Some application of the implicit function theorem to the stationary Navier-Stokes equations

Konstanty Holly

Annales Polonici Mathematici (1991)

  • Volume: 54, Issue: 2, page 93-109
  • ISSN: 0066-2216

Abstract

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We prove that - in the case of typical external forces - the set of stationary solutions of the Navier-Stokes equations is the limit of the (full) sequence of sets of solutions of the appropriate Galerkin equations, in the sense of the Hausdorff metric (for every inner approximation of the space of velocities). Then the uniqueness of the N-S equations is equivalent to the uniqueness of almost every of these Galerkin equations.

How to cite

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Konstanty Holly. "Some application of the implicit function theorem to the stationary Navier-Stokes equations." Annales Polonici Mathematici 54.2 (1991): 93-109. <http://eudml.org/doc/262241>.

@article{KonstantyHolly1991,
abstract = {We prove that - in the case of typical external forces - the set of stationary solutions of the Navier-Stokes equations is the limit of the (full) sequence of sets of solutions of the appropriate Galerkin equations, in the sense of the Hausdorff metric (for every inner approximation of the space of velocities). Then the uniqueness of the N-S equations is equivalent to the uniqueness of almost every of these Galerkin equations.},
author = {Konstanty Holly},
journal = {Annales Polonici Mathematici},
keywords = {weak solutions; stationary Navier-Stokes equations; incompressible fluid; implicit function theorem},
language = {eng},
number = {2},
pages = {93-109},
title = {Some application of the implicit function theorem to the stationary Navier-Stokes equations},
url = {http://eudml.org/doc/262241},
volume = {54},
year = {1991},
}

TY - JOUR
AU - Konstanty Holly
TI - Some application of the implicit function theorem to the stationary Navier-Stokes equations
JO - Annales Polonici Mathematici
PY - 1991
VL - 54
IS - 2
SP - 93
EP - 109
AB - We prove that - in the case of typical external forces - the set of stationary solutions of the Navier-Stokes equations is the limit of the (full) sequence of sets of solutions of the appropriate Galerkin equations, in the sense of the Hausdorff metric (for every inner approximation of the space of velocities). Then the uniqueness of the N-S equations is equivalent to the uniqueness of almost every of these Galerkin equations.
LA - eng
KW - weak solutions; stationary Navier-Stokes equations; incompressible fluid; implicit function theorem
UR - http://eudml.org/doc/262241
ER -

References

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  1. [1] M. S. Berger, Nonlinearity and Functional Analysis, Benjamin, 1977. Zbl0368.47001
  2. [2] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), p. 339. Zbl0219.46015
  3. [3] C. Foiaş et J. C. Saut, Remarques sur les équations de Navier-Stokes stationnaires, Ann. Scuola Norm. Sup. Pisa (5) 1 (1983), p. 186. Zbl0546.35058
  4. [4] C. Foiaş and R. Temam, Structure of the set of stationary solutions of the Navier-Stokes equations, Comm. Pure. Appl. Math. 30 (1977), 149-164. Zbl0335.35077
  5. [5] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris 1969. Zbl0189.40603
  6. [6] K. Maurin, Analysis II, PWN, Warszawa 1980. 
  7. [7] L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Institute, New York 1974. Zbl0286.47037
  8. [8] W. Rudin, Functional Analysis, McGraw-Hill, New York 1973, Chap. 4. 
  9. [9] J. C. Saut, Generic properties of nonlinear boundary value problems, in: Partial Differential Equations, Banach Center Publ. 10, PWN, Warszawa 1983, 331-351. 
  10. [10] R. Temam, Navier-Stokes Equations, North-Holland, Amsterdam 1979. 

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