On stable self-similar blow up for equivariant wave maps

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 $f_0$.

arXiv commentSome minor changes according to the referee's suggestions, to appear in Comm. Pure Appl. Math

References in corpus (14)

• Global regularity of wave maps II. Small energy in two dimensions
• Regularity of Wave-Maps in dimension 2+1
• Energy dispersed large data wave maps in 2+1 dimensions
• Equivariant self-similar wave maps from Minkowski spacetime into 3-sphere
• Local and global well-posedness of wave maps on ${\mathbb{R}}^{1+1}$ for rough data
• From resolvent bounds to semigroup bounds
• The cosmological gravitating $\sigma$ model: solitons and black holes
• On the Formation of Singularities in the Critical O(3) Sigma-Model
• Late-time tails of wave maps coupled to gravity
• SU(2) Cosmological Solitons
• A note on the eigenvalues for equivariant maps of the SU(2) sigma-model
• The Radial Wave Operator in Similarity Coordinates
• Spectral Properties and Linear Stability of Self-Similar Wave Maps
• Stable blow up dynamics for the critical co-rotational Wave Maps and equivariant Yang-Mills problems