BLOW-UP OF THE NONEQUIVARIANT ()-DIMENSIONAL WAVE MAP

On the existence of smooth self-similar blow-up profiles for the wave-map equation

Consider the equivariant wave map equation from Minkowski space to a rotationnally symmetric manifold N which has an equator (example: the sphere). In dimension 3, this article presents a necessary and sufficient condition on N for the existence of a smooth self-similar blow up profile. More generally, we study the relation between the minimizing properties of the equator map for the Dirichlet ...

Smooth type II blow up solutions to the four dimensional energy critical wave equation

We exhibit C∞ type II blow up solutions to the focusing energy critical wave equation in dimension N = 4. These solutions admit near blow up time a decomposiiton u(t, x) = 1 λ N−2 2 (t) (Q+ ε(t))( x λ(t) ) with ‖ε(t), ∂tε(t)‖Ḣ1×L2 ≪ 1 where Q is the extremizing profile of the Sobolev embedding Ḣ → L∗ , and a blow up speed λ(t) = (T − t)e− √ |log(T−t)|(1+o(1)) as t → T.

Capacity and blow - up for the 3 + 1 dimensional wave operator

Blow up for the Semilinear Wave Equation in Schwarzschild Metric

We study the semilinear wave equation in Schwarzschild metric (3 + 1 dimensional space time). First, we establish that the problem is locally well posed in H for any σ > 1; then we prove the blow up of the solution for every p > 1 and non negative initial data. The work is dedicated to prof. Yvonne Choquet Bruhat in occasion of her 80th year.

On the blow-up of four-dimensional Ricci flow singularities

In this paper we prove a conjecture by Feldman–Ilmanen–Knopf (2003) that the gradient shrinking soliton metric they constructed on the tautological line bundle over CP is the uniform limit of blow-ups of a type I Ricci flow singularity on a closed manifold. We use this result to show that limits of blow-ups of Ricci flow singularities on closed four-dimensional manifolds do not necessarily have...