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Approximations of Gol'dshtik's model

D. K. Potapov1

AnnotationIn this paper we consider continuous approximations of the Gol'dshtik problem for separated flows of incompressible fluid. An approximating problem is obtained from the initial problem by small perturbations of a spectral parameter (vorticity) and by the continuous approximations of discontinuous nonlinearity in the phase variable. Using a variational method under certain conditions, we prove the convergence of solutions of the approximating problems to the solutions of the initial problem. A modification for a one-dimensional analogue of the Gol'dshtik mathematical model is considered. The model is a nonlinear differential equation with a boundary condition. Nonlinearity in the equation is continuous and depends on a small parameter. We have a discontinuous nonlinearity, when this parameter tends to zero. The results of the solutions are in accord with the results obtained for the one-dimensional analogue of the Gol'dshtik model.
KeywordsGol'dshtik model, separated flows, nonlinear differential equation, discontinuous nonlinearity, continuous approximation.

1Associate Professor of Higher Mathematics Chair, Saint-Petersburg State University, Saint-Petersburg; potapov@apmath.spbu.ru.

Citation: D. K. Potapov, "[Approximations of Gol'dshtik's model]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,13:2 (2011) 100–107 (In Russian)