on subspace spanned by trajectory

Note

The problem was brought up from some earlier research activities. It seems there are still a few interesting things out there, though the field is already fully occupied.

🏷️Preliminaries

Let $\mathcal{A}$ be a strongly elliptic self-adjoint operator and one important observation is the Cauchy-Dunford integral:

$$e^{\mathcal{A}t}=\frac{1}{2\pi i}{\int }_C{e}^{-\lambda t}(\lambda -\mathcal{A}{)}^{-1}d\lambda ,$$

which permits an efficient quadrature rule on a parabolic curve $C$ surrounding the sector that contains the spectrum. Roughly speaking, there exists ${\lambda }_k\in \mathbb{C}$ that

$$\Vert e^{\mathcal{A}t}-\sum _{k=1}^N{c}_k{e}^{-{\lambda }_{k}t}({\lambda }_k-\mathcal{A}{)}^{-1}{\Vert }_{op}\le ϵ,t\in [\tau ,T],$$

where $N=O(|\log ϵ|\log (T/\tau ))$. Therefore, the solution trajectory $u(\cdot ,t)$ cannot expand to a large space (up to some tolerance $ϵ$). At a late stage $\tau \gg 1$, we can expect a much smaller subspace that “holds” the trajectory, this can be seen by a simple expansion of eigenfunctions but eigenfunctions are not the most efficient.

In general, we can find a weaker way to define the subspace $V$ such that

$${\int }_{\tau }^T\Vert u(\cdot ,t)-P_{V}u(\cdot ,t){\Vert }_{L^2(D)}dt\le ϵ\Vert u_0{\Vert }_{L^2(D)}.$$

🧩Selection of initial condition

🌀Extension to other operators

〰️Inertia manifolds

🌊Potential usages

Notes

Links