On stable self-similar blow up for equivariant wave maps

TL;DR

This paper proves the stability of a known self-similar blow-up solution for equivariant wave maps from 3+1 Minkowski space into the three-sphere, linking the problem to a spectral analysis of a linear ODE.

Contribution

It reduces the problem of stable blow-up to a spectral problem for a linear ODE and discusses the numerical evidence for mode stability of the self-similar solution.

Findings

Blow-up stability is linked to spectral properties of a linear operator.

Numerical evidence suggests no unstable modes for the self-similar solution.

The stability condition reduces to verifying the absence of eigenvalues in a certain region.

Abstract

We consider co--rotational wave maps from (3+1) Minkowski space into the three--sphere. This is an energy supercritical model which is known to exhibit finite time blow up via self-similar solutions. The ground state self--similar solution $f_0$ is known in closed form and based on numerics, it is supposed to describe the generic blow up behavior of the system. We prove that the blow up via $f_0$ is stable under the assumption that $f_0$ does not have unstable modes. This condition is equivalent to a spectral assumption for a linear second order ordinary differential operator. In other words, we reduce the problem of stable blow up to a linear ODE spectral problem. Although we are unable, at the moment, to verify the mode stability of $f_0$ rigorously, it is known that possible unstable eigenvalues are confined to a certain compact region in the complex plane. As a consequence, highly reliable numerical techniques can be applied and all available results strongly suggest the nonexistence of unstable modes, i.e., the assumed mode stability of $f_0$.