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A Nash-Moser-Hörmander implicit function theorem with applications to control and Cauchy problems for PDEs

Emanuele Haus
Tuesday, February 21, 2017 - 11:00

We prove an abstract Nash-Moser implicit function theorem which, when applied to control and Cauchy problems for PDEs in Sobolev class, is sharp in terms of the loss of regularity of the solution of the problem with respect to the data. The proof is a combination of: (i) the iteration scheme by Hörmander (ARMA 1976), based on telescoping series, and very close to the original one by Nash; (ii) a suitable way of splitting series in scales of Banach spaces, inspired by a simple, clever trick used in paradifferential calculus (for example, by Métivier). As an example of application, we apply our theorem to a control and a Cauchy problem for quasi-linear perturbations of KdV equations, improving the regularity of a previous result. The theorem has a fruitful application also in solving a quasi-periodic version of a transport equation, which is the first step towards the reduction to constant coefficients of the linearized operator associated to one-dimensional gravity water waves. This is a joint work with Pietro Baldi.

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