Long-time Sobolev stability for small solutions of quasi-linear Klein-Gordon equations on the circle

Abstract

We prove that higher Sobolev norms of solutions of quasi-linear Klein-Gordon equations with small Cauchy data on S 1 \mathbb S^1 remain small over intervals of time longer than the ones given by local existence theory. This result extends previous ones obtained by several authors in the semi-linear case. The main new difficulty one has to cope with is the loss of one derivative coming from the quasi-linear character of the problem. The main tool used to overcome it is a global paradifferential calculus adapted to the Sturm-Liouville operator with periodic boundary conditions.