Existence and stability theorems for abstract parabolic equations, and some of their applications

Gerhard Ströhmer; Wojciech Zajączkowski

Banach Center Publications (1996)

  • Volume: 33, Issue: 1, page 369-382
  • ISSN: 0137-6934

Abstract

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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.

How to cite

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Ströhmer, Gerhard, and Zajączkowski, Wojciech. "Existence and stability theorems for abstract parabolic equations, and some of their applications." Banach Center Publications 33.1 (1996): 369-382. <http://eudml.org/doc/262753>.

@article{Ströhmer1996,
abstract = {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.},
author = {Ströhmer, Gerhard, Zajączkowski, Wojciech},
journal = {Banach Center Publications},
keywords = {abstract parabolic equations},
language = {eng},
number = {1},
pages = {369-382},
title = {Existence and stability theorems for abstract parabolic equations, and some of their applications},
url = {http://eudml.org/doc/262753},
volume = {33},
year = {1996},
}

TY - JOUR
AU - Ströhmer, Gerhard
AU - Zajączkowski, Wojciech
TI - Existence and stability theorems for abstract parabolic equations, and some of their applications
JO - Banach Center Publications
PY - 1996
VL - 33
IS - 1
SP - 369
EP - 382
AB - 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.
LA - eng
KW - abstract parabolic equations
UR - http://eudml.org/doc/262753
ER -

References

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  1. [1] A. Friedman, Partial Differential Equations, Krieger, Malabar, Fla., 1983. 
  2. [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. 
  3. [3] M. W. Hirsch, Differential Topology, Springer, Berlin, 1976 
  4. [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. Zbl0103.19502
  5. [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. 
  6. [6] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flows, 2nd ed., Gordon and Breach, New York, 1969. Zbl0184.52603
  7. [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. 
  8. [8] L. Landau and E. Lifschitz, Electrodynamics of Continuous Media, Nauka, Moscow, 1957 (in Russian). 
  9. [9] J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math. 63 (1934), 193-248. 
  10. [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. Zbl0708.76044
  11. [11] G. Ponce, R. Racke, T. S. Sideris and E. S. Titi, Global stability of large solutions to the 3d Navier-Stokes equations, preprint. Zbl0795.35082
  12. [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). 
  13. [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). Zbl0164.12502
  14. [14] G. Ströhmer, About an initial-boundary value problem from magneto-hydrodynamics, Math. Z. 209 (1992), 345-362. Zbl0756.76095
  15. [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. 
  16. [16] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1988. Zbl0662.35001
  17. [17] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978. Zbl0387.46033
  18. [18] W. von Wahl, The Equations of Navier-Stokes and Abstract Parabolic Equations, Vieweg, Braunschweig, 1985. 

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