21 problems

Let a one-dimensional dispersive flow on the real line have a cubic defocusing nonlinearity, and let its initial data be sufficiently small. Non-localized data defocusing global we…

Global well-posedness and scattering conjecture. Then $u$ is global and scatters, and

Let $u$ solve the radial, defocusing, cubic wave equation … on $\mathbf{R}^{3}$ with initial data $(u_0,u_1)\in \dot{H}^{1/2}(\mathbf{R}^{3})\times\dot{H}^{-1/2}(\mathbf{R}^{3})$,…

Critical-norm global well-posedness and scattering conjecture. Under these assumptions, global well-posedness and scattering should hold under a priori control of the critical norm…

Critical-norm global well-posedness and scattering conjecture. If $s_c>0$, then $u$ is global and scatters as $t\to\pm\infty$.

Consider a cubic quasilinear dispersive flow with small initial data in spatial dimension at least two. Higher-dimensional non-localized data global well-posedness conjecture. The…

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.

Consider the defocusing septic nonlinear Schrödinger equation on the line … For initial data $u_0\in \dot{H}^{1/6}_x(\mathbb{R})$, critical-regularity conjecture. Global well-posed…

Consider quasilinear dispersive flows in two and higher spatial dimensions with cubic nonlinearities and sufficiently small initial data. Two-dimensional and higher-dimensional qua…

Consider a one-dimensional dispersive flow which is locally well-posed in some Sobolev space $H^s$, has phase rotation symmetry, and is conservative. For initial data $u_0$ satisfy…

Half-plus regularity propagation conjecture. The solution carries the $\frac12+$ regularity of the initial data up to time $T_0(s)$: for every $T\in(0,T_0(s))$,

Let $u_0\in L^2(\mathbb{R})$ be smooth initial data with a single hump for the defocusing modified fractional Korteweg–de Vries equation, with dispersion parameter $\alpha$. Defocu…

Consider the hyperbolic Yang–Mills equation in $4+1$ dimensions for finite-energy connections, with ground state energy $E_{\mathrm{GS}}$, the energy of the minimal nontrivial harm…

Global well-posedness conjecture. When $\mu=1$, the equation is globally well posed: for each $u_0\in Q(H)$ there is a unique global solution…

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…

Consider the defocusing cubic nonlinear Schrödinger equation … Here $s\geq 1/2$, and scattering means that the global solution approaches linear Schrödinger evolutions as time tend…

A Schrödinger map is a solution of the equation … from $\mathbb{R}^2$ into the sphere, with energy … A non-trivial harmonic map is a non-constant static solution of the correspondi…

Let $\mu=\pm1$. Let $(u_0,u_1)\in H^1_x\times L^2_x$. In the focusing case, assume additionally that $M(u_0)<M(Q)$ and $E(u_0,u_1)<E(Q)$, where $Q$ is the ground state. A solution…

A Schrödinger map is a time-dependent map from a one-dimensional domain into a compact Kähler manifold evolving according to the Schrödinger map flow. Ding's conjecture. The Schröd…

Let $u$ solve the derivative nonlinear Schrödinger equation considered in the paper, with initial datum $u_0$, let $s=[n/2]+1$, let $ugeq 2+4/n$, let $D^beta$ denote a spatial der…

Let $u$ solve the cubic nonlinear Schrödinger equation on $\mathbb{R}^2$ with initial data $u_0\in H^s_x(\mathbb{R}^2)$. An a priori bound is an estimate of the form … for every…