Stable self-similar blowup in energy supercritical Yang-Mills theory

We consider the Cauchy problem for an energy supercritical nonlinear wave equation that arises in $(1+5)$--dimensional Yang--Mills theory. A certain self--similar solution $W_0$ of this model is conjectured to act as an attractor for generic large data evolutions. Assuming mode stability of $W_0$, we prove a weak version of this conjecture, namely that the self--similar solution $W_0$ is (nonlinearly) stable. Phrased differently, we prove that mode stability of $W_0$ implies its nonlinear stability. The fact that this statement is not vacuous follows from careful numerical work by Bizo\'n and Chmaj that verifies the mode stability of $W_0$ beyond reasonable doubt

A vector field method on the distorted Fourier side and decay for wave equations with potentials

We study the Cauchy problem for the one-dimensional wave equation with an inverse square potential. We derive dispersive estimates, energy estimates, and estimates involving the scaling vector field, where the latter are obtained by employing a vector field method on the "distorted" Fourier side. In addition, we prove local energy decay estimates. Our results have immediate applications in the context of geometric evolution problems. The theory developed in this paper is fundamental for the proof of the co-dimension 1 stability of the catenoid under the vanishing mean curvature flow in Minkowski space.Comment: 74 pages; minor adjustments to match the published version, will appear in Memoirs of the AM

Decay estimates for the one-dimensional wave equation with an inverse power potential

We study the wave equation on the real line with a potential that falls off like $|x|^{-\alpha}$ for $|x| \to \infty$ where $2 < \alpha \leq 4$. We prove that the solution decays pointwise like $t^{-\alpha}$ as $t \to \infty$ provided that there are no resonances at zero energy and no bound states. As an application we consider the $\ell=0$ Price Law for Schwarzschild black holes. This paper is part of our investigations into decay of linear waves on a Schwarzschild background.Comment: 14 pages, added some details in order to match the published versio

A spectral mapping theorem for perturbed Ornstein-Uhlenbeck operators on L^2(R^d)

We consider Ornstein-Uhlenbeck operators perturbed by a radial potential. Under weak assumptions we prove a spectral mapping theorem for the generated semigroup. The proof relies on a perturbative construction of the resolvent, based on angular separation, and the Gearhart-Pr\"u{\ss} Theorem.Comment: 43 pages, improved presentation, suggestions by referees incorporated, will appear in Journal of Functional Analysi