20 problems

Consider the three-dimensional wave equation associated with the inverse-square potential parameter $a$, with $-\frac{1}{4}<a<0$, and also the critical case $a=-\frac{1}{4}$. Let t…

Wave maximizer conjecture. The set of maximizers for which equality holds in the sharp wave Strichartz inequality coincides with the set of initial data of solutions in the orbit o…

Spline approximation conjecture. The assumption is satisfied for all $C^{k_t}$ splines with polynomial degree $p_t\geq 2$ such that $p_t-k_t$ is odd. This would extend the stated v…

Müller–Thiele–Vallarino endpoint conjecture. The Sobolev estimate for solutions to the wave equation on a Damek–Ricci space remains valid at both endpoint regularity exponents abov…

Optimal threshold conjecture. The optimal limit for $s$ in the convergence theorem is actually $s>1$, which would yield the scaling $n\sim \omega^{1^+}$ as the frequency $\omega$ g…

Estimator-behaviour conjecture. The estimator behaviour reported in Tables 7 and 8 of Gorynina, Lozinski, and Picasso is due to the different a posteriori error estimator used ther…

Generic wave control conjecture. Generically with respect to the lengths $(\ell_i)_{i=1,\dots,N}$, (GGCC) is a necessary and sufficient condition for the internal control of the wa…

Avdonin–Zhao's watershed conjecture. The watershed description of Avdonin and Zhao provides the optimal time for the observation and control of the free wave equation on a graph.

Endpoint local smoothing conjecture. For every $p>\frac{2d}{d-1}$, the local smoothing estimate

Let $d \geq 2$, $q \in (2,\infty)$, and $r \in [2,\infty)$ satisfy … Let $\beta_{d/2}(q,r)$ denote the exponent appearing in the conjectured estimate, and let the estimate … holds…

$L^p$-$L^q$ local smoothing conjecture. The inequality

Compact modified-equation stability conjecture. This condition holds at every even order $2p=2,4,6,\ldots$, with a similar result in three-dimensional problems. The condition is kn…

Taylor time-stepping methods have local stability radii that do not yield an order-independent CFL number. Stability-enhanced Taylor-scheme conjecture. Stability-enhanced schemes c…

Let $v(x,t;k)$ be a superposition of Gaussian beams with frequency parameter $k$, and let $u_{exact}(x,t)$ denote the exact solution. For the initial data described above, suppose…

Optimal-constant conjecture. The optimal constant should be

Let $\Sigma_0$ be a complete $2$-ended asymptotically flat Cauchy hypersurface in a Kerr spacetime with $0<|a|<M$, and let $\psi$ solve the linear wave equation … The relevant assu…

Let $\Sigma_0$ be a Cauchy hypersurface in a subextremal rotating Kerr spacetime, and let $\psi$ solve the linear wave equation … Assume the initial data on $\Sigma_0$ are smooth a…

Let … where … and … with $x,y\in{\bf R}^n$, $k\ge1$, $1\le m<n$, and $0<h\ll1$. The oscillatory-integral conjecture. If $V$ satisfies condition (1.6), then … with a constant $C>0$…

The critical-regularity dispersive conjecture. The perturbed wave group $e^{it\sqrt{G}}$ should satisfy optimal dispersive estimates under these assumptions, possibly requiring onl…

Let $u(x,t)$ denote the shape of a vibrating string, where $x$ is the spatial variable and $t$ is time. A trigonometric series is a series formed from trigonometric functions such…