Global regularity of wave maps IV. Absence of stationary or self-similar solutions in the energy class

Abstract

Using the harmonic map heat flow, we construct an energy class for wave maps $\varphi$ from two-dimensional Minkowski space ${\mathbb{R}}^{1+2}$ to hyperbolic spaces $\H^m$, and then show (conditionally on a large data well-posedness claim for such wave maps) that no stationary, travelling, self-similar, or degenerate wave maps exist in this energy class. These results form three of the five claims required in our earlier paper (arXiv:0805.4666) to prove global regularity for such wave maps. (The conditional claim of large data well-posedness is one of the remaining claims required in that paper.)

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 Will not be published in current form, pending future reorganisation of the heatwave project

References in corpus (3)

• Global regularity of wave maps VII. Control of delocalised or dispersed solutions
• Global regularity of wave maps VI. Abstract theory of minimal-energy blowup solutions
• Global behaviour of nonlinear dispersive and wave equations

Cited by in corpus (9)

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• 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
• Universality of blow up profile for small blow up solutions to the energy critical wave map equation
• Nondispersive radial solutions to energy supercritical non-linear wave equations, with applications
• Bilinear Strichartz estimates for the Schr{ö}dinger map problem
• Long time solutions for wave maps with large data
• Continuous in time bubble decomposition for the harmonic map heat flow
• Strichartz type estimates for fractional heat equations