Title: Nonlinear interpolation and the flow of quasilinear equations
Abstract: I will present an abstract result showing that, for a quasilinear evolution problem, the continuity of the data-to-solution map follows automatically from the estimates that are usually established in the proof of existence of solutions. This result is in fact a consequence of an interpolation theorem for nonlinear functionals defined on scales of Banach spaces generalizing Besov spaces. Our analysis is independent of any prior results on interpolation theory. It relies solely on the concepts of dyadic decompositions and Friedrichs mollifiers, viewed through the formalism introduced by Hamilton for the study of the Nash-Moser scheme, combined with the frequency envelopes introduced by Tao. This is joint work with N. Burq, M. Ifrim, D. Tataru, and C. Zuily.