12 problems

On blowup for large data. For initial data with sufficiently large energy, the solutions of equation blow up in finite time in the sense that the derivative $u_r(t,0)$ diverges as…

Small-data critical well-posedness conjecture. The Cauchy problem is well-posed for small data in $\dot H^{\frac{n}{2}}$.

Let $(M,g)$ be a complete Riemannian manifold, and consider wave maps from $\mathbb R^2\times\mathbb R$ into $(M,g)$. The critical initial-data space is $H^1\times L^2$. Wave maps…

Let $(\mathcal{M},g)$ be a Riemannian manifold. The controlled wave maps equation is a map $\phi:\mathbb{R}\times\mathbb{S}^1\to\mathcal{M}$, with states in…

Let $\boldsymbol{u}_c(t)$ denote the constructed blow-up solution from Part (a) of Theorem, and let $\boldsymbol{u}(t) \in \mathcal{H}_1$ be any other solution that blows up backwa…

Critical-threshold conjecture. For $d\in\{3,4,5,6\}$, the threshold of blowup is given by the codimension-one stable manifold of the self-similar solution $f_1$.

Non-equivariant soliton resolution conjecture. The unique wave-map evolution associated with $(u_0,u_1)$ is globally regular and scatters to $(\mathcal{P}_\lambda,0)$ as…

Soliton resolution conjecture. The Cauchy problem is globally well-posed, and its solution scatters to $P_\lambda$ as $t\to\pm\infty$. The harmonic maps $P_\lambda$ minimize the en…

Bizoń et al.'s large-data blow-up conjecture. For initial data … blow up in finite time in this sense.

Consider equivariant wave-map solutions that are close to a soliton in a stronger topology, with additional regularity and decay at infinity. Type 1 solutions are finite-time blow-…

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 $\dot{\mathcal H}^1$ be the wave-map energy space, and let an energy-class solution be a restriction of a maximal Cauchy development to a subinterval. Such a solution…