Unstable invariant sets of semigroups of non-linear operators and their perturbations.(English. Russian original)Zbl 0624.47065

Russ. Math. Surv. 41, No. 4, 1-41 (1986); translation from Usp. Mat. Nauk 41, No. 4(250), 3-34 (1986).
An evolution $$u_ t=A(u)$$ defines usually a semigroup $$\{S_ t,t\geq 0\}$$ represented by $$S_ tu(0)=u(t)$$. The study on the asymptotic behavior of solutions u(t) then reduces to the study on $$S_ t$$. This article gives a good survey in this connection. It contains a number of valuable propositions and theorems with their proofs sketched or omitted. As illustrative examples the authors give a number of specific nonlinear evolution equations taken from fluid mechanics, such as the well-known Navier Stokes equations, chemical kinetics, viscous elasticity and so on.
Reviewer: Tu Gui-Zhang

MSC:

 47H20 Semigroups of nonlinear operators 35B40 Asymptotic behavior of solutions to PDEs 76D05 Navier-Stokes equations for incompressible viscous fluids 35Q30 Navier-Stokes equations 35B20 Perturbations in context of PDEs 47A55 Perturbation theory of linear operators
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