Global regularity of wave maps VI. Abstract theory of minimal-energy blowup solutions

Abstract

In the previous papers in this series, the global regularity conjecture for wave maps from two-dimensional Minkowski space ${\mathbb{R}}^{1+2}$ to hyperbolic space $\H^m$ was reduced to the problem of constructing a minimal-energy blowup solution which is almost periodic modulo symmetries in the event that the conjecture fails. In this paper, we show that this problem can be reduced further, to that of showing that solutions at the critical energy which are either frequency-delocalised, spatially-dispersed, or spatially-delocalised have bounded ``entropy''. These latter facts will be demonstrated in the final paper in this series.

arXiv comment36 pages, no figures. Will not be published in current form, pending future reorganisation of the heatwave project

References in corpus (4)

• Global regularity of wave maps II. Small energy in two dimensions
• Energy dispersed large data wave maps in 2+1 dimensions
• Global regularity of wave maps IV. Absence of stationary or self-similar solutions in the energy class
• Global regularity of wave maps VII. Control of delocalised or dispersed solutions

Cited by in corpus (17)

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• Energy dispersed large data wave maps in 2+1 dimensions
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• Global regularity of wave maps IV. Absence of stationary or self-similar solutions in the energy class
• Global regularity of wave maps III. Large energy from ${\mathbb{R}}^{1+2}$ to hyperbolic spaces
• Global regularity of wave maps V. Large data local wellposedness and perturbation theory in the energy class
• On stability of blow up solutions for the critical co-rotational Wave Maps problem
• Geometric renormalization below the ground state
• Quantization of time-like energy for wave maps into spheres
• An inverse theorem for the bilinear $L^2$ Strichartz estimate for the wave equation
• 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 $H^1$
• Asymptotic stability of harmonic maps between 2D hyperbolic spaces under the wave map equation. II. Small energy case
• Convergence to harmonic maps for the Landau-Lifshitz flows on two dimensional hyperbolic spaces
• Long time solutions for wave maps with large data
• Bilinear Strichartz estimates for the Schr{ö}dinger map problem
• The Cauchy Problem for Wave Maps on a Curved Background