31 problems

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…

Spherically symmetric scattering conjecture. In the defocusing case ($\mu=+1$) we have $m_{0,\operatorname{rad}}=+\infty$, while in the focusing case ($\mu=-1$) we have…

Scattering conjecture. In the defocusing case ($\mu=+1$) we have $m_0=+\infty$, while in the focusing case ($\mu=-1$) we have $m_0=M(Q)$. This predicts global well-posedness and sc…

Quadratic-phase conjecture. For the combined nonlinear Schrödinger equation, if $b>0$, the solution with initial condition $v_0$ exists globally and scatters, whereas if $b<0$ and…

Global existence and scattering conjecture. All maximal-lifespan solutions are globally well-posed and scatter.

Kenig–Merle critical norm conjecture. If $u$ has bounded critical Sobolev norm in this sense, then $u$ is global and scatters to a free solution.

One-dimensional dispersive flows with cubic defocusing nonlinearities and small initial data are considered. Global small-data scattering conjecture. Such flows should have global-…

Let $u$ be a solution of the cubic wave equation on its maximal interval $[0,t_+)$, with initial data in $\mathcal{H}^{1/2}$. Logarithmic critical-norm criterion. There exists…

Let $\mathcal{H}^s=\dot H^s(\mathbb{R}^3)\times\dot H^{s-1}(\mathbb{R}^3)$ denote the phase space, and consider initial data for the cubic wave equation. The self-similar profile i…

Global non-scattering solutions conjecture. Let $u$ be a solution of the cubic wave equation with the prescribed initial-data problem, defined for $t\in[0,\infty)$ and not scatteri…

Global scattering conjecture. Such initial data should give rise to a solution that exists globally in time and scatters.

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…

Let the Einstein vacuum equations be posed as a scattering problem with conformally regular data on an ingoing null hypersurface and no incoming radiation from past null infinity.…

Below-ground-state almost-periodic-solution conjecture. There does not exist a nonzero almost periodic solution $u$ to $(1.1)$ satisfying

Let $u$ solve the focusing, mass-critical generalized KdV equation $(1.1)$ with initial data $u(0,x)=u_0\in L^2(\mathbb{R})$, and let … be the ground state, which satisfies…

In dimension $d=3$, let $\phi_\omega$ denote the ground state solution of the cubic-quintic Schrödinger equation at frequency $\omega$, with $0<\omega<\tfrac{3}{16}$, and let…

Consider the time-dependent density-matrix equation with self-interaction satisfying condition (SI2), and let $d$ denote the spatial dimension and $\beta$ the exponent in that cond…

Consider the defocusing Schrödinger initial value problem for $u:I\times\mathbb{R}^{d}\to\mathbb{C}$, … where $s_c\geq 0$ and $p=\frac{4}{d-2s_c}$. The critical norm conjecture ass…

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…

Let $d\geq 3$, let $\mu\in\{1,-1\}$, and consider the energy-critical nonlinear Schrödinger equation without potential … For $\mu=-1$, define … where $W$ solves…

Let $d\geq 3$, let $I$ be the lifespan interval, and let $u:I\times\mathbf{R}^{d}\to\mathbf{C}$ solve the focusing energy-critical nonlinear Schrödinger equation with…

Cazenave–Kenig–Merle conjecture. When $\mu=1$, solutions exist globally and scatter: for every $u_0\in\dot{H}^1(\mathbb{R}^d)$, there is a unique global solution…

Scattering conjecture. Every global solution satisfying the conditions above has finite $S(\dot{H}^{1/2})$-norm, and it scatters in $H^1$.

Let $d\geq 1$ and $p>0$ satisfy … Let $u:I\times\mathbb{R}^d\to\mathbb{C}$ be a maximal-lifespan solution to the nonlinear Schrödinger equation, with…

Let $\mu=\pm1$, and let $w_0\in L^2_x(\mathbb{R}^2)$. In the focusing case assume $M(w_0)<M(Q)$. A solution scatters if there are $w_\pm\in L^2_x$ with…