Soliton Resolution for Equivariant Wave Maps on a Wormhole

Abstract

Abstract. In this paper, we initiate the study of finite energy equivariant wave maps from the (1 +3)-dimensional spacetime R × (R × S 2 ) → S 3 where the metric on R × (R × S 2 ) is given byThe constant time slices are each given by the Riemannian manifold M := R × S 2 with metricThe Riemannian manifold M contains two asymptotically Euclidean ends at r → ±∞ that are connected by a spherical throat of area 4π 2 at r = 0. The spacetime R × M is a simple example of a wormhole geometry in general relativity. In this work we will consider 1-equivariant or corotational wave maps. Each corotational wave map can be indexed by its topological degree n. For each n, there exists a unique energy minimizing corotational harmonic map Q n : M → S 3 of degree n. In this work, we show that modulo a free radiation term, every corotational wave map of degree n converges strongly to Q n . This resolves a conjecture made by Bizon and Kahl in [3] in the corotational case.

Soliton resolution for equivariant wave maps on a wormhole: I

“…We remark that in [3] Bizon and Kahl gave numerical evidence that soliton resolution holds in the more general ℓ-equivariant setting (here corotational corresponds to ℓ = 1). In the companion work [21] we prove this and completely resolve the soliton resolution conjecture for all equivariant wave maps on a wormhole.…”

Asymptotic stability of harmonic maps between 2D hyperbolic spaces under the wave map equation. II. Small energy case

“…The motivation is the so called soliton resolution conjecture in dispersive PDEs which claims that every global bounded solution splits into the superposition of divergent solitons with a radiation part plus an asymptotically vanishing remainder term as t → ∞. The version for wave maps and hyperbolic Yang-Mills has been verified by Cote [9] and Jia, Kenig [19] for equivariant maps along a time sequence, see also [20,21] for exotic-ball wave maps and [42] for wormholes. Recently Duyckaerts, Jia, Kenig, Merle [11] obtained the universal blow up profile for type II blow up solutions to wave maps u : R × R 2 → S 2 with initial data of energy slightly above the ground state.…”

Fluid Dynamics in the Ellis Wormhole

“…We shall first note that Eq. (3.7) admits a conserved energy E. It is easy to show that the energy E is given by [17]:…”