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

Babin, A. V.; Vishik, M. I.

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

Keywords:

regular attractor; Lyapunov function; local unstable manifolds near a critical point; semigroup; asymptotic behavior of solutions; nonlinear evolution equations; fluid mechanics; Navier Stokes equations; chemical kinetics; viscous elasticity