Global regularity of wave maps VII. Control of delocalised or dispersed solutions

Abstract

This is the final paper in the series \cite{tao:heatwave}, \cite{tao:heatwave2}, \cite{tao:heatwave3}, \cite{tao:heatwave4} that establishes global regularity for two-dimensional wave maps into hyperbolic targets. In this paper we establish the remaining claims required for this statement, namely a divisible perturbation theory, and a means of synthesising solutions for frequency-delocalised, spatially-dispersed, or spatially-delocalised data out of solutions of strictly smaller energy. As a consequence of the perturbation theory here and the results obtained earlier in the series, we also establish spacetime bounds and scattering properties of wave maps into hyperbolic space.

arXiv comment125 pages, one figure. Some references added. Will not be published in current form, pending future reorganisation of the heatwave project

References in corpus (3)

• Well-posedness and scattering for the KP-II equation in a critical space
• Global regularity of wave maps II. Small energy in two dimensions
• Regularity of Wave-Maps in dimension 2+1

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• Global regularity of wave maps V. Large data local wellposedness and perturbation theory in the energy class
• Global Well-posedness for the Three Dimensional Simplified Inertial Ericksen-Leslie Systems Near Equilibrium
• Geometric renormalization below the ground state
• On stability of blow up solutions for the critical co-rotational Wave Maps problem
• Quantization of time-like energy for wave maps into spheres
• The hyperbolic Yang--Mills equation in the caloric gauge. Local well-posedness and control of energy dispersed solutions
• Gauge choice for the Yang-Mills equations using the Yang-Mills heat flow and local well-posedness in $H^1$
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• Asymptotic stability of harmonic maps between 2D hyperbolic spaces under the wave map equation. II. Small energy case
• Bilinear Strichartz estimates for the Schr{ö}dinger map problem
• Convergence to harmonic maps for the Landau-Lifshitz flows on two dimensional hyperbolic spaces
• Long time solutions for wave maps with large data
• The Cauchy Problem for Wave Maps on a Curved Background
• Unconditional well-posedness for wave maps