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  "body": "\n## Definition\n\nLet $H$ be a real [[Hilbert space|HM2HHE3J51]] and let $(S(t))_{t \\ge 0}$ be a [[semigroup|0XMEM406JM]] of continuous maps $S(t) \\colon H \\to H$ with $S(0) = \\id$ and $S(t + s) = S(t) S(s)$.\nAn *inertial manifold* for $S$ is a subset $\\mathcal{M} \\subseteq H$ such that:\n\n1. $\\mathcal{M}$ is a finite-dimensional Lipschitz manifold: there are an orthogonal projection $P$ of finite rank and a [[Lipschitz|0AW1E6YTJY]] map $\\Phi \\colon PH \\to (I - P)H$ with $\\mathcal{M} = \\set{p + \\Phi(p) : p \\in PH}$;\n2. $\\mathcal{M}$ is positively invariant: $S(t) \\mathcal{M} \\subseteq \\mathcal{M}$ for all $t \\ge 0$;\n3. $\\mathcal{M}$ attracts exponentially: there is $\\gamma \u003e 0$ and, for each $u_0 \\in H$, a constant $C(u_0)$ with $\\operatorname{dist}(S(t) u_0, \\mathcal{M}) \\le C(u_0) e^{-\\gamma t}$ for all $t \\ge 0$.\n\n## Remarks\n\nIf the semigroup has a [[global attractor|4SH2ZB984E]] $\\mathcal{A}$, then $\\mathcal{A} \\subseteq \\mathcal{M}$.\nThe dynamics on $\\mathcal{M}$ are given by an ordinary differential equation for $p = P u$ in the finite-dimensional space $PH$, the inertial form.\nSome authors ask in clause 3 for exponential tracking: each solution approaches some solution that lies on $\\mathcal{M}$.\n\nExample: let $H = \\R^2$ and $S(t)(x, y) = (e^{-t} x, e^{-10 t} y)$.\nTake $P(x, y) = (x, 0)$ and $\\Phi = 0$, so $\\mathcal{M}$ is the $x$-axis.\nIt is invariant, and $\\operatorname{dist}(S(t)(x, y), \\mathcal{M}) = e^{-10 t} \\abs{y}$, so all clauses hold with $\\gamma = 10$.\n\nNon-example: for the same semigroup, the parabola $\\set{(x, x^2)}$ is the graph of a function, but it is not invariant.\nThe point $(1, 1)$ goes to $(e^{-t}, e^{-10 t})$, which is not on the parabola for $t \u003e 0$, so clause 2 fails.\n\nConsider a dissipative equation $u' + A u = F(u)$ with $A$ self-adjoint with compact inverse and $F$ globally Lipschitz.\nAn inertial manifold exists when the eigenvalues of $A$ have large enough gaps compared with the Lipschitz constant of $F$.\n",
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