Slow Blow up solutions for certain critical wave equations

We describe in this article two recent results [11], [12], obtained by the author jointly with W. Schlag and D. Tataru, about singular solutions for the critical wave maps equation, as well as the critical focussing semilinear wave equation. Specifically, the first result [11] establishes for the first time the conjectured formation of singularities for co-rotational wave maps into the sphere $S^2$ in $n = 2$ spatial dimensions, while the 2nd result [12] establishes the existence of one-point blow-up solutions for the energy critical semilinear equation $u = −u^5$ in $n = 3$ spatial dimensions. Singularity formation for wave maps was previously only known in dimensions $n\ge3$, while for the semilinear equation above, all previously known blow up solutions become singular along a hyper surface. Furthermore, both results show that there exists a continuum of blowup rates for both problems. This is a new observation in the context of either equation, having previously been observed for a wave maps type equation in $n = 1$ spatial dimension in [19].