ERC Starting grant – Waves propagation in real life physics occur in media which are neither homogeneous or spatially infinity. The birth of radar/sonar technologies (and the raise of computed tomography) greatly motivated numerous developments in microlocal analysis and the linear theory. Only recently toy nonlinear models have been studied on a curved background, sometimes compact or rough. Understanding how to extend such tools, dealing with wave dispersion or focusing, will allow us to significantly progress in our mathematical understanding of physically relevant models. There, boundaries appear naturally and most earlier developments related to propagation of singularities in this context have limited scope with respect to crucial dispersive effects. Despite great progress over the last decade, driven by the study of quasilinear equations, our knowledge is still very limited. Going beyond this recent activity requires new tools whose development is at the heart of this proposal, including good approximate solutions going over arbitrarily large numbers of caustics, sharp pointwise bounds on Green functions, development of efficient wave packets methods, quantitative refinements of propagation of singularities (with direct applications in control theory), only to name a few important ones.

Abstract exterior ball

Dispersion estimates for the wave and the Schrödinger equations outside a ball and counterexamples

Abstract : We consider the wave and the Schrödinger equations with Dirichlet boundary conditions in the exterior of a ball in R^d. In dimension d=3 we construct a sharp, global in time parametrix and then proceed to obtain sharp dispersive estimates, matching the R³ case, for all frequencies (low and high). If d ≥4 we provide an explicit solution to the wave equation localized at large frequency 1/h with data a Dirac mass at a point Q_0at large distance s from the center of the ball : taking s~1/h^1/3, the decay rate of that solution exhibits a (t/h)^(d-3)/4 loss with respect to the boundary less case, that occurs at t~ 2s with an observation point being symmetric to Q_0 with respect to the center of the ball (at the Poisson Arago spot). A similar counterexample is also obtained for the Schrödinger flow.

The question about whether or not dispersion did hold outside general strictly convex obstacles was raised more than 20 years ago : in this work we give sharp answers which highlight the importance of diffractive effects, especially in higher dimensions where we provide unexpected counterexamples.