9 problems

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

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…

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…

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…

Global well-posedness and scattering conjecture. For $d\geq 1$, the defocusing, mass-critical nonlinear Schrödinger initial value problem $$ is globally well-posed for $u0in L^2(ma…

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…