22 problems

Let $\mathcal G$ be a compact Lie group with Lie algebra $\mathbf g$, let $s_0=\frac{n-2}{2}$ be the scaling index, and consider the Yang–Mills equation … for a connection with cur…

Let $(M,g)$ be a complete Riemannian manifold, and consider wave maps from $\mathbb R^2\times\mathbb R$ into $(M,g)$. The critical initial-data space is $H^1\times L^2$. Wave maps…

Let $n\geq 4$, let $0\leq s\leq \frac12$, and consider the power nonlinear wave equation … The initial data lie in $H^s\times H^{s-1}$. Local well-posedness conjecture. The equatio…

For a semilinear wave equation, let $H^s \times H^{s-1}$ denote the Sobolev data space, and let $s_0$ be the scaling index determined by the equation's scaling. Below-scaling ill-p…

The systems under consideration are basic field theories with critical well-posedness exponent $s_c$, meaning that the $dot H^{s_c} \times dot H^{s_c-1}$ norm of the initial data i…

Let $\mathcal{X}=(\{v\leq 0\}\cup \{t\leq -0.9\})\cap\{-1\leq u<0\}$ and let $\mathcal{C}$ be the backward light cone. Let $\phi$ solve the relevant wave-map or semilinear equation…

Let $\mathcal{X}=(\{v\leq 0\}\cup \{t\leq -0.9\})\cap\{-1\leq u<0\}$ and let $\mathcal{C}$ be the backward light cone. The symbols $\phi$, $r$, $\nu$, $d$, $p$, and $k$ have the me…

Let $\mathcal{X}=(\{v\leq 0\}\cup \{t\leq -0.9\})\cap\{-1\leq u<0\}$, and let $\phi$ be a solution arising from smooth initial data in the stated Type I self-similar stability resu…

Consider the focusing nonlinear wave equation $P[\phi]=0$ with $p=3$ in spatial dimension $n\geq 5$. Let $(\phi_{0,\omega},\phi_{1,\omega})$, $\omega\in[0,1]$, be a one-parameter f…

Consider the nonlinear wave equation … with $p\in\mathbb{N}$ and $p\geqslant3$. Expand the field in spherical harmonics, … At fixed finite radius $r$, the modes are expected to dec…

Reducible-tree conjecture. The infinite reducible tree and its rescalings according to the relevant scaling provide an effective description for the structure of time-periodic solu…

Let $\alpha$ be sufficiently close to zero, let $\mathfrak{M}_\alpha$ be a blow-up of the compactification of Minkowski space, and let $W$ denote the soliton profile. A function is…

Weak null conjecture. The original system of wave equations also admits global solutions for sufficiently small initial data.

Let $(f,g)\in H^1\times L^2$ satisfy … Here $\mathcal{E}_Z$ denotes the Zakharov energy and $Q$ is the ground state. Scattering below the ground state conjecture. There exists a un…

Monotone-profile conjecture. There exist $\gamma,m>0$ such that

Uniqueness conjecture. There exist $\gamma,m,z\in\mathbb{R}$ such that

For a defocusing nonlinear Schrödinger equation or nonlinear wave equation, let $u$ be a solution whose critical Sobolev norm remains bounded in time. Bounded critical Sobolev norm…

De Giorgi's conjecture. There exists a limit

Let $s_c\geq0$, $mu=-1$, and let $(u_0,u_1)\in\dot{H}^{s_c}\times\dot{H}^{s_c-1}$. Low regularity conjecture. The corresponding solution $u$ is global and scatters: there exist uni…

Consider a bounded solution $u$ of the focusing energy-critical nonlinear wave equation (NLW). A soliton resolution conjecture predicts that, as time approaches the maximal time of…

First-singularity conjecture. Spherically symmetric solutions of this equation have their first singularities on the event horizon $\mathcal{H}^{+}$.

Consider the critical wave equation in three spatial dimensions, and let $u$ be a radial energy class solution with energy close to the ground-state energy $E(W,0)$, where $W$ deno…