Objective
The three-dimensional Navier-Stokes equations are the fundamental mathematical model of fluid flow. However, currently we only know that unique solutions exist for all time for 'small' data (initial condition and forcing); for 'large' data they can only be guaranteed to exist for a short time. The existence of unique smooth solutions that exist for all time for any choice of data is one of the Clay Foundation's Million Dollar Millennium Prize Problems, and is exceedingly hard. The goal of this proposal is to address the problem of uniqueness of solutions in a way that does not require a solution of this problem in full. We aim to show that the solutions are unique for certain large classes of data. More concretely, we aim to prove the following three results: (i) it is possible to verify uniqueness numerically (at least in theory) for any set of initial conditions that is bounded in H^1; (ii) for a fixed initial condition, a prevalent set of forcing functions give rise to unique solutions; and (iii) for a fixed forcing function, a dense set of complex initial conditions give rise to unique solutions. The result of (i) relies on showing that the property of uniqueness is in some sense robust, which we will prove in a way that generalises previous results obtained by the host. In (ii), "prevalence" is a probabilistic notion, introduced for various problems in dynamical systems, which means that one can describe something as happening "with probability one". It is already know that this result is true if one replaces "prevalent" by "dense", but a result valid "almost surely" will be more practically relevant. Objective (iii), density of initial conditions giving rise to unique solutions, is a high-profile problem, which we will treat using results from the statistical theory of the equations developed in the 1980s.
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 applied mathematics dynamical systems
- natural sciences mathematics applied mathematics mathematical model
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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.
Topic(s)
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.
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.
Call for proposal
Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
FP7-PEOPLE-2009-IEF
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Funding Scheme
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.
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.
Coordinator
CV4 8UW COVENTRY
United Kingdom
The total costs incurred by this organisation to participate in the project, including direct and indirect costs. This amount is a subset of the overall project budget.