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GEOEDP

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Geometry, Control and Genericity for Partial Differential Equations

Many physics models are described by waves or more generally dispersive equations (Schrödinger equations) with propagation in a non homogeneous and bounded medium. Toy models (mostly in flat backgrounds) have been developed by mat... ver más

30/09/2028

UNIVERSITE PARISSA...

2M€

Presupuesto del proyecto: 2M€

Líder del proyecto

UNIVERSITE PARISSACLAY No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

Fecha límite participación Sin fecha límite de participación.

Financiación concedida El organismo HORIZON EUROPE notifico la concesión del proyecto el día 2023-04-24

ERC-2022-ADG: ERC ADVANCED GRANTS Scope:Objectives

I+D

Cerrada hace 2 años

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1 Participantes

UNIVERSITE PARISSACLAY

1.65M€ | Lider

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Duración del proyecto: 65 meses Fecha Inicio: 2023-04-24
Fecha Fin: 2028-09-30

Líder del proyecto

UNIVERSITE PARISSACLAY No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

Presupuesto del proyecto 2M€

Fecha límite de participación Sin fecha límite de participación.

Descripción del proyecto Many physics models are described by waves or more generally dispersive equations (Schrödinger equations) with propagation in a non homogeneous and bounded medium. Toy models (mostly in flat backgrounds) have been developed by mathematicians. However, many questions remain open even on these simplified models in the presence of inhomogeneities and boundaries. In particular, the works of mathematicians in the last decade have allowed to exhibit some pathological behaviours which appear to be quite unstable. A first point in this proposal will be to expand the understanding of the influence of the geometry (inhomogeneities of the media, boundaries) on the behaviour of solutions to dispersive PDE’s. When these behaviours appear to be unstable, a natural question is whether they are actually rare. The last years have seen the emergence of a new point of view on these questions: random data Cauchy theories. The idea behind is that for random initial data, the solution’s behaviours are better than expected (deterministically). The second point of this project is precisely to go further in this direction. After identifying these pathological behaviours, is it possible to show that for almost all initial data, almost all geometries, they do not happen? Understanding how to combine the powerful techniques from micro-local and harmonic analysis with a probabilistic approach in this context should allow a much better understanding of these physically relevant models. Summarising, the purpose of my project is to develop tools and give answers to the following questions in the context of dispersive PDE’s (and to some extent fluids mechanics) Can we understand the influence of the geometric background (and boundaries) on concentration properties and the the behaviour of solutions to dispersive evolution PDE’s? Can we define generic behaviours for solutions to waves and fluids PDE's ? Can we show that some very pathological behaviours (which do happen) are actually very rare?

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