The Rothe method and time periodic solutions to the Navier-Stokes equations and equations of magnetohydrodynamics

Dana Lauerová

Aplikace matematiky (1990)

  • Volume: 35, Issue: 2, page 89-98
  • ISSN: 0862-7940

Abstract

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The existence of a periodic solution of a nonlinear equation z ' + A 0 z + B 0 z = F is proved. The theory developed may be used to prove the existence of a periodic solution of the variational formulation of the Navier-Stokes equations or the equations of magnetohydrodynamics. The proof of the main existence theorem is based on Rothe method in combination with the Galerkin method, using the Brouwer fixed point theorem.

How to cite

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Lauerová, Dana. "The Rothe method and time periodic solutions to the Navier-Stokes equations and equations of magnetohydrodynamics." Aplikace matematiky 35.2 (1990): 89-98. <http://eudml.org/doc/15614>.

@article{Lauerová1990,
abstract = {The existence of a periodic solution of a nonlinear equation $z^\{\prime \} + A_0z + B_0z=F$ is proved. The theory developed may be used to prove the existence of a periodic solution of the variational formulation of the Navier-Stokes equations or the equations of magnetohydrodynamics. The proof of the main existence theorem is based on Rothe method in combination with the Galerkin method, using the Brouwer fixed point theorem.},
author = {Lauerová, Dana},
journal = {Aplikace matematiky},
keywords = {Navier-Stokes equations; periodic solutions; existence of generalized solutions; nonlinear operator equation; variational formulation; equations of magnetohydrodynamics; Galerkin method; Brouwer fixed point theorem; existence of a periodic solution; nonlinear operator equation; variational formulation; Navier-Stokes equations; equations of magnetohydrodynamics; Galerkin method; Brouwer fixed point theorem},
language = {eng},
number = {2},
pages = {89-98},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The Rothe method and time periodic solutions to the Navier-Stokes equations and equations of magnetohydrodynamics},
url = {http://eudml.org/doc/15614},
volume = {35},
year = {1990},
}

TY - JOUR
AU - Lauerová, Dana
TI - The Rothe method and time periodic solutions to the Navier-Stokes equations and equations of magnetohydrodynamics
JO - Aplikace matematiky
PY - 1990
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 35
IS - 2
SP - 89
EP - 98
AB - The existence of a periodic solution of a nonlinear equation $z^{\prime } + A_0z + B_0z=F$ is proved. The theory developed may be used to prove the existence of a periodic solution of the variational formulation of the Navier-Stokes equations or the equations of magnetohydrodynamics. The proof of the main existence theorem is based on Rothe method in combination with the Galerkin method, using the Brouwer fixed point theorem.
LA - eng
KW - Navier-Stokes equations; periodic solutions; existence of generalized solutions; nonlinear operator equation; variational formulation; equations of magnetohydrodynamics; Galerkin method; Brouwer fixed point theorem; existence of a periodic solution; nonlinear operator equation; variational formulation; Navier-Stokes equations; equations of magnetohydrodynamics; Galerkin method; Brouwer fixed point theorem
UR - http://eudml.org/doc/15614
ER -

References

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  1. J. Nečas, Application of Rothe's method to abstract parabolic equations, Czechoslovak Math. J. 24 (1974), p. 496-500. (1974) Zbl0311.35059MR0348571
  2. J. Kačur, Application of Rothe's Method to Nonlinear Equations, Mat. čas. 25 (1975), p. 63-81. (1975) Zbl0298.34058MR0394344
  3. J. Kačur, Method of Rothe and Nonlinear Parabolic Boundary Value Problems of Arbitrary Order, Czechoslovak Math. J. 28 (1978), p. 507-524. (1978) Zbl0402.35053MR0506431
  4. K. Rektorys, The method of Discretization in Time and Partial Differential Equations, D. Reidel Publishing Company, Holland a SNTL Praha 1982. (1982) Zbl0522.65059MR0689712
  5. R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland Publishing Company - Amsterodam, New York, Oxford 1977. (1977) Zbl0383.35057MR0609732
  6. O. Vejvoda, al., Partial Differential Equations. Time Periodic Solutions, Sijthoff Noordhoff 1981. (1981) 
  7. J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris 1969. (1969) Zbl0189.40603MR0259693
  8. D. Lauerová, Proof of existence of a weak periodic solution of Navier-Stokes equations and equations of magnetohydrodynamics by Rothe's method, (Czech.) Dissertation, Praha 1986. (1986) 

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