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

Jean-Marc Delort (1)

1 LAGA - Laboratoire Analyse, Géométrie et Applications

We prove that higher Sobolev norms of solutions of quasi-linear Klein-Gordon equations with small Cauchy data on $\mathbb{S}^1$ 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.

Mots clés: Quasi-linear Klein-Gordon equation; Long-time stability; Paradifferential calculus

Domaines: Equations aux dérivées partielles [math.AP]

Jean-Marc Delort. Long-time Sobolev stability for small solutions of quasi-linear Klein-Gordon equations on the circle. 2008. ⟨hal-00157765v3⟩