# A Stability Technique for Evolution Partial Differential Equations

## A Dynamical Systems Approach

Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 56)

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common feature is that these evolution problems can be formulated as asymptoti cally small perturbations of certain dynamical systems with better-known behaviour. Now, it usually happens that the perturbation is small in a very weak sense, hence the difficulty (or impossibility) of applying more classical techniques. Though the method originated with the analysis of critical behaviour for evolu tion PDEs, in its abstract formulation it deals with a nonautonomous abstract differ ential equation (NDE) (1) Ut = A(u) + C(u, t), t > 0, where u has values in a Banach space, like an LP space, A is an autonomous (time-independent) operator and C is an asymptotically small perturbation, so that C(u(t), t) ~ ° as t ~ 00 along orbits {u(t)} of the evolution in a sense to be made precise, which in practice can be quite weak. We work in a situation in which the autonomous (limit) differential equation (ADE) Ut = A(u) (2) has a well-known asymptotic behaviour, and we want to prove that for large times the orbits of the original evolution problem converge to a certain class of limits of the autonomous equation. More precisely, we want to prove that the orbits of (NDE) are attracted by a certain limit set [2* of (ADE), which may consist of equilibria of the autonomous equation, or it can be a more complicated object.

Navier-Stokes equation continuum mechanics differential equation fluid dynamics functional analysis nonlinear analysis ordinary differential equation partial differential equation pdes

- DOI https://doi.org/10.1007/978-1-4612-2050-3
- Copyright Information Birkhäuser Boston 2004
- Publisher Name Birkhäuser Boston
- eBook Packages Springer Book Archive
- Print ISBN 978-1-4612-7396-7
- Online ISBN 978-1-4612-2050-3
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