11 problems

Hong–Yang conjecture. The integer

Let $E$ be a measurable set in the periodic setting, and let $E^{*\operatorname{per}}$ denote its periodic rearrangement, defined by applying Steiner symmetrization in the $x_1$-di…

Generalized Yang's conjecture. If

Let ${\boldsymbol \tau}$ be a uniform knot sequence such that the dimension of $\mathcal{S}_{p,{\boldsymbol \tau},\operatorname{per}}$ is $2m$. For any $q\geq 0$, let $Q_p^q$ be th…

Let $g$ be a positive integer, let $u,\alpha\in\operatorname{\mathbb{C}}^g$, and let $u_{g+1}\in\operatorname{\mathbb{C}}$. Let $\sigma$ be the genus-$g$ hyperelliptic sigma-functi…

Let $C: \mathbb{N} \to \mathbb{C}$ be periodic with period $k$, and let $s(t)$ be a real polynomial of positive degree $d$ with positive leading coefficient $a_d$. Then, as $q$ app…

Let $(u_0,0,T)$ be a smooth periodic set of data. Global regularity for homogeneous periodic data with normalised pressure. There exists a smooth periodic solution $(u,p,u_0,0,T)$…

Let $(u,p,u_0,0,T,L)$ be a smooth periodic, homogeneous normalised-pressure solution, with $0<T<T_0<\infty$ and periodic data norm ${\mathcal H}^1(u_0,0,T,L)\leq A<\infty$. A prior…

Let $(u,p,u_0,f,T,L)$ be a smooth periodic, normalised-pressure solution, with periodic $H^1$ data norm ${\mathcal H}^1(u_0,f,T,L)$ finite. A priori periodic $H^1$ bound. There exi…

Let $(u_0,f,T)$ be a smooth periodic set of data. A periodic solution has normalised pressure when its pressure satisfies the normalisation used in the periodic Navier–Stokes formu…

Let $(u_0,f,T)$ be a smooth periodic set of data. Global regularity for periodic data. There exists a smooth periodic solution $(u,p,u_0,f,T)$ with the indicated data. This is the…