For a class of semi-abstract evolution equations for sections on vector bundles on a three-dimensional compact manifold we prove that for initial values with certain symmetries strong solutions exist for all times. In case these solutions become small after some time, strong solutions exist also for small perturbations of these initial values. Many systems from fluid mechanics are included in this class.
@article{bwmeta1.element.bwnjournal-article-bcpv33z1p369bwm,
author = {Str\"ohmer, Gerhard and Zaj\k aczkowski, Wojciech},
title = {Existence and stability theorems for abstract parabolic equations, and some of their applications},
journal = {Banach Center Publications},
volume = {37},
year = {1996},
pages = {369-382},
zbl = {0851.35016},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv33z1p369bwm}
}
Ströhmer, Gerhard; Zajączkowski, Wojciech. Existence and stability theorems for abstract parabolic equations, and some of their applications. Banach Center Publications, Tome 37 (1996) pp. 369-382. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv33z1p369bwm/
[000] [1] A. Friedman, Partial Differential Equations, Krieger, Malabar, Fla., 1983.
[001] [2] J. M. Ghidaglia, Etude d'écoulements de fluides visqueux incompressibles: comportement pour les grands temps et applications aux attracteurs, Thèse de 3e cycle, Université de Paris Sud, Orsay, 1984.
[002] [3] M. W. Hirsch, Differential Topology, Springer, Berlin, 1976
[003] [4] O. A. Ladyzhenskaya, Solution in the large of the nonstationary boundary value problem for the Navier-Stokes system with two space variables, Comm. Pure Appl. Math. 12 (1959) 427-433. | Zbl 0103.19502
[004] [5] O. A. Ladyzhenskaya, On unique global solvability of three dimensional Cauchy problem for Navier-Stokes equations with rotational symmetry, Zap. Nauchn. Sem. LOMI 7 (1968) 155-177.
[005] [6] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flows, 2nd ed., Gordon and Breach, New York, 1969. | Zbl 0184.52603
[006] [7] L. Landau and E. Lifschitz, Mechanics of Continuous Media, Nauka, Moscow, 1954 (in Russian); English transl.: Pergamon Press, Oxford, 1959; new edition: Hydrodynamics, Nauka, Moscow, 1986 (in Russian), English transl.: Fluid Mechanics, Pergamon Press, Oxford, 1987.
[007] [8] L. Landau and E. Lifschitz, Electrodynamics of Continuous Media, Nauka, Moscow, 1957 (in Russian).
[008] [9] J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math. 63 (1934), 193-248.
[009] [10] A. Mahalov, E. S. Titi and S. Leibovich, Invariant helical subspaces for the Navier-Stokes equations, Arch. Rational Mech. Anal. 112 (1990), 193-222. | Zbl 0708.76044
[010] [11] G. Ponce, R. Racke, T. S. Sideris and E. S. Titi, Global stability of large solutions to the 3d Navier-Stokes equations, preprint. | Zbl 0795.35082
[011] [12] V. A. Solonnikov, Estimates of the solutions of a non-stationary linearized system of Navier-Stokes equations, Trudy Mat. Inst. Steklov. 70 (1964), 213-317 (in Russian).
[012] [13] V. A. Solonnikov, On boundary problems for linear parabolic systems of differential equations of general type, Trudy Mat. Inst. Steklov. 83 (1965) (in Russian); English transl.: Proc. Steklov Inst. Math. 83 (1967). | Zbl 0164.12502
[013] [14] G. Ströhmer, About an initial-boundary value problem from magneto-hydrodynamics, Math. Z. 209 (1992), 345-362. | Zbl 0756.76095
[014] [15] G. Ströhmer, An existence result for partially regular weak solutions of certain abstract evolution equations, with an application to magneto-hydrodynamics, ibid. 213 (1993), 373-385.
[015] [16] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1988. | Zbl 0662.35001
[016] [17] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978. | Zbl 0387.46033
[017] [18] W. von Wahl, The Equations of Navier-Stokes and Abstract Parabolic Equations, Vieweg, Braunschweig, 1985.