9 problems

Let $\rho$ be the doping profile in the three-dimensional nonlinear Schrödinger–Poisson problem, and let $\|\rho\|_{L^{\frac{6}{5}}}$ denote its $L^{\frac{6}{5}}$-norm. Large-dopin…

Let $u(t)$ solve the discrete nonlinear Schrödinger Cauchy problem in $l^p_h$, where $1\leq p\leq +\infty$, and let $h>0$, $d\geq 1$. The sharp $l^p_h$ estimate. A finer estimate s…

Weak-strong uniqueness. The weak and strong solutions are identical almost everywhere in $[0,T]\times\Omega$. This is presented as an open question for the Navier–Stokes–nonlinear-…

Three-dimensional blow-up conjecture. In three dimensions, all initial data sufficiently large in some sense blow up.

Lebowitz–Rose–Speer conjecture. The invariant measures associated with these formal Gibbs measures undergo a phase transition.

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…

Monotone-profile threshold conjecture. For initial data $u_0(x)=\alpha\psi(x)$, the solution scatters if $\alpha<\alpha_0$ and blows up if $\alpha>\alpha_0$.

Real-data scattering conjecture. Under this assumption, $u(t)$ scatters as $t\to-\infty$ and as $t\to+\infty$.

Conjecture 1. For each $epsilon>0$, there exists such initial data for which $u(t)$ scatters as $t\to-\infty$ and blows up in finite forward time.