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Topological approximation methods for evolutionary problems of nonlinear hydrodynamics. (English) Zbl 1155.76004

de Gruyter Series in Nonlinear Analysis and Applications 12. Berlin: de Gruyter (ISBN 978-3-11-020222-9/hbk). xii, 230 p. (2008).
The book examines Navier-Stokes equations for viscoelastic and nonlinear-viscous media. In the case of non-Newtonian fluid, the deviatoric strain velocity tensor of the fluid and the deviatoric stress tensor are related by a linear equation with a negative coefficient \(\eta\) called viscosity. This investigation starts from a well-known book by V. T. Dmitrienko and V. G. Zvyagin [Approximate-topological approach to investigation of problems of hydrodynamics. Navier-Stokes system. (in Russian). Moscow (2004)]. Solutions of the Navier-Stokes equations are studied by means of finite-dimensional approximation of the corresponding operator in Banach space. Such operators appear in non-evolutionary and evolutionary equations of hydrodynamics. Topological methods are the Leray-Schauder degree methods in the proof of the existence of solutions. Finally, the authors give a construction of attractors for the weak solutions to the initial-boundary value problems in the dynamics of viscoelastic media.

MSC:

76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76A10 Viscoelastic fluids
35Q30 Navier-Stokes equations
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