18 problems

Let a nonlinear dispersive equation have dispersion of order $l$ and a nonlinearity of order $k\ge 3$ with a loss of $m$ derivatives. Let $s_0$ be such that, for $s>s_0$, the $H^s$…

Semiclassical transport conjecture. There exists $\beta<3/4$ such that, under the assumptions in, the solution $\Psi_t$ of

Classification of minimal-mass blowup solutions. Only the following two cases occur:

Consider the periodically forced Dirac equation with mass $m=1$, and let $M^n$ denote its $n$-period propagator. For $\beta>0$, write…

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

Klein–Roudenko–Stoilov conjecture. The solution blows up at a finite time $T$, and, as $t\to T$,

Let $s$ denote the Sobolev regularity exponent, and let Theorems and be the stated local well-posedness results for the two-dimensional and $n$-dimensional cubic problems. Large-da…

Let $s$ denote the Sobolev regularity exponent, and let Theorems and be the local well-posedness results stated in the source for the two-dimensional and $n$-dimensional cubic prob…

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

Uniform two-thirds decay conjecture. For any $\zeta\in\mathcal{C}_c^{\infty}(\mathbb{R}^d)$,

Let a focusing flow satisfy the hypotheses of the paper, and let the result in Theorem provide its $\epsilon^{-8}$ lifespan bound and accompanying estimates. Sharpness conjecture fo…

Let a one-dimensional dispersive flow on the real line have a cubic nonlinearity, and let its initial data have size $\epsilon$ in a suitable Sobolev norm. Non-localized data focusi…

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

Klein–Saut conjecture. The following assertions hold:

Let $U(t)$ denote the linear evolution operator and let $u_0$ be initial data, with the mixed-norm estimate … Here $d$ is the spatial dimension, $p$ is the spatial integrability ex…

Let $u0 in L2(\mathbb{R})$ be smooth initial data with a single hump. The fKdV equation is conjectured to have the following behavior. For $\alpha0.5$, solutions remain smooth for al…

Let $H^s$ denote the Sobolev space of order $s$, let $H^\infty$ denote the smooth-data space, and let $S_T^\infty$ be the solution map for the real-valued modified Benjamin–Ono equ…

Let $\psi=1+u$ be a global solution of the Gross–Pitaevskii equation in $C(\mathbb{R};1+F_1)$, and let $E_1$ denote its renormalized energy. Fix the threshold $\mathcal{E}_0>0$ fro…