---
tag: W1F5C7RZ0M
kind: concept
name: Inertial manifold
msc: [37L25, 35B42]
sources:
  - url: https://en.wikipedia.org/wiki/Inertial_manifold
same_as:
  wikidata: Q17098174
  wikipedia: Inertial manifold
---

## Summary

An inertial manifold of a dissipative evolution equation is a finite-dimensional Lipschitz manifold in the infinite-dimensional phase space.
It is invariant and attracts all solutions exponentially fast.
It contains the [[global attractor|4SH2ZB984E]].
Restricted to it, the equation becomes a system of finitely many ordinary differential equations, the inertial form, which carries the full long-time dynamics.
Inertial manifolds were introduced by Foias, Sell and Temam in the 1980s.
They exist for many dissipative PDEs whose linear part has large gaps in its spectrum.

## Explanations

### L1 human

Many physical systems, like a fluid or a chemical reaction spread over space, need infinitely many numbers to describe their state.
Yet after a short time their behaviour often looks as if only a few numbers mattered.
An inertial manifold makes this precise: it is a finite-dimensional surface in the space of states that all solutions approach very fast.
Once a solution is near it, the motion is governed by finitely many equations.