9 problems

Let … . KAM conjecture. If the Diophantine equation has only finitely many solutions, then a KAM type result holds for … , depending on a parameter which takes value in a Cantor-ty…

Let $(n_0,j_0)$ be a pair of natural numbers with $n_0\geq2$, $j_0\geq1$, and set … Suppose that the only natural number solution of the Diophantine equation … is $(j_0,n_0)$. Peri…

The spherical capillary water waves system is considered on the sphere, with small-amplitude time-periodic solutions as the objects of interest. Periodic-solution conjecture. There…

Let $d>0$ and let $\beta_d$ be the kernel of the periodic Hilbert transform on the strip of depth $d$, defined for $s\in\mathbb{R}\setminus 2\pi\mathbb{Z}$ by the kernel formulas i…

Let … denote the spherical water waves equation. Lifespan conjecture. If the Diophantine equation … , the lifespan goes beyond $\varepsilon^{-(1+\alpha)}$ as $\varepsilon\to0$. Fin…

Let $a\in(-1,1)$ and let $u$ be a bounded, smooth solution of the generalized fractional Allen–Cahn equation … Assume that $\partial_{x_n}u>0$. Generalized fractional De Giorgi con…

Let $\varphi$ be the profile of a highest cusped periodic traveling-wave solution of Whitham's equation, with speed $\mu>0$, satisfying … where $L$ has Fourier symbol…

Graph-to-splash existence conjecture. There exist initial data $z_0(\alpha),\omega_0(\alpha)$ of solutions of the water wave equations such that at time $0$ the curve $z_0(\alpha)$…

Let $\epsilon$ and $\delta$ be the perturbation parameters, and let $\mathcal{H}_{nm}$ denote the Hamiltonian contribution of bidegree $(n,m)$ in the grading with respect to…