40 problems

Soliton resolution conjecture. Any smooth, finite-energy solution $W(t,r)$ tends as $t\to\infty$, modulo sign, either to the half-kink or to a rescaled half-kink together with an a…

Generic transverse blow-up conjecture. For generic data on $\Sigma$, the transverse derivatives of $\phi$ blow up on $\mathcal C\mathcal H^+$; in fact, $\phi$ is not in…

Scattering conjecture. If $u$ is the solution with data $(u_0,u_1)$, then $u$ exists for all time $t$ and there is a constant…

Fix soliton parameters $A$, $\boldsymbol{\lambda}$, $\nu$, and $\sigma$. Let $\phi_{\mathrm{rad}}$ be a smooth solution of the wave equation $\Box\phi_{\mathrm{rad}}=0$ satisfying…

Let $n\geq2$ and $\frac12<s<\frac n2$. Let $(f_j)_j$ be an orthonormal system in $H^s(\mathbb{R}^n)$, and let $\beta$ be the exponent in the maximal estimate denoted by … holds for…

Let $n\geq2$, $q\geq \frac{2(n+1)}{n-1}$, and let $(f_j)_j$ be an orthonormal system in the homogeneous Sobolev space $\dot{H}^s(\mathbb{R}^n)$. For a sequence of coefficients…

Let $u$ solve the wave equation Cauchy problem … with $u(x,0)=f(x)$ and $\partial_tu(x,0)=0$ on $\mathbb{R}^d\times\mathbb{R}$. For $s_p=\frac{d-1}{2}-\frac{d-1}{p}$, Sogge's conje…

Let $N\in\mathbb N_{\geq1}$, let $\phi$ solve $\Box_{\mathbb R^{2N+1,1}}\phi=0$ with no incoming radiation, and let…

Let $\mathcal{D}^{\mathrm{par}}$ be a compactification of Minkowski space, let $W^{\mathrm{sup}}$ be the supercritical ground-state soliton, and let $z\in\mathbb{R}^3$. Define … A…

Fix distinct soliton speeds, and let $f_{\mathrm{start}}$ denote the starting error whose leading behavior at one soliton is $t^{-1}$. Let $W_a$ be ground-state solitons, let…

Let $\phi$ solve the three-dimensional energy-critical wave equation and satisfy the no-outgoing-radiation condition, and consider multi-solitons whose ground-state components are…

Let $d\geq 1$, let $2<p<\infty$, let $0<T<\infty$, and let $u$ solve the Hermite wave equation … with $u(0)=u_0$ and $\dot u(0)=0$. The local smoothing conjecture for the Hermite w…

Let $\psi^{\mathrm{dat}}$ be data as in the paper's scattering-data definition, with $d>0$ and $N>5$, for the linearised equation on the hypersurface $\Sigma_1$. Let…

Late-time asymptotics conjecture. If $0<|qe|<\frac{1}{2}$, then a solution $\phi$ with generic, regular initial data obeys the asymptotics stated in the paper's sharp asymptotic re…

Let $n\geqslant3$, let $y\in[1/C_\infty,1/C_0)$, and let $\widetilde{C}_1\leqslant\cdots\leqslant\widetilde{C}_{n+1}$ be the nondecreasing reordering of the parameters…

Let $(\mathcal{M},g)$ be a Riemannian manifold. The controlled wave maps equation is a map $\phi:\mathbb{R}\times\mathbb{S}^1\to\mathcal{M}$, with states in…

Decay conjecture in the hyperbolic case.

Decay conjecture in the flat case.

Consider solutions $\phi$ of the wave equation on a fixed Schwarzschild background. Let $\Gamma$ be the timelike initial hypersurface, let $t$ be a time coordinate, let $u$ and $v$…

Let $d\geq 3$ and, for $s>0$, define … where … and $\dim_H$ denotes Hausdorff dimension. Barceló–Bennett–Carbery–Rogers conjecture. … The quantity $\mathcal D_d(s)$ measures the la…

Suppose $M=\mathbb R^d$ has the flat metric, $u$ solves the Cauchy problem … and define $s_p=\frac{d-1}{2}-\frac{d-1}{p}$ and $p_d=\frac{2d}{d-1}$. Euclidean local smoothing conjec…

Let $u$ solve the wave equation on $\mathbb{R}\times\mathbb{R}^n$ with initial data $u(0,x)=0$ and $u_t(0,x)=f(x)$. The solution is expressed through the half-wave oscillatory inte…

Well-posedness hypothesis. There exists a unique solution $\boldsymbol z\in\mathcal C^1([0,\infty);\mathcal X_\kappa)\cap\mathcal C([0,\infty);\mathcal Z_\kappa)$ with…

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…

Let $s=1$ and consider an obstacle problem of the form of the wave equation with obstacle, with obstacle $g$, and let $u$ be the weak solution obtained through the convex minimizat…