Smooth self-similar blow-up profiles for the wave map equation

2008/06/25 by Pierre Germain, Germain, Pierre
Mathematics · Physics and Astronomy · #35L70 #35Q80 #53C44 #58E20 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.0806.4148

openalex publication_date 2008/06/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider the equivariant wave map equation from Minkowski space to a rotationnally symmetric manifold which has an equator (example: the sphere). In dimension 3, this article gives a necessary and sufficient condition for the existence of a smooth self-similar blow up profile. More generally, we study the relation between 1. the minimizing properties of the equator map for the (elliptic) Dirichlet energy and 2. the existence of a smooth blow-up profile for the (hyperbolic) wave map problem. Several applications of this approach are described.

Citations

• Weak solutions and development of singularities of the SU(2) σ‐model
• Strichartz estimates for the Wave and Schrodinger Equations with the Inverse-Square Potential

Related

• Stable blowup for supercritical wave maps into perturbed spheres
• Globally stable blowup profile for supercritical wave maps in all dimensions
• Existence and stability of discretely self-similar blowup for a wave maps type equation
• Global regularity of wave maps IV. Absence of stationary or self-similar solutions in the energy class
• Hyperboloidal similarity coordinates and a globally stable blowup\n profile for supercritical wave maps