Abstract
We proveL 2-decay rates of suitable weak solutions to the Navier-Stokes equations in exterior domains. The results for the order of decay are the same as for the solutions to the Cauchy problem of the Navier-Stokes equations. Finally in the case ofω=R 3 the decay rate order is sharp in the class of solutions considered by us.
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Caffarelli, L., Kohn, R., Nirenberg, L.: Partial regularity of suitable weak solutions of the Navier-Stokes equations, Commun. Pure Appl. Math.35, 771–831 (1982)
Friedman, A.: Partial differential equations. New York: Holt, Rinehart and Winston 1969
Galdi, G. P., Rionero, S.: Weighted energy methods in fluid dynamics and elasticity. Lectures Notes in Mathematics, Vol.1134. Berlin, Heidelberg, New York: Springer 1985
Galdi, G. P., Maremonti, P.: Monotonic decreasing and asymptotic behaviour of the kinetic energy for weak solutions of the Navier-Stokes equations in exterior domains. Arch. Rat. Mech. Anal.94, 253–266 (1986)
Heywood, J. G.: The Navier-Stokes equations: On the existence, regularity and decay of solutions. Indiana Univ. Math. J.29, 639–681 (1980)
Hopf, E.: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen Math. Nachr.4, 213–231 (1951)
Kajikiva, R., Miyakawa, T.: OnL 2 decay of weak solutions of the Navier-Stokes equations inR n. Math. Z.192, 135–148 (1986).
Kato, T.: StrongL p-solutions of the Navier-Stokes equation inR n, with applications to weak solutions. Math. Z.187, 471–480 (1984)
Kiselev, A. A., Ladyzhenskaya, D. A.: On the existence and uniqueness of the solution of the nonstationary problem for a viscous incompressible fluid. Izv. Akad. Nauk SSSR Ser. Mat.21, 655–680 (1957)
Knightly, G. H.: On a class of global solutions of the Navier-Stokes equations. Arch. Rat. Mech. Anal.21, 211–245 (1966)
Leray, J.: Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math.63, 193–248 (1934)
Maremonti, P.: Asymptotic stability of incompressible viscous fluid motion in exterior domains. Rend. Sem. Mat. Univ. di Padova71, 35–72 (1984)
Maremonti, P.: Stabilità asintotica in media per moti fluidi viscosi in domini esterni., Ann. Mat. Pura Appl.142, 57–75 (1985)
Maremonti, P.:L 2 decay of suitable weak solutions to the Navier-Stokes equations in three-dimensional exterior domains. Proc. Workshop “Energy stability and convection”, Capri May 1986, to be published by Longman
Maremonti, P.: Partial regularity of a generalized solution to the Navier-Stokes equations in exterior domain. Commun. Math. Phys.110, 75–87 (1987)
Masuda, K.: Weak solutions of the Navier-Stokes equations. Tohoku Math. J.36, 623–646 (1984)
Miranda, C.: Istituzioni di analisi funzionale lineare, Vol.I–II, Unione Matem. Italiana and C.N.R., Tip. Oderisi (ed.). Gubbio, Italy 1978
Miyakawa, T.: On nonstationary solutions of the Navier-Stokes equations in an exterior domain. Hiroshima Math. J.12, 115–140 (1982)
Schonbek, M. E.:L 2-decay for weak solutions of the Navier-Stokes equations. Arch. Rat. Mech. Anal.89, 209–222 (1985)
Schonbek, M. E.: Large time behaviour of solutions to the Navier-Stokes equations. Commun. Part. Differ. Equations11, 733–763 (1986)
Solonnikov, V. A.: Estimates for solutions of nonstationary Navier-Stokes equations. J. Sov. Math.8, 467–528 (1977)
Sohr, H., von Wahl, W.: On the regularity of the pressure of weak solutions of Navier-Stokes equations. Arch. Math.46, 428–439 (1986)
Sohr, H. von Wahl, W., Wiegner, M.: Zur asymptotik der gleichungen von Navier-Stokes. Nachr. Akad. Wiss. Göttingen3, 45–59 (1986)
Wiegner, M.: Decay results for weak solutions of the Navier-Stokes equations onR n. J. Lond. Math. Soc.35, 303–313 (1987)
Galdi, G. P., Maremonti, P.: Sulla regolarità delle soluzioni deboli al sistema di Navier-Stokes in arbitrari domini diR n, per pern=2, 3, 4. (forthcoming)
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Communicated by C. H. Taubes
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Maremonti, P. On the asymptotic behaviour of theL 2-norm of suitable weak solutions to the Navier-Stokes equations in three-dimensional exterior domains. Commun.Math. Phys. 118, 385–400 (1988). https://doi.org/10.1007/BF01466723
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DOI: https://doi.org/10.1007/BF01466723