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Semi-Classical Analysis and Partial Differential Equations

Final Report Summary - SCAPDE (Semi-Classical Analysis and Partial Differential Equations)

The project was focused on developing sharp mathematical tools related to wave propagation and applied them successfully to a wide range of issues: wave dispersion in heterogeneous media, control theory, probability theory.

the unifying theme underlying our work is a systematic use of microlocal analysis, which may be seen as a sophisticated version of Fourier analysis. Standard objects like functions, solutions to partial differential equations may be decomposed both in suitable regions of space(-time) and their frequency counterpart. One then infers global properties from properties of these simpler buildings blocks that are localised and more amenable to classical tools. In the particular case of waves or related equations, the sophisticated analysis of an highly oscillatory solution may often be reduced to a suitable version of classical hamiltonian dynamics in phase space (the reunion of physical space and frequency space).

Wave propagation in complex media, in the presence of interfaces, boundaries or obstacles is a notoriously difficult subject. One of the great successes of microlocal analysis establishes so called propagation of singularities results, that predict how energy is transported by waves. However, predicting how waves decay (the so-called dispersion) turns out to be much more difficult and requires to analyse their amplitude anywhere in space-time. During the project we obtained the first sharp results for waves in two opposite situations: inside a strictly convex domain, where waves can glide along the boundary (as observed by Lord Rayleigh in the St-Paul cathedral dome, or by the builders of the temple of Heaven in Beijing much earlier), and outside a strictly convex obstacle, where diffraction behind the obstacle can lead to refocalisation (as observed by Arago and predicted by Fresnel). Our results are expected to generalise beyond these extreme cases and pave the way for refinements of the previously known propagation or singularities results, in turn potentially leading to improvements in control theory of partial differential equations.

Control theory is indeed another area where microlocal analysis has proved to be successful and the project made several intestins contributions to this area. We single out two: by refining the analysis of a degenerate parabolic operator, one exibits situations were controllability does not and cannot hold ; a new result on propagation of singularities for the Stokes operator (of crucial importance in fluid mechanics) was obtained.

The third and final topic of interest to the project was probability theory and its interplay with partial differential equations and related tools. The analysis of Markov processes, random walks or diffusion processes is known to be strongly related to partial differential equations, and it turns out that microlocal methods developed for these are very powerful to deal with certain probabilistic questions.
random walks and diffusion processes in different environnements which were previously out of reach by known methods. The project investigated further some of these questions, most notably the analysis of Markov processes and random walks in increasingly complicated geometrical setting, as well as the refined analysis of Fokker-Planck models.

To summarise, the project has obtained several groundbreaking results, not only answering long lasting opened questions but also opening new venues for development.