22 problems

Ivrii's conjecture. If every singular point of $p=0$ is effectively hyperbolic, then the Cauchy problem is $C^{\infty}$ well posed for any lower order term.

De Giorgi's conjecture. There exists a limit

Let $(\Box_{N+1}+u(x))\varphi=0$ be a hyperbolic equation, and suppose that its sequence of Hadamard coefficients terminates. A locus configuration is a configuration of hyperplane…

Let $u(x_1,\ldots,x_n)$ be a potential and consider the hyperbolic equation … A potential is called huygensian when the corresponding equation satisfies Huygens' Principle. Complet…

Let $D\subseteq\Omega$ be an open set with continuous boundary satisfying the geometric conditions in the semi-uniform stability setup of ${}^{\text{su}}$, and let $\1_D$ denote mu…

Assume $(B,T)$, $(u_{0},u_{1})$, $d$, and $\Omega$ are given, and let $u$ be the unique maximal solution to the hyperbolic problem. Critical-threshold conjecture. There exist criti…

Fully discrete methods use a compact stencil in both space and time when each update depends only on a bounded local set of spatial and temporal degrees of freedom. Roe et al.'s co…

Let $Omega a$ be the spatial domain, let $f$ be the initial source, let $c$ be the sound speed, and let $q(x')$ depend on the first two spatial variables $x'$. Write $chiOmega$ for…

The modified shock–turbulence interaction is posed for the one-dimensional Euler equations on $\Omega=[-5,5]$, with the initial density, velocity, and pressure given by … with infl…

Let a high-order linear or nonlinear evolution PDE be given, together with the general first-order hyperbolic relaxation, or hyperbolization, constructed in the paper to approximat…

Let $U_{\rm sh}$ be a shear flow, and consider nonlinear hyperbolic equations admitting shear-flow solutions and the setting of Theorem. Gevrey-3 stability conjecture. Theorem hint…

Let the characteristic fields of the regularized reduced model be the fields corresponding to its characteristic speeds $\lambda^{(N)}_k$, with the fields corresponding to…

The obstacle problem concerns a weak solution produced by the proposed convex minimization and semi-discrete approximation scheme; in cases where weak solutions are nonunique, comp…

Let $P$ be a hyperbolic operator whose principal symbol has triple characteristic points only on the initial plane $t=0$, and suppose that $P$ is effectively hyperbolic at every tr…

Consider a traveling front for a nonlinear hyperbolic equation of bistable type, and let the corresponding linearized operator act on the phase space used for the spectral stabilit…

Sign conjecture for $K_0$. The kernel satisfies

Effective hyperbolicity conjecture. A hyperbolic operator with at most triple characteristics is strongly hyperbolic if and only if it is effectively hyperbolic.

General-kernel controllability conjecture. The same equivalence should remain true without the assumption $g(x,y)=g(x)$; namely, for kernels satisfying…

The sharp-interface limit is described by a displacement–crack set pair $u_0(t,x)$ and $S_{u^0(t)}$, with the wave equation coupling elastic waves to the evolving fracture path. Th…

For the scalar semilinear equation, let $n$ denote the spatial dimension, $m$ the physical mass, and let the nonlinearity exponent satisfy $\alpha\in(0,\infty)$. Forbidden physical…

The propagation of analyticity concerns solutions of a weakly hyperbolic nonlinear system … Assume that the solution $U(t,\cdot)$ is bounded in a functional class $\mathcal X$ in w…

Let $Q$ be the periodic domain under consideration, with boundary $4\partial Q4$, and let $\sigma_+(t)$ and $\sigma_-(t)$ be the characteristic curves starting from…