Project description
Analysing the qualitative properties of waves in fluids
The ability to predict the behaviour of waves in fluids is one of the key challenges in mathematics. The EU-funded HamDyWWa project aims to change this. To that end, it will develop Kolmogorov–Arnold–Moser as well as normal form methods for partial differential equations in higher dimensional spaces. A key focus will be on oscillatory motion and stability properties for partial differential equations of fluid dynamics and quantum mechanics, using techniques from the Kolmogorov–Arnold–Moser theorem, the normal form theory, as well as harmonic and micro-local analysis.
Objective
KAM and normal form methods are very powerful tools for analyzing the dynamics of nearly integrable finite dimensional Hamiltonian systems. In the last decades, the extension of these methods to infinite dimensional systems, like Hamiltonian PDEs (partial differential equations), has attracted the interest of many outstanding mathematicians like Bourgain, Craig, Kuksin, Wayne and many others. These techniques provide some tools for describing the phase space of nearly integrable PDEs. More precisely they give a way to construct special global solutions (like periodic and quasi-periodic solutions) and to analyze stability issues close to equilibria or close to special solutions (like solitons). In the last seven years, I developed new methods for proving the existence of quasi-periodic solutions of quasi-linear, one-dimensional PDEs. This is an important step towards treating many of the fundamental equations from physics since most of these equations are quasi-linear. In particular, this is the case for the equations in fluid dynamics, the water waves equation being a prominent example. These novel techniques are based on a combination of pseudo-differential and para-differential calculus, with the classical perturbative techniques and they allowed to make significant advances of the KAM and normal form theory for one-dimensional PDEs. On the other hand, many challenging problems remain open and the purpose of this proposal is to investigate some of them. The main goal of this project is to develop KAM and normal form methods for PDEs in higher space dimension, with a particular focus on equations arising from fluid dynamics, like Euler, Navier-Stokes and water waves equations. By extending the novel approach, developed for PDEs in one space dimension, I have already obtained some preliminary results on PDEs in higher space dimension (like the Euler equation in 3d), which makes me confident that the proposed project is feasible.
Fields of science (EuroSciVoc)
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
- natural sciences mathematics pure mathematics mathematical analysis differential equations partial differential equations
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Keywords
Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
Programme(s)
Multi-annual funding programmes that define the EU’s priorities for research and innovation.
Multi-annual funding programmes that define the EU’s priorities for research and innovation.
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HORIZON.1.1 - European Research Council (ERC)
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Topic(s)
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Calls for proposals are divided into topics. A topic defines a specific subject or area for which applicants can submit proposals. The description of a topic comprises its specific scope and the expected impact of the funded project.
Funding Scheme
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Funding scheme (or “Type of Action”) inside a programme with common features. It specifies: the scope of what is funded; the reimbursement rate; specific evaluation criteria to qualify for funding; and the use of simplified forms of costs like lump sums.
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Call for proposal
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Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
(opens in new window) ERC-2021-STG
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20122 Milano
Italy
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