Energy dispersed large data wave maps in 2+1 dimensions

Abstract

In this article we consider large data Wave-Maps from ${\mathbb{R}}^{2+1}$ into a compact Riemannian manifold $(\mathcal{M},g)$, and we prove that regularity and dispersive bounds persist as long as a certain type of bulk (non-dispersive) concentration is absent. In a companion article we use these results in order to establish a full regularity theory for large data Wave-Maps.

89 pages

References in corpus (8)

• Regularity of Wave-Maps in dimension 2+1
• 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 VI. Abstract theory of minimal-energy blowup solutions
• Global regularity of wave maps V. Large data local wellposedness and perturbation theory in the energy class
• An inverse theorem for the bilinear $L^2$ Strichartz estimate for the wave equation
• On the Formation of Singularities in the Critical O(3) Sigma-Model
• Geometric renormalization of large energy wave maps

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• Equivariant Schrödinger Maps in two spatial dimensions
• Global well-posedness and scattering of the (4+1)-dimensional Maxwell-Klein-Gordon equation
• Soliton resolution for critical co-rotational wave maps and radial cubic wave equation
• Global regularity of wave maps VI. Abstract theory of minimal-energy blowup solutions
• Transference of Bilinear Restriction Estimates to Quadratic Variation Norms and the Dirac-Klein-Gordon System
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• Global well-posedness for the Maxwell-Klein Gordon equation in 4+1 dimensions. Small energy
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• Quantization of time-like energy for wave maps into spheres
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• Global well-posedness for the massless cubic Dirac equation
• An asymptotic expansion of two-bubble wave maps
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• Conditional large data scattering results for the Dirac-Klein-Gordon system
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• Global solutions of nonlinear wave equations with large energy
• Global regularity of critical Schrödinger maps: subthreshold dispersed energy
• Global wellposedness of hedgehog solutions for the (3+1) Skyrme model
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• Uniqueness of two-bubble wave maps
• Local well-posedness of the (4+1)-dimensional Maxwell-Klein-Gordon equation at energy regularity
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• The almost optimal multilinear restriction estimate for hypersurfaces with curvature: the case of $n-1$ hypersurfaces in R^n
• Optimal blowup stability for three-dimensional wave maps
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• Sharp universal rate for stable blow-up of corotational wave maps
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• The Cauchy Problem for Wave Maps on a Curved Background
• Non-Uniqueness of Bubbling for Wave Maps
• Scattering for the Equivariant U(1) Problem
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• Global results for a Cauchy problem related to biharmonic wave maps
• The time-like minimal surface equation in Minkowski space: low regularity solutions