12 problems

Nonlinear stability conjecture. These conditions should be relaxable, and space-times with a supersymmetric compactification and $n=3$ should be nonlinearly stable.

Null-condition well-posedness conjecture. If the null condition holds, then (GNLW) is well-posed in $H^s\times H^{s-1}$ for some $s<\frac n2+\frac34$ when $n=2$, respectively for s…

Let a shock hypersurface be a hypersurface across which a weak solution develops a shock in a quasilinear wave equation posed in multiple spatial dimensions. Dynamic singularity fo…

For a quasilinear wave equation whose nonlinearity has the form … and has a more favorable nonlinear structure, the associated local well-posedness should hold for Cauchy data with…

Let $k\geq 2$, let $\varepsilon_k^*=\frac{k}{k+1}$, and consider the two-dimensional quasilinear wave equation … . Suppose that … and … for $\xi_0=-1$ and…

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

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

Consider the scalar quasilinear wave equation … in $\mathbb{R}^{1+3}_{t,x}$ with small initial data … Here $g^{\alpha\beta}(u)$ are smooth, symmetric coefficients satisfying…

Let the top-order energy be the highest-order energy associated with a three-dimensional quasilinear wave equation enjoying a weak null structure. Alinhac's blowup-at-infinity conj…

Let $u$ solve a quasilinear wave system of the form … with the weak null condition (WNC) defined by the existence of a global asymptotic-PDE solution for the ansatz … whose derivat…

The equations under study are quasilinear wave equations with derivative-quadratic nonlinearities, and their solutions may develop a blowup surface in spacetime. Denote by…

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