7 problems

Let $k\geq 3$ and consider the $k$-generalized Korteweg-de Vries equation on the periodic domain, with initial data in the Sobolev space $H^s(\mathbb T)$. Unconditional well-posedn…

Wild non-uniqueness conjecture. For any such Riemann initial data, there exists a smooth pressure law $p$ with $p'>0$ such that the corresponding system admits infinitely many boun…

Higher-dimensional decomposition conjecture. For every $D\in\mathcal{C}^{m,\gamma}(\bar\omega,\mathbb{R}^{d\times d}_{\mathrm{sym}})$,

Non-uniqueness conjecture. Under the stated integrability condition and suitable assumptions on $\beta$ and $T$, such a density $\rho$ and incompressible Sobolev vector field $u$ e…

Isett–Oh conjecture. For every $\theta<\frac{1}{3}$, there exists a weak Euler solution $v\in C^\theta(\mathbb R\times\mathbb T^n)$ such that

Let $\Sigma$ be a $2$-dimensional compact Riemannian manifold, possibly with boundary, with positive Gauss curvature. A $C^{1,\alpha}$ isometric embedding is an isometric embedding…

Let $\Sigma_a$ be the polar cap and let $\mathscr{I}^\alpha_1(\Sigma_a)$ denote the class of $C^{1,\alpha}$ isometric embeddings of $\Sigma_a$ into the relevant codimension-one Euc…