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Periodic solutions of the first boundary value problem for a linear and weakly nonlinear heat equation

Věnceslava ŠťastnováOtto Vejvoda — 1968

Aplikace matematiky

One investigates the existence of an ω -periodic solution of the problem u t = u x x + c u + g ( t , x ) + ϵ f ( t , x , u , u x , ϵ ) , u ( t , 0 ) = h 0 ( t ) + ϵ χ 0 ( t , u ( t , 0 ) , u ( t , π ) ) , u ( t , π ) = h 1 ( t ) + ϵ χ 1 ( t , u ( t , 0 ) , u ( t , π ) ) , provided the functions g , f , h 0 , h 1 , χ 0 , χ 1 are sufficiently smooth and ω -periodic in t . If c k 2 , k natural, such a solution always exists for sufficiently small ϵ > 0 . On the other hand, if c = l 2 , l natural, some additional conditions have to be satisfied.

Equations of magnetohydrodynamics of compressible fluid: Periodic solutions

Milan ŠtědrýOtto Vejvoda — 1985

Aplikace matematiky

The authors prove the global existence and exponential stability of solutions of the given system of equations under the condition that the initial velocities and the external forces are small and the initial density is not far from a constant one. If the external forces are periodic, then solutions periodic with the same period are obtained. The investigated system of equations is a bit non-standard - for example the displacement current in the Maxwell equations is not neglected.

Small time-periodic solutions of equations of magnetohydrodynamics as a singularly perturbed problem

Milan ŠtědrýOtto Vejvoda — 1983

Aplikace matematiky

This paper deals with a system of equations describing the motion of viscous electrically conducting incompressible fluid in a bounded three dimensional domain whose boundary is perfectly conducting. The displacement current appearing in Maxwell’s equations, ϵ E t is not neglected. It is proved that for a small periodic force and small positive there exists a locally unique periodic solution of the investigated system. For ϵ 0 , these solutions are shown to convergeto a solution of the simplified (and...

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