12 problems

The Whitham equation is … for $(x,t)\in\mathbb{R}\times\mathbb{R}^+$, where $\zeta(x,t)\in\mathbb{R}$ is the surface elevation and $\sqrt{\mathcal{K}_{\mu}}(D)$ is the square root…

Cauchy-horizon conjecture. The perturbed solution has a Cauchy horizon that is a perturbation of the Cauchy horizon of the background solution.

Crease-and-singular-boundary conjecture. The perturbed solution has a crease and singular boundary that are perturbations of the crease and singular boundary of the background solu…

Time-of-first-blowup conjecture. The perturbed solution forms a shock at a time that is a perturbation of $T_{\mathrm{Shock}}$; its first blowup time is the smallest value of $t$ a…

The one-dimensional Burgers equation admits finite-codimension stable shock formation under suitable initial-data conditions, as established by the preceding proposition. Finite-co…

Asymptotic expansion conjecture. We conjecture that $U_{\ell}^{\mathrm{in}}$ admits an asymptotic expansion about infinity of the form

Let $u_0\in L^2(\mathbb{R})$ be smooth initial data with a single hump for the defocusing modified fractional Korteweg–de Vries equation, with dispersion parameter $\alpha$. Defocu…

Let $(\rho,u_r,u_\uptheta)$ be the solution to the two-dimensional isentropic compressible Euler equations given in Theorem general, and let $\Omega(t)$ be the material curve defin…

The equations of compressible gas dynamics can form shocks, singularities across which the physical variables change discontinuously. A caustic is a singular structure formed by wa…

Pearcey asymptotics conjecture. Near the first singularity, the solution has the asymptotic expansion

First-singularity conjecture. Spherically symmetric solutions of this equation have their first singularities on the event horizon $\mathcal{H}^{+}$.

Let $T_{(Lifespan);\uplambda}$ denote the classical lifespan of the solution to the quasilinear wave equation with initial data…