12 problems

The shallow water equations are quasilinear hyperbolic equations, and their singular solutions may include point singularities of the type of the square root of a quadratic form. M…

Let $n\geqslant 3$ and $\alpha\in(0,n)$. Consider a singular positive solution $u\in \mathcal{C}^{2}(\mathbb R^n\setminus\{0\})\cap L^{2^{*}_{\alpha}}(\mathbb{R}^{n})$ of the local…

Landau-solution classification conjecture. Then $u$ is one of the Landau solutions, which are minus-one homogeneous and axisymmetric without swirl.

Higher-$\sigma_k$ gluing conjecture. Although the method is computationally restricted in the paper to $k=2$, the existence result for infinitely many solutions in the positive…

Asymptotic classification conjecture. If $p\in(1,2_m^*-1)$, then

Let $m\in\mathbb N^+$, set $N=2m$, and let $n>N$. Consider positive singular solutions $u\in C^N(\mathbb R^n\setminus\{0\})$ of … where…

Let $u$ be a positive solution of the local critical equation … and let $u_0\in C^6(\mathbb R^n\setminus\{0\})$ be the corresponding solution satisfying…

Let $\Omega\subset\mathbb R^N$ be a bounded domain containing $0$, let $N\geq 3$, $p=\frac{N}{N-2}$ and $M>0$, and let $u$ be a positive solution of … denote the asymptotic conditi…

Let $\Omega\subset\mathbb R^N$ be a bounded domain containing $0$, let $p>\frac{N}{N-2}$, and let $u$ be a positive solution of … in $\Omega$; or … or … The conjecture strengthens…

Consider the one-dimensional system … with Biot–Savart law … and singular steady solution … where $c>0$ is suitable. Let $\omega_0$ be a smooth profile compactly supported inside…

Sharp global estimate conjecture. Estimate should hold for all both-singular solutions not only when $N=4$ with $0<\beta<3\max\{\mu_1,\mu_2\}$, but also when $N=3$ without assuming…

Upper-branch fold conjecture. The upper solution branch undergoes infinitely many folds, and $\lambda$ tends to a finite value bounded away from zero.