6 problems

Let the initial data be non-trapping. Global well-posedness conjecture for non-trapping initial data. The result in Theorem holds for all non-trapping initial data. The source give…

Let $s$ denote the Sobolev regularity exponent in Theorem. Sharpness conjecture for the local well-posedness Sobolev range. The range of $s$ in Theorem is sharp for generic problem…

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…

Consider a one-dimensional dispersive flow on the real line with a cubic conservative nonlinearity and initial data of size $\epsilon \ll 1$. Non-localized data long-time well-posed…

Let the cubic problems … be as in Theorem … also holds in the large-data case. The paper considers only small data, so this extension is stated as expected rather than proved here.

Consider the cubic problems denoted by … . Sharpness conjecture. Generically, the problem … is ill-posed in $H^s$ for $s<2$. The theorem establishes local well-posedness above thes…