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

TL;DR

This paper constructs and analyzes a class of smooth, globally existing solutions for the equivariant wave maps equation from 4D spacetime to 3-sphere, demonstrating their stability and infinite Sobolev norm.

Contribution

It introduces a perturbative method around approximate self-similar solutions to establish the existence and stability of large, smooth solutions with infinite critical Sobolev norm.

Findings

Existence of smooth solutions with infinite Sobolev norm

Solutions can have arbitrarily large initial amplitude outside the unit ball

Solutions are stable under certain perturbations

Abstract

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 $\|u(0, \cdot)\|_{L^\infty(|x|\geq 1)}>M$ 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.