45 problems

Glassey's critical-threshold conjecture. There should exist a global-in-time small-data solution when $C_{\mathrm{Gla}}=0$.

Let $d\geq 3$ and consider the mass-critical nonlinear Schrödinger equation with nonlinearity $F(z)=+|z|^{4/d}z$. For initial data $u_0\in L^2_x(\mathbf{R}^d)$ and initial time…

Let initial data for the Novikov–Veselov equation be classified as critical, subcritical, or supercritical according to the behavior of the associated Schrödinger operator at the z…

Let a nonlinear symmetric hyperbolic system in $n$ spatial dimensions have totally linearly degenerate structures. Suppose a smooth solution belongs to … with $s>\frac{n}{2}+1$. Ma…

Let $u^I$ be a solution to a system of semilinear wave equations whose nonlinearities satisfy the weak null condition: the associated asymptotic system … has a global solution for…

For the $2$-D semilinear wave equation with scale-invariant damping … where $t\ge 1$, $\mu>0$, and $p>1$, a global small-data weak solution is a weak solution $u$ arising from suff…

Klein–Saut conjecture. The following assertions hold:

Let $({\mathbb H}^2,h)$ be the standard hyperbolic plane. A classical wave map originating on ${\mathbb R}^{2+1}$ is a map from $(2+1)$-dimensional Minkowski spacetime into…

Small-data asymptotic-freedom conjecture. If

Let $k>3/2$. In temporal gauge $A_0=0$, let … be the Banach space of Cauchy data $(A^{\operatorname{Cau}},\dot A^{\operatorname{Cau}},\Psi^{\operatorname{Cau}})$, and let…

Kraichnan's assertion. For every given $bcp,bcn\fteq 0$, sufficiently small $b5$ should ensure the existence of global-in-time strong solutions for large perturbations of the equil…

Let $\mathcal{B}=\{x\in\mathbb{R}^n:|x|\leq 1\}$ be the closed unit ball and consider the exterior problem on $\mathcal{B}^c$: … with $u=0$ on $\partial\mathcal{B}^c$ and initial d…

Let a system of hyperbolic conservation laws have totally linearly degenerate characteristics, and let its initial data belong to $H^s(\mathbb R^n)$ with $s\geq 1+\frac{n}{2}$. Maj…

Weak null condition conjecture. The weak null condition is sufficient for global existence of small-data solutions.

Let $D\geq 4$, and consider arbitrary asymptotically flat initial data for the Einstein equations in a spacetime with Bondi coordinates $(u,r)$, where $u$ is a timelike coordinate…

Integral criterion conjecture. Small-data global existence holds if

Chen–Reissig's conjecture. If $C_{\mathrm{Str}}>0$, then the solution blows up at finite time for some initial data; if $C_{\mathrm{Str}}=0$, then small-data global existence holds…

Tzvetkov's conjecture. The nonlinear Dirac equation has a small-data global solution when its nonlinearity takes such special forms as $N_{1}$ and $N_{2}$. This conjecture concerns…

Small-stress global-estimate conjecture. Global estimates should be obtainable when $\tau$ is sufficiently small.

Critical-mass global-existence conjecture. The solution $(n,c,\mathbf{u})$ also exists globally in time in the critical mass case.

Let $\mu$ be a modulus of continuity satisfying the global-existence condition … For the three-dimensional semilinear classical wave equation with radial small initial data, the st…

Critical-index conjecture. All solutions exist for all time if $k\geq \frac{n}{2}+1$, whereas for $k<\frac{n}{2}+1$ there are always some solutions that break down in finite time.…

Weak null conjecture. The original system of wave equations also admits global solutions for sufficiently small initial data.

Weak null condition conjecture. The weak null condition should be sufficient for global existence of solutions arising from sufficiently small and localized initial data.

Global existence conjecture. If the prescribed integral relation exists for all times, then the regulator problem has a unique global solution.