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

Abstract

In this paper, we prove that the small energy harmonic maps from H 2 to H 2 are asymptotically stable under the wave map equation in the subcritical perturbation class. This result may be seen as an example supporting the soliton resolution conjecture for geometric wave equations without equivariant assumptions on the initial data. In this paper, we construct Tao's caloric gauge in the case when nontrivial harmonic map occurs. With the "dynamic separation" the master equation of the heat tension field appears as a semilinear magnetic wave equation. By the endpoint and weighted Strichartz estimates for magnetic wave equations obtained by the first author [38], the asymptotic stability follows by a bootstrap argument.

Asymptotic Stability of Harmonic Maps on the Hyperbolic Plane Under the Schrödinger Maps Evolution

“…Finally, we mention again the work of Li-Ma-Zhao [46] and Li [42,43] in the closely related subject of perturbations of the wave equation on hyperbolic space. In their proof of stability of harmonic maps from H 2 into H 2 under the wave maps evolution, they established a global-in-time local energy decay estimate (yet with exponentially decaying weights) and L 2 t L p x -Strichartz estimates for the first and zeroth order perturbations of the wave equation arising from linearization around such harmonic maps.…”

“…The present work may be thought of as the Schrödinger maps analogue of the work [40], which concerns the asymptotic stability of finite energy equivariant harmonic maps from H 2 to N = S 2 or H 2 under the equivariant wave maps evolution, and of the works [42,43,46], which are on the same problem without any symmetry assumptions when N = H 2 . However, analysis of the Schrödinger maps equation without symmetry around a nonconstant harmonic map brings on new challenges in comparison with the previous A. Lawrie was supported by NSF grant DMS-1700127 and a Sloan Research Fellowship.…”

“…A related caloric gauge construction based on the Yang-Mills heat flow is a key ingredient of the recent proof of the threshold conjecture for the energy-critical hyperbolic Yang-Mills equation [53][54][55][56][57][58]. Finally, the caloric gauge played a fundamental role in the proof of asymptotic stability (in a subcritical Sobolev space) of nonconstant harmonic maps Q : H 2 → H 2 under the wave maps evolution in [42,43,46] and for the Landau-Lifshitz flow (but not the limiting case of Schrödinger maps) in [47].…”

Local smoothing estimates for Schrödinger equations on hyperbolic space

“…Finally, we mention an interesting series of work by Li-Ma-Zhao [32] and Li [30,31] in the closely related subject of perturbations of the wave equation on hyperbolic space. More precisely, in the context of their proof of stability of harmonic maps from H 2 into H 2 under the wave map evolution, they established a global-in-time local energy decay estimate (yet with exponentially decaying weights) and L 2 t L p x -Strichartz estimates for the first and zeroth order perturbations of the wave equation arising from linearization around such harmonic maps.…”

“…For the opposite inequality, note that for each ǫ > 0 we can find a function α ǫ : (0, 4] → A so that (32) , and hence…”