosel’skii, Ya. B. Rutitskii: Convex Functions and Orlicz Spaces. Translated from the first Russian edition. P. Noordhoff, Groningen, 1961.
[14] P. G. Lemarie-Rieusset: Recent Developments in the Navier-Stokes Problem. Chapman and Hall/CRC, Boca Raton, 2002.
[15] P. G. Lemarie-Rieusset: Further remarks on the analyticity of mild solutions for the Navier-Stokes equations in \(\mathbb{R}\)3. C. R. Math. Acad. Sci. Paris 338 (2004), 443–446. (In French.)
[16] 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. · Zbl 1090.35146
[17] G. Prodi: Un teorema di unicita per le equazioni di Navier-Stokes. Ann. Mat. Pura Appl. 48 (1959), 173–182. (In Italian.) · Zbl 0148.08202 · doi:10.1007/BF02410664
[18] 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.
[19] J. Serrin: On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Ration. Mech. Anal. 9 (1962), 187–195. · Zbl 0106.18302 · doi:10.1007/BF00253344
[20] H. Sohr: Zur Regularitatstheorie der instationaren Gleichungen von Navier-Stokes. Math. Z. 184 (1983), 359–375. · Zbl 0506.35084 · doi:10.1007/BF01163510
[21] R. Temam: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, second edition. Applied Mathematical Sciences Vol. 68. Springer-Verlag, New York, 1997. · Zbl 0871.35001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.