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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.)
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##### 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
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