6 problems

Infinite-dimensional attractor conjecture. There are a nonlinearity $f$ and right-hand side $g$ satisfying those assumptions such that $$ mathcal A($eta) has infinite Hausdorff and…

Global attraction conjecture. For generic such equations, every finite-energy solution admits the asymptotics

Let $G$ be a Lie symmetry group and consider a $G$-invariant Hamiltonian nonlinear PDE … where $\Psi(\cdot,t)$ belongs to $L^2(\mathbb R^3)\otimes\mathbb R^d$,…

Inertial-manifold obstruction conjecture. The absence of an inertial manifold is the only natural obstruction to the limit dynamics being finite-dimensional; equivalently, whenever…

An inertial manifold (IM) is a finite-dimensional, invariant, exponentially attracting manifold for the dynamics generated by a dissipative partial differential equation. Zelik's c…

Let $\mathcal H=\{u\in L^2_{per}(\mathbb T^2)^2: \operatorname{div}u=0,\ \int_{\mathbb T^2}u\,dx=0\}$ be the phase space for the two-dimensional periodic Navier–Stokes equations, a…