A construction of blow up solutions for co-rotational wave maps

TL;DR

This paper constructs finite-time blow-up solutions for co-rotational wave maps from 2+1 dimensions into surfaces of revolution, demonstrating specific solution forms and energy behaviors near blow-up time.

Contribution

It provides a novel construction of blow-up solutions for co-rotational wave maps with detailed solution structure and energy decay properties.

Findings

Existence of finite-time blow-up solutions proven.

Solutions of specific form with energy localized near blow-up.

Parameter  influences blow-up rate and solution behavior.

Abstract

The existence of co-rotational finite time blow up solutions to the wave map problem from R^{2+1} into N, where N is a surface of revolution with metric d\rho^2+g(\rho)^2 d\theta^2, g an entire function, is proven. These are of the form u(t,r)=Q(\lambda(t)t)+R(t,r), where Q is a time independent solution of the co-rotational wave map equation -u_{tt}+u_{rr}+r^{-1}u_r=r^{-2}g(u)g'(u), \lambda(t)=t^{-1-\nu}, \nu>1/2 is arbitrary, and R is a term whose local energy goes to zero as t goes to 0.