7 problems

Let $(M,J, \omega)$ be a compact almost-Kähler manifold, where $J$ is an almost-complex structure and $\omega$ is its fundamental $2$-form. Let $\sigma$ be a smooth volume form on…

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

Complete classification conjecture. Either $u\equiv 0$, or there exist $\lambda:=u(0)^{\frac{p}{N-p}}>0$ and $x_{0}\in\mathbb{R}^{N}$ such that

Thomas–Fermi dynamical neutrality conjecture. For every initial state $(\varphi_0,\rho_0)$ satisfying $0\leqslant\rho_0\in L^1(\mathbb{R}^3)\cap L^{5/3}(\mathbb{R}^3)$ and…

Dynamical Hartree conjecture. For every initial state $u_0\in H^1(\mathbb{R}^3)$ and every $R>0$,

Let $k=1$, let $u_0\in H^s(\mathbb{R})$ with $s>3/2$, and let $u$ be the corresponding solution of equation (8.0.1), with lifespan $T$. Finite-time wave breaking conjecture. The so…

Let $f$ satisfy (F1) with $q\geq 1$, suppose that $f'(u)F(u)\leq q$ for all sufficiently large $u>0$, and let $\phi\geq 0$. Set … If … then (S1E1) has a local-in-time solution. Exi…