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Sufficient conditions for the regularity of the solutions of the Navier-Stokes equations

1999; Wiley; Volume: 22; Issue: 13 Linguagem: Inglês

10.1002/(sici)1099-1476(19990910)22

1099-1476

Luigi C. Berselli,

Advanced Mathematical Physics Problems

Mathematical Methods in the Applied SciencesVolume 22, Issue 13 p. 1079-1085 Research Article Sufficient conditions for the regularity of the solutions of the Navier–Stokes equations Luigi C. Berselli, Corresponding Author Luigi C. Berselli [email protected] Dipartimento di Matematica “L.Tonelli”, Università degli Studi di Pisa. Via F. Buonarroti 2, 56127 Pisa, ItalyDipartimento di Matematica “L. Tonelli”, Università degli Studi di Pisa. Via F. Buonarroti 2, 56127 Pisa, Italy===Search for more papers by this author Luigi C. Berselli, Corresponding Author Luigi C. Berselli [email protected] Dipartimento di Matematica “L.Tonelli”, Università degli Studi di Pisa. Via F. Buonarroti 2, 56127 Pisa, ItalyDipartimento di Matematica “L. Tonelli”, Università degli Studi di Pisa. Via F. Buonarroti 2, 56127 Pisa, Italy===Search for more papers by this author First published: 29 July 1999 https://doi.org/10.1002/(SICI)1099-1476(19990910)22:13 3.0.CO;2-4Citations: 6AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onEmailFacebookTwitterLinkedInRedditWechat Abstract In this paper we find sufficient conditions, involving only the pressure, that ensure the regularity of weak solutions to the Navier–Stokes equations. Conditions involving only the pressure were previously obtained in [7,4]. Following a remark in this last reference we improve, in particular, Kaniel's result [7]. Our condition can be seen at the light of the heuristic idea that the pressure behaves similarly to the modulus squared of the velocity. Copyright © 1999 John Wiley & Sons, Ltd. References 1 Adams, R., Sobolev Spaces, Academic Press, New York, 1975. Google Scholar 2 Beirão da Veiga, H., ‘Existence and asymptotic behavior for strong solutions of the Navier–Stokes equations in the whole space’, Indiana Univ. Math. J., 36(1), 149–166 (1987). 10.1512/iumj.1987.36.36008 Web of Science®Google Scholar 3 Beirão da Veiga, H., ‘Concerning the regularity of the solutions to the Navier–Stokes equations via the truncation method. Part I’, Differential Integral Equations, 10(6), 1149–1156 (1997). Google Scholar 4 Beirão da Veiga, H., ‘ Concerning the regularity of the solutions to the Navier–Stokes equations via the truncation method. Part II’, In Équations aux dérivées partielles et applications, pp. 127–138. Gauthier-Villars, Éd. Sci. Méd. Elsevier, Paris, 1998. Google Scholar 5 Beirão da Veiga, H., ‘Regarding the effect of the pressure on the regularity of the solutions to the Navier–Stokes equations’, Math. Meth. Appl. Sci., in press. Google Scholar 6 Giga, Y. and T. Miyakawa, ‘Solutions in Lr of the Navier–Stokes initial value problem’, Arch. Rat. Mech. Anal., 89, 267–281 (1985). 10.1007/BF00276875 Web of Science®Google Scholar 7 Kaniel, S., ‘A sufficient condition for smoothness of solutions of Navier–Stokes equations’, Isr. J. Math., 6, 354–358 (1968). 10.1007/BF02771213 Web of Science®Google Scholar 8 Leray, J., ‘Essai sur le Mouvement d'un liquide Visquex Emplissant l'Espace’, Acta Math., 63, 193–248 (1934). 10.1007/BF02547354 Web of Science®Google Scholar 9 Lions, J. L., Quelque Méthodes de résolution des Problemès aux Limites Non-lineaires. Dunod Gauthier, Villars, 1969. Google Scholar 10 Miyakawa, T., ‘On the initial value problem for the Navier–Stokes equations in Lp spaces’, Hiroshima Math. J., 11, 9–20 (1981). Google Scholar 11 Prodi, G., ‘Un Teorema di Unicità per le Equazioni di Navier–Stokes’, Ann. Mat. Pura Appl. IV Ser., 48, 173–182 (1959). 10.1007/BF02410664 Google Scholar 12 Serrin, J., ‘On the interior regularity of weak solutions of the Navier–Stokes equations’, Arch. Rat. Mech. Anal., 9(3), 187–195 (1962). Web of Science®Google Scholar 13 Sohr, H., ‘Zur Regularitaetstheorie der instationaeren Gleichungen von Navier–Stokes’, Math. Z., 184, 359–375 (1983). 10.1007/BF01163510 Web of Science®Google Scholar 14 von Wahl, W., ‘ Regularity questions for the Navier–Stokes quations’, in: Approximation Methods for Navier–Stokes Problems R. Rautmann, ed., Proc. Symp. IUTAM, Paderborn, 1979, number 771 in Lect. Notes Math., pp. 538–542, Springer, Berlin, 1980. 10.1007/BFb0086929 Google Scholar Citing Literature Volume22, Issue1310 September 1999Pages 1079-1085 ReferencesRelatedInformation