Geometric renormalization of large energy wave maps

Abstract

There has been much progress in recent years in understanding the existence problem for wave maps with small critical Sobolev norm (in particular for two-dimensional wave maps with small energy); a key aspect in that theory has been a renormalization procedure (either a geometric Coulomb gauge, or a microlocal gauge) which converts the nonlinear term into one closer to that of a semilinear wave equation. However, both of these renormalization procedures encounter difficulty if the energy of the solution is large. In this report we present a different renormalization, based on the harmonic map heat flow, which works for large energy wave maps from two dimensions to hyperbolic spaces. We also observe an intriguing estimate of "non-concentration" type, which asserts roughly speaking that if the energy of a wave map concentrates at a point, then it becomes asymptotically self-similar.

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“…The caloric gauge was introduced by Tao [2004] in the setting of wave maps into hyperbolic space. In a series of unpublished papers [2008a; 2008b; 2008c; 2009a; 2009b], Tao used this gauge in establishing global regularity of wave maps into hyperbolic space.…”

“…We begin with some constructions that are valid for any smooth function φ : ‫ޒ‬ 2 × (−T, T ) → ‫ޓ‬ 2 . For a more general and extensive introduction to the gauge formalism we now introduce, see [Tao 2004]. Space and time derivatives of φ are denoted by ∂ α φ(x, t), where α = 1, 2, 3 ranges over the spatial variables x 1 , x 2 and time t with ∂ 3 = ∂ t .…”

“…These proofs more naturally account for magnetic nonlinearities, and we believe the technique developed here to be of independent interest and applicable to other settings. For local smoothing in the context of Schrödinger equations, see [Kenig et al 1993;2004;Ionescu and Kenig 2005;2006;2007b]. For other Strichartz and smoothing results for magnetic Schrödinger equations, see [Stefanov 2007;D'Ancona and Fanelli 2008;D'Ancona et al 2010;Erdogan et al 2008;Fanelli and Vega 2009] and the references therein.…”

“…A significant observation made in the same paper is that it is important that these spaces take into account a local smoothing effect; the authors crucially use this effect to help bring under control the worst term of the nonlinearity. Another novelty of [Bejenaru et al 2011c] is its implementation of the caloric gauge, which was first introduced by Tao [2004] and subsequently recommended by him for use in studying Schrödinger maps [Tao 2006a]. As the caloric gauge is defined using harmonic map heat flow, it can be thought of as an intrinsic and nonlinear analogue of classical Littlewood-Paley theory.…”

“…For other Strichartz and smoothing results for magnetic Schrödinger equations, see [Stefanov 2007;D'Ancona and Fanelli 2008;D'Ancona et al 2010;Erdogan et al 2008;Fanelli and Vega 2009] and the references therein. We also use in a fundamental way the subthreshold caloric gauge of [Smith 2012a], which is an extension of a construction introduced in [Tao 2004]. …”