Calculus of Variations and Geometric Measure Theory

E. Chiodaroli - J. Krieger

A class of large global solutions for the Wave--Map equation

created by chiodaroli on 20 Apr 2015
modified on 12 Aug 2015


Accepted Paper

Inserted: 20 apr 2015
Last Updated: 12 aug 2015

Journal: Transactions of the AMS
Year: 2015


In this paper we consider the equation for equivariant wave maps from $R^{3+1}$ to $S^3$ and we prove global in forward time existence of certain $C^\infty$-smooth solutions which have infinite critical Sobolev norm $\dot{H}^{\frac{3}{2}}(R^3)\times \dot{H}^{\frac{1}{2}}(R^3)$. Our construction provides solutions which can moreover satisfy the additional size condition $\
u(0, \cdot)\
\geq 1)}>M$ for arbitrarily chosen $M>0$. These solutions are also stable under suitable perturbations. Our method is based on a perturbative approach around suitably constructed approximate self--similar solutions.