On Stable Self-Similar Blow up for Equivariant Wave Maps: The Linearized Problem

Abstract

Abstract. We consider co-rotational wave maps from (3 + 1) Minkowski space into the threesphere. 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 f0 is known in closed form and based on numerics, it is supposed to describe the generic blow up behavior of the system. In this paper we develop a rigorous linear perturbation theory around f0. This is an indispensable prerequisite for the study of nonlinear stability of the self-similar blow up which is conducted in the companion paper [11]. In particular, we prove that f0 is linearly stable if it is mode stable. Furthermore, concerning the mode stability problem, we prove new results that exclude the existence of unstable eigenvalues with large imaginary parts and also, with real parts larger than 1 2 . The remaining compact region is well-studied numerically and all available results strongly suggest the nonexistence of unstable modes.

On stable self-similar blowup for equivariant wave maps

“…Furthermore, in [10] it is rigorously proved that λ = 1 is the only eigenvalue with Reλ ≥ 1 and in [9] we show that there do not exist real unstable eigenvalues (except λ = 1). Finally, in [12] it is shown that λ = 1 is the only eigenvalue with real part greater than 1 2 . All these results leave no doubt that ψ T is indeed mode stable although a completely rigorous proof of this property is still not available.…”

“…If the latter is true, then how does the breakdown (blow up) occur? There exists a heuristic principle which gives a hint for scaling invariant equations that possess a positive energy (such as the wave maps equation, see [12]). If the scaling behavior is such that shrinking of the solution is energetically favorable, then one expects finite time blow up.…”

“…This model can be described by the single semilinear wave equation (1.2) ψ tt − ψ rr − 2 r ψ r + sin(2ψ) r 2 = 0 where r ≥ 0 is the standard radial coordinate on Minkowski space. We refer to [12] and references therein for more details on the underlying symmetry reduction which is a special case of equivariance. Global well-posedness of the Cauchy problem for Eq.…”

“…Our result is conditional in the sense that it depends on a certain spectral property for a linear ordinary differential equation which cannot be verified rigorously so far. However, very reliable numerics and partial theoretical results leave no doubt that it is satisfied, see [12] for a thorough discussion of this issue. In other words, our result reduces the question of (nonlinear) stability of ψ T to a linear ODE spectral problem.…”

“…(1.9) has a solution u λ ∈ C ∞ [0, 1]. It is a consequence of [12] that only smooth solutions are relevant here. Furthermore, the eigenvalue is said to be unstable if Reλ ≥ 0 and stable if Reλ < 0.…”