374 problems

Long-time equilibrium conjecture. The one-dimensional solution should satisfy

Schläfli's conjecture. The system is locally solvable provided that the target Euclidean space has dimension $d\geq n_{\ast}$. This concerns the existence of local isometric immers…

Schonbek's sharp decay-rate conjecture. The sharp decay rate of the velocity in $L^2$ should be

The von Neumann sonic conjecture. There exists a supersonic regular shock reflection-diffraction configuration when

Let $D\subset\mathbb{R}^2$ be a bounded simply connected domain with sufficiently smooth boundary, and let $v\in C^4(D)\cap C^1(\overline D)$. The boundary conditions use the inwar…

Let $h(t,m;s)$ denote the solution of the PDE … with terminal condition $h(s,m;s)=1$. Quadratic-exponential form conjecture. The solution has the form … This ansatz is intended to…

Let $\mathsf{M}$ be a smooth manifold and let $\mathsf{P}$ be a properly supported pseudodifferential operator on $\mathsf{M}$ with complex coefficients. Local solvability means th…

Coercivity conjecture. The condition above holds for every $g>1$.

Let $(M,g)$ be an asymptotically static spacetime and let $\square_g$ denote its wave operator. Essential self-adjointness conjecture. The operator $\square_g$ is essentially self-…

Treves' conjecture. The operator $P$ is analytic hypoelliptic if and only if every stratum in the above-described Poisson–Treves stratification is symplectic.

Let $D$ be a two- or three-dimensional obstacle, let $x$ lie in its exterior, and write $r=|x|$ and $\alpha'=x/r$. Let $u(x,\alpha)$ be the acoustic field for an incident plane wav…

Lions's conjecture. The compressible Navier–Stokes system may exhibit propagation of oscillations; Lions provided an example for a variant of the system that includes a forcing ter…

High-noise uniqueness and stability conjecture. In the high-noise case, there is at most one stationary state which is homogeneous in time and globally asymptotically stable.

Consider the non-autonomous wave PDE … on a domain $\mathcal X$, with the stated edge conditions and assumptions on $f$ and $g$. Partition the space into $N$ elements, let…

Consider the two-dimensional incompressible isotropic Hookean elastodynamics system for a smooth flow map $X(t,y)$ satisfying … and … For small initial displacements, let the highe…

An equation is called integrable when it belongs to the class of integrable partial differential equations, and the Painlevé test is the algebraic test for the Painlevé property of…

Let $\rho$ be a solution of the two-dimensional pseudo-gravity model with diffusion coefficient $D$, and let $M_0$ denote the conserved mass. The entropy estimate indicates dissipa…

Forbidden mass interval conjecture. The interval

Let ${\mathcal S}$ be the space of classical data $(\phi_0,\phi_1)$, where $\phi_0:{\mathbf R}^2\to{\mathbf H}$ is Schwartz modulo constants and…

Let $\mathcal{E}$ be a system of partial differential equations with fundamental algebras $\mathfrak{f}_k$ obtained from its coverings and higher-order prolongation structure. Fini…

For $m\geq3$, let $P^{(m)}$ denote the higher-point generating series in the variables $w_1,\ldots,w_m$. Recursive rational differential-equation conjecture. After multiplying the…

Let $(t,x)\in\mathbb{R}_+\times\mathbb{R}^3$. For each $N$, let $\mathbf{u}_N$ denote the corresponding approximating solution, and let $U_{t,x}$ be a neighborhood of $(t,x)$. Loca…

Existence conjecture. A solution exists if and only if $L_{A,\sigma} f\geq 0$ for every convex function $f$ having $L^1$ boundary values, with strict inequality whenever $f$ is not…

Optimality conjecture. The Lipschitz $\widetilde{\frac{1}{2}}$-regularity corresponding to the case $p=\infty$ in Theorem 1.1 is optimal.

Unique-solvability conjecture. The boundary value problem is uniquely solvable if and only if $\mathcal{N}^\Omega_\zeta=\emptyset$.