Inertial manifold

Remarks

If the semigroup has a global attractor $\mathcal{A}$, then $\mathcal{A} \subseteq \mathcal{M}$. The dynamics on $\mathcal{M}$ are given by an ordinary differential equation for $p = P u$ in the finite-dimensional space $PH$, the inertial form. Some authors ask in clause 3 for exponential tracking: each solution approaches some solution that lies on $\mathcal{M}$.

Example: let $H = \R^2$ and $S(t)(x, y) = (e^{-t} x, e^{-10 t} y)$. Take $P(x, y) = (x, 0)$ and $\Phi = 0$, so $\mathcal{M}$ is the $x$-axis. It is invariant, and $\operatorname{dist}(S(t)(x, y), \mathcal{M}) = e^{-10 t} \abs{y}$, so all clauses hold with $\gamma = 10$.

Non-example: for the same semigroup, the parabola $\set{(x, x^2)}$ is the graph of a function, but it is not invariant. The point $(1, 1)$ goes to $(e^{-t}, e^{-10 t})$, which is not on the parabola for $t > 0$, so clause 2 fails.

Consider a dissipative equation $u' + A u = F(u)$ with $A$ self-adjoint with compact inverse and $F$ globally Lipschitz. An inertial manifold exists when the eigenvalues of $A$ have large enough gaps compared with the Lipschitz constant of $F$.