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Mathematical analysis of fluid flows: the challenge of randomness

Project description

Maths face the challenge to reveal the role of randomness in fluid dynamics

The EU-funded FluFloRan project aims to address several, long lasting, open problems in the mathematical theory of fluid flows. The project will work on finding the right solutions to nonlinear partial differentiation equations that describe flows in liquids and gases. Major attention will be dedicated to the iconic example, the Navier-Stokes system for incompressible fluids and the corresponding Millennium Problem. The project assumes that a probabilistic description is indispensable for modelling fluid flows – to capture the chaotic behaviour of deterministic systems after blow-up – and describing model uncertainties due to high sensitivity to input data or parameter reduction.

Objective

The main goal of the present project is to make substantial contributions to the understanding of fundamental problems in the mathematical theory of fluid flows. This theory is formulated in terms of systems of nonlinear partial differential equations (PDEs). Major attention has been paid to the iconic example, the Navier-Stokes system for incompressible fluids, and the corresponding Millennium Problem. Despite joint efforts and a substantial progress for various models in fluid dynamics, fundamental questions concerning existence and uniqueness of solutions as well as long time behavior remain unsolved.

This project is based on the conviction that a probabilistic description is indispensable in modeling of fluid flows to capture the chaotic behavior of deterministic systems after blow-up, and to describe model uncertainties due to high sensitivity to input data or parameter reduction. For a set of selected models, we investigate different aspects of the underlying deterministic and stochastic PDE dynamics. In particular, we are concerned with the question of solvability and well-posedness or alternatively ill-posedness. For some models including the incompressible stochastic Navier-Stokes system we investigate non-uniqueness in law. For the compressible counterpart we aim to prove existence of a unique ergodic invariant measure.

The guiding theme of this research program is a core question in the field, namely, how to select physically relevant solutions to PDEs in fluid dynamics. The project lies at the challenging frontiers of PDE theory and probability theory and it will tackle several long standing open problems. The results will have an impact in the deterministic PDE theory, stochastic partial differential equations and from a wider perspective also in mathematical physics.

Host institution

UNIVERSITAET BIELEFELD
Net EU contribution
€ 1 500 000,00
Address
UNIVERSITAETSSTRASSE 25
33615 Bielefeld
Germany

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Region
Nordrhein-Westfalen Detmold Bielefeld, Kreisfreie Stadt
Activity type
Higher or Secondary Education Establishments
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Total cost
€ 1 500 000,00

Beneficiaries (1)