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 : We prove that the range of Strichartz estimates on a model 2D convex domain may be further restricted compared to the known counterexamples. Our new family ofcounterexamples is built on the parametrix construction from our recent paper “Dispersion for the wave equation inside strictly convex domains”. Interestingly enough, it is sharp in at least some regions of phase space.
Abstract : We consider the wave equation with Dirichlet boundary conditions in the exterior of a cylinder in R3 and we construct a sharp global in time parametrix to derive sharp dispersive estimates for all frequencies (low and high) and, as a corollary Strichartz estimates matching the R3 case.
Abstract : We prove sharper Strichartz estimates than expected from the optimal dispersion bounds. This follows from taking full advantage of the space-time localization of caustics. Several improvements on the parametrix construction from our previous works are obtained along theway and are of independent interest.
Abstract : We prove global in time dispersion estimates for the wave and the Klein-Gordon equation inside the Friedlander domain by taking full advantage of the space-time localization of caustics and a precise estimate of the number of waves that may cross at a given, large time. Moreover, we uncover a significant difference between Klein-Gordon and the wave equation in the low frequency, large time regime, where Klein-Gordon exhibits a worse decay that the wave, unlike in the flat space.
Abstract : We consider an anisotropic model case for a strictly convex domain of dimension d>1 with smooth, non-empty boundary and we describe dispersion for the semi-classical Schrödinger equation with Dirichlet boundary condition. More specifically, we obtain the following fixed time decay rate for the semi-classical Schrödinger flow : a loss of 1/4 occurs with respect to the boundary less case due to repeated swallowtail type singularities, and is proven optimal. Corresponding Strichartz estimates allow to solve the cubic nonlinear Schrödinger equation on such a 3D model convex domain, hence matching known results on generic compact boundary less manifolds.
Abstract : We consider the wave and the Schrödinger equations with Dirichlet boundary conditions in the exterior of a ball in Rd. 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 counterexamples …
Abstract : In this paper, we consider the wave equation on a strictly convex domain Ω of dimension d ≥ 2 with smooth boundary and with Dirichlet boundary conditions. We construct a sharp local in time parametrix and then proceed to obtain dispersion estimates.
Abstract : The authors prove a newsmoothing type property for solutions of the 1D quintic Schrödinger equation. As a consequence, they prove that a family of natural gaussian measures are quasi-invariant under the flow ofthis equation. In the defocusing case,they prove globalin time quasi-invariance while in the focusing case they only get local in time quasi-invariance because of a blow-up obstruction. Their results extend as well to generic oddpower nonlinearities.
An. Sc Norm. Super Pisa Cl. Sci.(5), vol. XXII (2021), 109-182
Abstract : The authors prove long time existence for a large class of fully nonlinear, reversible and parity preserving Schrödinger equations on the one dimensional torus. They show that for any initial condition even in x, regular enough and of size ε sufficiently small, the life span of the solution is of order ε-N for any N∈ℕ if some non resonance conditions are fulfilled. After a paralinearization of the equation we perform several para-differential changes of variables which diagonalize the system up to a very regularizing term. Once achieved the diagonalization, they construct modified energies for the solution by means of Birkhoff normal forms techniques.
Abstract : We consider the quantum hydrodynamic system on a d-dimensional irrational torus with d=2,3. We discuss the behaviour, over a “non trivial” time interval, of the Hs -Sobolev norms of solutions. More precisely we prove that, for generic irrational tori, the solutions, evolving from ε-small initial conditions, remain bounded in Hs for a time scale of order O(ε-1-1/(d-1)), which is strictly larger with respect to the time-scale provided by local theory. We exploit a Madelung transformation to rewrite the system as a nonlinear Schrödinger equation. We therefore implement a Birkhoff normal form procedure involving small divisors arising from three waves interactions. The main difficulty is to control the loss of derivatives coming from the exchange of energy between high Fourier modes. This is due to the irrationality of the torus which prevent to have “good separation” properties of the eigenvalues of the linearized operator at zero.
Journal of dynamics and differential equations (2021)
Abstract : The authors consider the semi-linear beam equation on the d dimensional irrational torus with smooth nonlinearity of order n – 1 with n ≥ 3 and d ≥ 2. If ϵ ≪ 1 is the size of the initial datum, they prove that the lifespan Tϵ of solutions is O(ϵ-A(n-2) ) where A = A(d, n) = 1 + 3/(d-1) when n is even and A = 1 + 3/(d-1) + max((4-d)/(d-1) , 0) when n is odd. For instance for d = 2 and n = 3 (quadratic nonlinearity) they obtain Tϵ = O(ϵ-6), much better than O(ϵ-1), the time given by the local existence theory. The irrationality of the torus makes the set of differences between two eigenvalues of (Δ2 + 1)1/2 accumulate to zero, facilitating the exchange between the high Fourier modes and complicating the control of the solutions over long times. Our result is obtained by combining a Birkhoff normal form step and a modified energy step.
Abstract : The authors prove a local in time well-posedness result for quasi-linear Hamiltonian Schrödinger equations on Td for any d ≥ 1. For any initial condition in the Sobolev space Hs, with s large, we prove the existence and unicity of classical solutions of the Cauchy problem associated to the equation. The lifespan of such a solution depends only on the size of the initial datum. Moreover we prove the continuity of the solution map.
Abstract : The authors consider quasilinear, Hamiltonian perturbations of the cubic Schrödinger and of the cubic (derivative) Klein-Gordon equations on the d dimensional torus. If ε ≪ 1 is the size of the initial datum, we prove that the lifespan of solutions is strictly larger than the local existence time ε-2. More precisely, concerning the Schrödinger equation we show that the lifespan is at least of order O(ε-4), in the Klein-Gordon case we prove that the solutions exist at least for a time of order O(ε-8/3 ) as soon as d ≥ 3. Regarding the Klein-Gordon equation, our result presents novelties also in the case of semi-linear perturbations: we show that the lifespan is at least of order O(ε-10/3 ), improving, for cubic non-linearities and d ≥ 4, the general results
Abstract : In this paper we consider an abstract class of quasi-linear para-differential equations on the circle. For each equation in the class we prove the existence of a change of coordinates which conjugates the equation to a diagonal and constant coefficient para-differential equation. In the case of Hamiltonian equations we also put the system in Poincaré-Birkhoff normal forms. We apply this transformation to quasi-linear perturbations of the Schrödinger and Beam equations, obtaining a long time existence result without requiring any symmetry on the initial data. We also provide the local in time well-posedness for quasi-linear perturbations of the Benjamin-Ono equation.
This project has received funding from the ERC under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 757 996)