Joachim Krieger

Renormalization and blow up for charge one equivariant critical wave maps

We prove the existence of equivariant finite time blow-up solutions for the wave map problem from ℝ2+1→S2 of the form

On the focusing critical semi-linear wave equation

u(t,r)=Q(\lambda(t)r)+\mathcal{R}(t,r)

Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension

where u is the polar angle on the sphere,

Global dynamics away from the ground state for the energy-critical nonlinear wave equation

Q(r)=2\arctan r

Nonscattering solutions and blowup at infinity for the critical wave equation

is the ground state harmonic map, λ(t)=t-1-ν, and

Non-generic blow-up solutions for the critical focusing NLS in 1-D

\mathcal{R}(t,r)

Global Well-Posedness For The Maxwell-Klein-Gordon Equation In 4+1 Dimensions: Small Energy

is a radiative error with local energy going to zero as t→0. The number

ON STABILITY OF PSEUDO-CONFORMAL BLOWUP FOR L 2 -CRITICAL HARTREE NLS

\nu>\frac{1}{2}

A non-local inequality and global existence

can be prescribed arbitrarily. This is accomplished by first “renormalizing” the blow-up profile, followed by a perturbative analysis.

On Stability of Pseudo-Conformal Blowup for L2-critical Hartree NLS

The wave equation ∂ttψ − Δψ − ψ5 = 0 in ℝ3 is known to exhibit finite time blowup for data of negative energy. Furthermore, it admits the special static solutions φ(x, a) = (3a) ¼ (1 + a|x|2)−½ for all a > 0 which are linearly unstable. We view these functions as a curve in the energy space ˙H1 ×L2. We prove the existence of a family of perturbations of this curve that lead to global solutions possessing a well-defined long time asymptotic behavior as the sum of a bulk term plus a scattering term. Moreover, this family forms a co-dimension one manifold M of small diameter in a suitable topology. Loosely speaking, M acts as a center-stable manifold with the curve Φ(·, a) as an attractor in M.