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On the Stability of Solutions of the Navier-Stokes Equations and Convergence to Steady State

1967; Society for Industrial and Applied Mathematics; Volume: 15; Issue: 2 Linguagem: Inglês

10.1137/0115035

ISSN

1095-712X

Autores

L. E. Payne,

Tópico(s)

Advanced Mathematical Physics Problems

Resumo

Previous article Next article On the Stability of Solutions of the Navier-Stokes Equations and Convergence to Steady StateL. E. PayneL. E. Paynehttps://doi.org/10.1137/0115035PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] J. H. Bramble and , L. E. Payne, Bounds for solutions of second-order elliptic partial differential equations, Contributions to Differential Equations, 1 (1963), 95–127 MR0163049 0141.09801 Google Scholar[2] J. H. Bramble and , L. E. Payne, Bounds in the Neumann problem for second order uniformly elliptic operators, Pacific J. Math., 12 (1962), 823–833 MR0146504 0111.09701 CrossrefISIGoogle Scholar[3] D. E. Edmunds, On the uniqueness of viscous flows, Arch. Rational Mech. Anal., 14 (1963), 171–176 10.1007/BF00250698 MR0154487 0126.42302 CrossrefISIGoogle Scholar[4] Gaetano Fichera, Su un principio di dualità per talune formole di maggiorazione relative alle equazioni differenziali, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8), 19 (1955), 411–418 (1956) MR0079705 0071.31801 Google Scholar[5] Robert Finn, On the steady-state solutions of the Navier-Stokes equations. III, Acta Math., 105 (1961), 197–244, see also On the Stokes paradox and related questions, Proc. Symp. Nonlinear Problems, University of Wisconsin, 1963, pp. 99–115 MR0166498 0126.42203 CrossrefISIGoogle Scholar[6] H. Fujita and , T. Kato, On the Navier-Stokes initial value problem, I. Technical Note, 121, Stanford University, Stanford, California, 1963 Google Scholar[7] Eberhard Hopf, Ein allgemeiner Endlichkeitssatz der Hydrodynamik, Math. Ann., 117 (1941), 764–775 10.1007/BF01450040 MR0005003 0024.13505 CrossrefGoogle Scholar[8] Seizô Itô, The existence and the uniqueness of regular solution of non-stationary Navier-Stokes equation, J. Fac. Sci. Univ. Tokyo Sect. I, 9 (1961), 103–140 (1961) MR0163082 0116.17905 Google Scholar[9] J. Kampé de Fériet, Sur la décroissance de l'énergie cinétique d'un fluide visqueux incompressible occupant un domaine borné ayant pour frontière des parois solides fixes, Ann. Soc. Sci. Bruxelles. Sér. I., 63 (1949), 36–45 MR0029606 0035.41605 Google Scholar[10] J. L. Lions, Sur la régularité et l'unicité des solutions turbulentes des équations de Navier Stokes, Rend. Sem. Mat. Univ. Padova, 30 (1960), 16–23 MR0115018 0098.17205 Google Scholar[11] O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Revised English edition. Translated from the Russian by Richard A. Silverman, Gordon and Breach Science Publishers, New York, 1963xiv+184 MR0155093 0121.42701 Google Scholar[12] L. E. Payne, Uniqueness criteria for steady state solutions of the Navier-Stokes equations, Atti del Simp. Inter. sulle Appl. dell' Anal. alla Fis. Mat., Cagliari-Sassari, (1964), 130–153 Google Scholar[13] L. E. Payne and , H. F. Weinberger, New bounds for solutions of second order elliptic partial differential equations, Pacific J. Math., 8 (1958), 551–573 MR0104047 0093.10901 CrossrefGoogle Scholar[14] L. E. Payne and , H. F. Weinberger, A stability bound for viscous flows, Symp. Nonlinear Problems, Mathematics Research Center, U.S. Army, University of Wisconsin, Madison, 1962 Google Scholar[15] Giovanni Prodi, Un teorema di unicità per le equazioni di Navier-Stokes, Ann. Mat. Pura Appl. (4), 48 (1959), 173–182 MR0126088 0148.08202 CrossrefGoogle Scholar[16] J. Serrin, Mathematical principles of classical fluid mechanicsEncyclopedia of Physics, VIII, Vol. 11, Springer-Verlag, Berlin, 1959, On the stability of viscous fluid flows, Arch. Rational Mech. Anal., 3 (1959), pp. 1–13 CrossrefGoogle Scholar[17] James Serrin, The initial value problem for the Navier-Stokes equations, Nonlinear Problems (Proc. Sympos., Madison, Wis., Univ. of Wisconsin Press, Madison, Wis., 1963, 69–98 MR0150444 0115.08502 Google Scholar[18] James Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187–195 10.1007/BF00253344 MR0136885 0106.18302 CrossrefISIGoogle Scholar[19] T. Y. Thomas, On the uniform convergence of the solutions of the Navier-Stokes equations, Proc. Nat. Acad. Sci. U. S. A., 29 (1943), 243–246 MR0008523 0061.43808 CrossrefGoogle Scholar[20] W. Velte, Über ein Stabilitätskriterium der Hydrodynamik, Arch. Rational Mech. Anal., 9 (1962), 9–20 10.1007/BF00253330 MR0155501 0108.39403 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails IntroductionNon-Standard and Improperly Posed Problems | 1 Jan 1997 Cross Ref Continuous Dependence on Modeling Forward in TimeNon-Standard and Improperly Posed Problems | 1 Jan 1997 Cross Ref BibliographyNon-Standard and Improperly Posed Problems | 1 Jan 1997 Cross Ref Structural Stability for the Brinkman Equations of Porous MediaMathematical Methods in the Applied Sciences, Vol. 19, No. 16 | 10 Nov 1996 Cross Ref Best constants in Korn-Poincaré's inequalities on a slabMathematical Methods in the Applied Sciences, Vol. 17, No. 7 | 10 Jun 1994 Cross Ref Double diffusive porous penetrative convection—thawing subsea permafrostInternational Journal of Engineering Science, Vol. 26, No. 8 | 1 Jan 1988 Cross Ref THE ASYMPTOTIC BEHAVIOUR OF SOME SEMI-DISCRETE AND FULLY-DISCRETE FINITE ELEMENT SCHEMES FOR THE EVOLUTION NAVIER-STOKES EQUATIONSMathematical Analysis and its Applications | 1 Jan 1988 Cross Ref On Korn’s Inequality for Incompressible MediaCornelius O. 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