10 problems

Consider the non-autonomous wave PDE … on a domain $\mathcal X$, with the stated edge conditions and assumptions on $f$ and $g$. Partition the space into $N$ elements, let…

Let $\Gamma$ and $H$ be smooth functions with all derivatives bounded, and let the initial data be correspondingly regular and compatible. Let $\mathrm{Red}_\varepsilon$ denote the…

The slow-manifold foliation conjecture. For $0<\varepsilon\ll1$, the level sets of $\phi_{{\rm i}\omega}^\varepsilon$ coincide with the leaves of the foliation of the attracting sl…

The Vlasov–Maxwell system has a slow manifold identified in the article, with slow-fast parameter $\epsilon$, and an optimally truncated approximation to that manifold. Normal stab…

Optimality conjecture. The bound given by Theorem $$ cannot be improved in general.

Riemannian metrization conjecture. An appropriate metrization of the solution manifold in extended phase space will allow a complete geometric characterization of the SIM.

Covariant SIM characterization conjecture. A covariant formulation of SIM criteria should be at the root of a geometric characterization of an object in phase space that is reasona…

Extrinsic-curvature characterization conjecture. The SIM could be characterized by extremals or zeros of an appropriate extrinsic curvature notion.

Consider the stable periodic solutions of the system under discussion, parametrized by sufficiently small $epsilon$, with stability understood in the sense established above. Unifo…

Let $\mathcal{X}$ be an open subset of $\mathbb{R}^n$, and let $\mathcal{S}(T)$ be a finite-time slow manifold, meaning a submanifold such that…