53 problems

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…

Bizoń et al.'s blow-up profile conjecture. The rescaled solution converges to the static profile:

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 $(M,g)$ 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…

Stable-manifold threshold conjecture. The codimension-one stable manifold of the solution $f_1$ plays the role of the threshold of singularity formation for a large set of initial…

Let $epsilon$ parameterize the initial data, and let $epsilon^{*}$ be a non-zero value of this parameter. A solution is said to turn around when it changes from initially outgoing,…

Let $\epsilon$ parameterize the initial data, let $\chi_s(r)$ denote the static solution, and let $a$ be an $\epsilon$-dependent parameter. Static-solution threshold conjecture. Fo…

The solution $f_0$ denotes the fundamental self-similar wave map in the sequence of connecting-orbit solutions constructed above. Asymptotic stability conjecture. The solution…

Let $f_0$ be the closed-form self-similar solution for co-rotational wave maps from Minkowski space to the three-sphere, whose existence was established for the blow-up profile und…

Let $\alpha$ be the dimensionless coupling constant, and let $\alpha_0<\alpha_1$ denote the parameter values associated with the CSS solutions. Let the $n=1$ solution be the contin…

Ionescu's Kerr wave-map stability conjecture. For any subextremal Kerr metric with $|a|<M$, this stationary solution is future asymptotically stable in the domain of outer communic…

Let $(M,g)$ be a compact Riemannian manifold, and consider the wave maps equation from the circle to $M$. Two states are homotopic when they lie in the same homotopy class as maps…

Let $\kappa$ be the dimensionless curvature parameter of the constant-curvature target manifold, and let $\Delta$ denote the echoing period of a discretely self-similar (DSS) solut…

Let $J$ be a positive integer, and consider the stable manifolds associated with $2J$-chains and $(2J+1)$-chains of alternating kinks and anti-kinks. The stable-manifold codimensio…

Equivariant wave maps from the $2+1$-dimensional wormhole to the $2$-sphere admit explicit harmonic map solutions which, in suitable coordinates, are sine-Gordon kinks and anti-kin…

Consider wave maps in dimensions $n=3,4,5,6$ and the countable family of self-similar solutions described in the source, ordered so that the ground state is followed by the excited…

Let corotational wave maps and equivariant Yang–Mills connections be considered in the settings described above, with the explicit self-similar blowup solutions and profiles introd…

Let $(\mathcal{M},g)$ be a Riemannian manifold, and let $\phi:\mathbb{R}\times\mathbb{S}^1\to\mathcal{M}$ solve the controlled wave maps equation. For initial and final states…

Let $(\mathcal{M},g)$ be a general Riemannian manifold target for the controlled wave maps equation. The controllable space (or accessible space) is the set of target states reacha…

For equivariant wave maps into the sphere in spatial dimension $d\geq 7$, let a corotational self-similar profile mean a solution of the self-similar wave-map equation in the corot…

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 $d=2$, and consider the energy-critical wave maps equation with initial data whose energy is below the energy of any non-trivial finite-energy harmonic map into the target mani…

The soliton resolution conjecture concerns solutions of a broad class of nonlinear dispersive equations. In the setting of (quasi-)equivariant wave maps, let…

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…

Let $\{Q_{\nu}\}$ be the 1-equivariant harmonic map class parameterized by $\nu\in(0,1)$ from $\Bbb H^2$ to $\Bbb H^2$. Let $(u_0,u_1)$ be finite-energy initial data for the wave-m…