29 problems

Let $(M,\mathsf{g})$ be a closed, compact Riemannian manifold without boundary, and let $(u,p)$ be a smooth solution of the incompressible Euler equations on $M$. Tao's conjecture.…

Consider the inviscid Hou–Li model with $C^{\epsilon}$ initial data for small positive $\epsilon$, where the data belong to $C^{\epsilon}$ but to no higher Hölder class. The preced…

Fix soliton parameters $A$, $\boldsymbol{\lambda}$, $\nu$, and $\sigma$. Let $\phi_{\mathrm{rad}}$ be a smooth solution of the wave equation $\Box\phi_{\mathrm{rad}}=0$ satisfying…

The energy-critical wave equation is an evolutionary dispersive equation whose maximally extended solutions are expected to exhibit soliton resolution. A dynamically scaled soliton…

Let $\mathcal{X}=(\{v\leq 0\}\cup \{t\leq -0.9\})\cap\{-1\leq u<0\}$, and let $\phi$ be a solution arising from smooth initial data in the stated Type I self-similar stability resu…

Consider the fractional stochastic Navier–Stokes system studied in the paper, but replace its Gaussian white-noise forcing by discontinuous forcing generated by a Lévy process with…

Consider a vortex patch whose boundary has a single corner of acute angle, and define the order of a cusp at $x=x_0$ by … where $f_1$ and $f_2$ represent the graphs of the two side…

Consider a vortex patch whose boundary has a single corner of acute angle, without any imposed symmetry. Cohen–Danchin's conjecture. Any such acute single corner would cusp immedia…

Let $\Omega(t)\subset\mathbb R^2$ be the evolving region of a vortex patch, with boundary $\partial\Omega(t)$, and suppose that the initial boundary $\partial\Omega(0)$ is a smooth…

In $3+1$ dimensions, the original BKL conjecture predicts highly oscillatory behaviour near the Big Bang for vacuum solutions and most matter sources. Scalar field matter is expect…

In $3+1$ dimensions, vacuum solutions are expected to become highly oscillatory as they approach the Big Bang, with this behaviour persisting in the presence of most matter sources…

Let an entropy-maximizing solution of the Navier–Stokes equation have strictly turbulent initial conditions, meaning that the initial data are orthogonal to the smooth data at each…

Let $G_{\mathrm{FS}}$ be the Fubini–Study Einstein metric, and consider unstable perturbations that are conformal but not Kähler. Let $\mathcal L^2_{-1}$ be the Kähler blowdown sol…

Finite-time cusp blow-up conjecture. For sufficiently small $calpha<calpha_c(omega)$, initial data of sufficiently large mass lead to a blow-up of the fCH solution in finite time.…

Consider the nonlocal transport equation with parameters $\alpha$ and $\gamma$, where the equation and its solution class are as specified in the source. Li–Rodrigo's singularity c…

Let the Whitham equation be considered with initial data $u_0$ of sufficiently large mass, and let $t_c$ denote the characteristic breaking time. A cusp at time $t^*>t_c$ has a spa…

Let $\mathfrak{S}$ be the sign function defined on the energy sublevel region near the equilibria, and consider solutions of the strong-force Hill lunar problem with $\alpha\geq 2$…

Let $\boldsymbol{u}_c(t)$ denote the constructed blow-up solution from Part (a) of Theorem, and let $\boldsymbol{u}(t) \in \mathcal{H}_1$ be any other solution that blows up backwa…

The approximate SQG front equation describes a front through a graph $y=\varphi(x,t)$, with singularity formation referring to the breakdown of the corresponding solution. Breaking…

Okamoto–Sakajo–Wunsch full-range singularity conjecture. Finite-time singularity formation occurs throughout the range $a<2$.

Okamoto–Sakajo–Wunsch threshold conjecture. For smooth initial data, solutions are globally regular when $a\geq2$, while finite-time singularity formation occurs when $a<2$.

De Gregorio's smoothness dichotomy conjecture. Somewhat smooth strong solutions can develop a singularity in finite time, whereas very smooth strong solutions remain smooth for all…

Okamoto–Sakajo–Wunsch threshold conjecture. When $a<1$, finite-time singularities occur, whereas when $a\geq1$, solutions are globally regular.

Let $\theta(y,t)$ be a smooth, even solution of the Córd​​oba–Córd​​oba–Fontelos equation … with initial condition having a single nondegenerate maximum and decaying sufficiently q…

Consider the generalized KP equation … with $\epsilon$-independent initial data $u(x,y,0;\epsilon)=u_0(x,y)$, and assume its solution is at least $C^4$ for $t>0$. Let…