Multilinear Fourier Analysis and Nonlinear PDEs – AFoMEN

This proposal aims to establish connections and collaborations between several people coming from different mathematical communities in order to pursue the growth of the understanding of the multilinear (or nonlinear) Fourier analysis and the applications for some PDEs. More precisely, we are specially interested in boundedness of several multilinear quantities (operators, oscillatory integrals ...), requiring an accurate time-frequency analysis or to study how standard inequalities are modified by a composition with a measure-preserving map. All these problems are motivated by applications for global-time existence problems for nonlinear PDEs. This team is constituted by people with experience in real Harmonic Analysis and other ones with a (linear and nonlinear) PDEs background. By this way, we pay attention to focus on problems appearing at the interface of the multilinear Fourier analysis and the theoretical study of PDEs (scattering problems, global time well-posedness results, ...). These two topics are fashionable and have given rise to numerous works.
In a one hand, the multilinear time-frequency analysis has attracted a great deal of attention since the breakthrough of Lacey and Thiele who answered to the famous Calderon's conjecture concerning bilinear Hilbert transforms and the development of the multilinear Calderon-Zygmund theory. In another hand, theoretical study of PDEs (scattering and global existence) is a topic on which many people are working since there is not general theory.
Our proposal fits into these two topics and aims to make a link between between them : use the multilinear (or nonlinear) analysis to solve difficult problems in the study of PDEs due to the presence of nonlinear terms. We also hope to bring new insights and new approaches in both these two mathematical trends by developing interactions between them. Indeed, the current proposal will be split into several tasks. One of them are focused on problems coming from the the bilinear time-frequency analysis. A second will be devoted to the study of multilinear integrals appearing in the Space-time resonances method in order to get a general approach for studying PDEs with a nonlinearity. The last task concerns specific nonlinear problems coming from fluids mechanic, which have to be studied by an accurate analysis.