Research activity and projects

My main research interests lie in the field of Partial Differential Equations, especially evolution nonlinear equations of parabolic type. The models I have been studying mainly arise from Fluid Mechanics and Geophysics, or from Mathematical Biology and Probability theory. Much of my work is devoted to a better understanding of the asymptotic properties of the solutions: the study of their behavior for large time and in the far-field under different perspectives (energy dissipation, time decay or growth, asymptotic profiles, selfsimilarity, spatial spreading, localization issues. . . ), as well as the asymptotic analysis when a physical parameter inside the models goes to zero, represent my most important contribution. It should be stressed that the Navier–Stokes equations constitute an outstanding challenge for all these issues. Not surprisingly, these equations take an important place in my work. A couple of my papers [12, 13], deal with functional analysis (in particular, with the theory of multipliers in Sobolev spaces) or harmonic analysis and approximation theory, in connection with wavelet bases. These two contributions to Analysis are quite independent from the rest of my scientific production, though they are not completely unrelated. Indeed harmonic analysis is present throughout my work: the classical Fourier analysis (for example, the Fourier splitting method) often yields to sharp results on the long time behavior. The Littlewood-Paley analysis is a powerful tool for obtaining existence results. Moreover, the use of a large panel of function spaces (some of them considered as somewhat exhotic in the PDE community until recently) provides valuable information on the solutions. For example, Besov spaces with negative regularity can be used to measure their oscillating character, weighed spaces tell us about their localization; Morrey and Lorentz spaces are a good setting where to study selfsimilarity, etc. In addition, many physical models have a non-local nature: a perturbation of a quantity in some region of the space affects the whole system even far from that region. We usually attack these models by recasting the initial PDE as a pseudodifferential equation. The theory of singular integrals, well-suited for the study of non-local phenomena, will also play an important role. Below, I will describe a selection of my results.