Article
MSC:
35A15,
35Q10,
35Q30,
65J15,
65M20,
65N40,
76D05,
76M10 | MR 1042846 | Zbl 0703.76021 | DOI: 10.21136/AM.1990.104392
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Keywords:
Navier-Stokes equations; periodic solutions; existence of generalized solutions; nonlinear operator equation; variational formulation; equations of magnetohydrodynamics; Galerkin method; Brouwer fixed point theorem
Navier-Stokes equations; periodic solutions; existence of generalized solutions; nonlinear operator equation; variational formulation; equations of magnetohydrodynamics; Galerkin method; Brouwer fixed point theorem
Summary:
The existence of a periodic solution of a nonlinear equation $z' + A_0z + B_0z=F$ 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.
References:
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[5] R. Temam: Navier-Stokes Equations. Theory and Numerical Analysis. North-Holland Publishing Company - Amsterodam, New York, Oxford 1977. MR 0609732 | Zbl 0383.35057
[6] O. Vejvoda, al.: Partial Differential Equations. Time Periodic Solutions. Sijthoff Noordhoff 1981.
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[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.
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