REGULARITY CRITERION ON WEAK SOLUTIONS TO THE NAVIER-STOKES EQUATIONS

• Gala, Sadek (UNIVERSITY OF MOSTAGANEM DEPARTMENT OF MATHEMATICS)
• Published : 2008.03.31

Abstract

Consider a weak solution u of the Navier-Stokes equations in the class $L^2((0,T);X_1(\mathbb{R}^d)^d)$. We establish a new approach to treat the regularity problem for the Navier-Stokes equation in term of the multiplier space $X_1(\mathbb{R}^d)$.

References

1. H. Beirao da Veiga, A new regularity class for the Navier-Stokes equations in $R^n$, Chinese Ann. Math. Ser. B 16 (1995), no. 4, 407-412
2. R. Coifman, P.-L. Lions, Y. Meyer, and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. (9) 72 (1993), no. 3, 247-286
3. P. Constantin, Remarks on the Navier-Stokes Equations, New perspectives in turbulence (Newport, RI, 1989), 229-261, Springer, New York, 1991
4. C. Foias, Une remarque sur l'unicite des solutions des equations de Navier-Stokes en dimension n, Bull. Soc. Math. France 89 (1961), 1-8
5. C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137-193 https://doi.org/10.1007/BF02392215
6. E. Hopf, Uber die Anfangswertaufgabe fur die hydrodynamischen Grundgleichungen, Math. Nachr. 4 (1951), 213-231
7. T. Kato, Strong solutions of the Navier-Stokes equation in Morrey spaces, Bol. Soc. Brasil. Mat. (N.S.) 22 (1992), no. 2, 127-155 https://doi.org/10.1007/BF01232939
8. H. Kozono and H. Sohr, Regularity criterion of weak solutions to the Navier-Stokes equations, Adv. Differential Equations 2 (1997), no. 4, 535-554
9. H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations, Math. Z. 235 (2000), no. 1, 173-194 https://doi.org/10.1007/s002090000130
10. P. G. Lemarie-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics, 431, Chapman & Hall/CRC, Boca Raton, FL, 2002
11. P. G. Lemarie-Rieusset and S. Gala, Multipliers between Sobolev spaces and fractional differentiation, J. Math. Anal. Appl. 322 (2006), no. 2, 1030-1054 https://doi.org/10.1016/j.jmaa.2005.07.043
12. J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math. 63 (1934), no. 1, 193-248 https://doi.org/10.1007/BF02547354
13. F. Murat, Compacite par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), no. 3, 489-507
14. K. Masuda, Weak solutions of Navier-Stokes equations, Tohoku Math. J. (2) 36 (1984), no. 4, 623-646 https://doi.org/10.2748/tmj/1178228767
15. L. Tartar, Compensated Compactness and Applications to Partial Differential Equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, pp. 136-212, Res. Notes in Math., 39, Pitman, Boston, Mass.-London, 1979
16. J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal. 9 (1962), 187-195
17. E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, With the assistance of Timothy S. Murphy. Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993
18. E. M. Stein and G.Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, No. 32. Princeton University Press, Princeton, N.J., 1971
19. M. E. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. Partial Differential Equations 17 (1992), no. 9-10, 1407-1456 https://doi.org/10.1080/03605309208820892
20. R. Temam, Navier-Stokes Equations, Studies in Mathematics and its Applications, Vol. 2. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977

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