Small data global existence and decay for two dimensional wave maps

Abstract

We prove the small-data global existence for the wave-map equation on ${\mathbb{R}}^{1,2}$ using a variant of the vector field method. The main innovations lie in the introduction of two new linear estimates. First is the control of the dispersive decay of the solution $\varphi$ itself (as opposed to its derivatives), via a logarithmic weighted Hardy inequality. This control has not been previously established using purely physical space methods in two spatial dimensions. Second is a point-wise decay estimate for a twisted derivative of $\varphi$ associated to the Morawetz $K$ multiplier, that cannot be reduced to point-wise decay estimates associated to the standard commutator vector fields. As the linear theory is largely similar between dimensions, and in view of the novelty of the second innovation even in higher dimensions, we include a discussion of the method for ${\mathbb{R}}^{1,d}$ in general. Both linear estimates are used crucially in our study: the control of the wave-map equation in the small data regime necessarily requires understanding the dispersive behavior of the bare solution $\varphi$ itself, by virtue of the equation it satisfies. The point-wise decay for the twisted derivative allows us to avoid certain top-order logarithmic energy growths; this is indispensable for extending our argument from the case of compactly supported initial data (to which our methods are most naturally adapted) to initial data that are strongly localized but not necessarily of compact support, via an iterative construction.

v2: edits to address referee comments. Significant changes: estimate of twisted derivative in (4.9), which allows removal of log energy growth in Sect 6. This enables Sects 7+8 (new) proving (a) nonlinear stability for generic, large, dispersive solutions and (b) extension to strongly localized (but not necessarily compactly supported) initial data

References in corpus (1)

• The Euclidian-hyperboidal foliation method and the nonlinear stability of Minkowski spacetime

Cited by in corpus (8)

• Global solutions of nonlinear wave-Klein-Gordon system in two spatial dimensions: weak coupling case
• Global nearly-plane-symmetric solutions to the membrane equation
• Global solution for massive Maxwell-Klein-Gordon equations
• Global stability of some totally geodesic wave maps
• Two dimensional wave--Klein-Gordon equations with semilinear nonlinearities
• Stability of a class of semilinear waves in $2+1$ dimension under null condition
• Global versions of Gagliardo-Nirenberg-Sobolev inequality and applications to wave and Klein-Gordon equations
• Dispersive estimates for Klein-Gordon equations via a physical space approach