18 problems

The paper considers Anti-de Sitter spacetime with either reflective or optimally dissipative boundary conditions. Nonlinear instability and dissipative stability conjecture. Anti-d…

Nonlinear stability conjecture. These conditions should be relaxable, and space-times with a supersymmetric compactification and $n=3$ should be nonlinearly stable.

Generic phase-plateau behavior conjecture. This behavior should be generic across Hamiltonian models with symmetry: localized perturbations should produce phase profiles whose expa…

Ionescu's Kerr wave-map stability conjecture. For any subextremal Kerr metric with $|a|<M$, this stationary solution is future asymptotically stable in the domain of outer communic…

Let eRN denote an extremal Reissner–Nordström black hole. Nonlinear orbital stability means that sufficiently nearby initial data remain close to the reference exterior and settle…

Let AdS denote anti-de Sitter space-time, and consider the Einstein-scalar field equations with fully reflective boundary conditions at the conformal boundary. AdS instability conj…

Let $\nu$ denote the viscosity, and consider linear instabilities of an exponential shear-layer profile or a Blasius profile. Saturation-size conjecture. These linear instabilities…

Let $\nu$ denote the viscosity, and consider small perturbations of the exponential or Blasius shear-layer profile in the viscous boundary-layer problem. The perturbations generate…

Nonlinear stability conjecture. All $N$-spot ring patterns are nonlinearly stable unless the spot-spot distance is the smallest binding distance.

Uniqueness hypothesis conjecture. The hypothesis $$ should be valid for all $cfrac{1}{2}$.

Let anti-de Sitter space be a solution of the Einstein–Klein–Gordon system, and consider small perturbations of this solution arising from initial data. Bizoń–Rostworowski islands-…

Consider the lapse-scalar field system and its linearization around the backgrounds studied in the paper. The asymptotic data are functions naturally associated with the contractin…

Consider the initial amplitudes $\alpha_n$ of the residual components, with $n\ne j$, and the energy threshold of nonlinear instability for the corresponding prevailing mode. The i…

Let $\mathbb{E}_{12}:[0,1]\to[0,+\infty]$ denote the energy threshold of instability before time $T$ for the truncated beam model, defined by … Here $a$ is the pier-location parame…

The higher-order mKdV Lyapunov functional conjecture. The functional $\mathcal{H}_{2n+1}$ generates the associated nonlinear ODE

In an asymptotically flat spacetime, consider the long-time evolution of self-gravitating matter arising from arbitrarily small perturbations of oscillating soliton stars. Dispersi…

Let $\mathcal{F}:B\to B$ be a map with linearization $L=d\mathcal{F}|_{u=0}$, and suppose that $L$ is exponentially unstable. Nonlinear stabilization means that the nonlinear map f…

Let $d\geq 4$ and consider the co-rotational wave maps equation from $(d+1)$-dimensional Minkowski space to the sphere $\mathbb{S}^d$. Bizoń and Biernat's explicit self-similar sol…