A surjection theorem for singular perturbations with loss of derivatives

Ivar Ekeland (1) , Eric Séré (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

In this paper we introduce a new algorithm for solving nonlinear functional equations which admit a right-invertible linearization, but such that the inverse loses derivatives. The main difference with the by now classical Nash-Moser algorithm is that, instead of using a regularized Newton scheme, we solve a sequence of Galerkin problems thanks to a topological argument. As a consequence, in our estimates there are no quadratic terms. We apply our method to a singular perturbation problem with loss of derivatives studied by Texier-Zumbrun. We will compare the two results and we will show that ours improves significantly on theirs, when applied, in particular, to a nonlinear Schrödinger Cauchy problem with highly oscillatory initial data.

Domaines: Equations aux dérivées partielles [math.AP]

Ivar Ekeland, Eric Séré. A surjection theorem for singular perturbations with loss of derivatives. 2018. ⟨hal-01924328v1⟩