The Navier-Stokes equations: functional analysis and dynamical systems

Lead Research Organisation: University of Warwick
Department Name: Mathematics

Abstract

The Navier-Stokes equations are well established as the mathematical model for the flow of fluids. But while they are used extensively in both theoretical and computational analyses of every aspect of fluid flow, their mathematical foundations are still uncertain.In the year 2000, the Clay Mathematics Institute announced a list of Seven Millennium problems, solutions for each of which will attract a prize of one million dollars. Included in this list are 'classic problems' such as the Riemann Hypothesis and the Poincar conjecture (now solved by the work of Perelman); but here one can also find the question of the existence (or otherwise) of unique solutions for the three-dimensional Navier-Stokes equations.The point of a mathematical model is that it enables prediction: if you know what happens at an initial time, you can predict what will happen in the future. However, being able to make a 'prediction' relies on the model having only one solution: two (or more) solutions starting from the same initial setup make prediction a matter of divination rather than science.This is the 'uniqueness problem' (which can be formulated precisely given the correct mathematical language) that remains unresolved for the three-dimensional Navier-Stokes equations: although used routinely, there is no mathematical proof that they have any predictive power. Part of this proposal focuses on questions related to this fundamental difficulty, which is a fault line running through mathematical fluid dynamics. The formation of a 'singularity' is the process by which predictive power can be lost, and this project will consider how one can limit the formation of these singularities (should they actually occur). Related to this is the question of how the Navier-Stokes equations relate to the Euler equations, an older and some sense simpler model that neglects the effect of viscosity.The other half of the proposal considers questions that arise when one considers the two-dimensional Navier-Stokes equations. The two-dimensional model has less physical relevance, but does not suffer from the fundamental problems that bedevil its three-dimensional counterpart: this makes it a useful testbed for techniques that could eventually be applied in the three-dimensional case.The theory of dynamical systems (of which 'chaos theory' forms a part) can be applied to the two-dimensional equations. In this context, it is possible to show that the equations have an attractor that is finite-dimensional. In a very loose way this says that 'what happens in the long run should be relatively easy to describe'; in the language of physics one might express this as 'fully-developed two-dimensional turbulence has a finite number of degrees of freedom'.Giving a rigorous (and mathematically concrete) interpretation of this idea forms the other half of this proposal.

Publications

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Blömker D (2015) Rigorous numerical verification of uniqueness and smoothness in a surface growth model in Journal of Mathematical Analysis and Applications

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Bosia S (2014) A Weak-L p Prodi-Serrin Type Regularity Criterion for the Navier-Stokes Equations in Journal of Mathematical Fluid Mechanics

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Carvalho A (2012) Structure and bifurcation of pullback attractors in a non-autonomous Chafee-Infante equation in Proceedings of the American Mathematical Society

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Carvalho A (2009) On the continuity of pullback attractors for evolution processes in Nonlinear Analysis: Theory, Methods & Applications

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CARVALHO A (2009) Lower semicontinuity of attractors for non-autonomous dynamical systems in Ergodic Theory and Dynamical Systems

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Carvalho A (2010) Finite-dimensional global attractors in Banach spaces in Journal of Differential Equations

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Chemin J (2016) Local existence for the non-resistive MHD equations in Besov spaces in Advances in Mathematics

 
Description I have investigated (i) when abstract collections of mathematical objects (metric spaces) can be realised in more concrete settings, which has applications to many problems including understand the long-term behaviour of fluid flows and computational questions and (ii) how one can view the flow of fluids via the movement of (notional) individual fluid particles, even when at the macroscopic level the flow seem very irregular. My work towards (i) has resulted in a collection of strong results, categorised by various different notions of the dimension of the original metric space, and led in particular to a research monograph in the Cambridge Tracts in Mathematics Series. My work towards (ii) has shown that even for seemingly irregular and turbulent fluid flows it is possible to follow particle trajectories, which opens up a new way to try to understand (or rule out) possible singularity formation in the Navier-Stokes model, a long-standing mathematical question.
Exploitation Route Primarily within academia. I have obtained results in the theory of embeddings of finite-dimensional sets into Euclidean spaces that have implications for dynamical systems theory, and the Lagrangian approach to the Navier-Stokes equations should serve to stimulate further research and open new directions in the study of this model.
Sectors Education

 
Description Assouad dimension and dynamical systems 
Organisation University of Manchester
Country United Kingdom 
Sector Academic/University 
PI Contribution Joint research
Collaborator Contribution Joint research
Impact Two papers, currently under review
Start Year 2013
 
Description Assouad dimension and dynamical systems 
Organisation University of Nevada
Country United States 
Sector Academic/University 
PI Contribution Joint research
Collaborator Contribution Joint research
Impact Two papers, currently under review
Start Year 2013
 
Description Flow around obstacles 
Organisation State University of Campinas
Country Brazil 
Sector Academic/University 
PI Contribution Joint research
Collaborator Contribution Joint research
Impact One published paper, one submitted, one in preparation.
Start Year 2009
 
Description Flow around obstacles 
Organisation University of Sussex
Country United Kingdom 
Sector Academic/University 
PI Contribution Joint research
Collaborator Contribution Joint research
Impact One published paper, one submitted, one in preparation.
Start Year 2009
 
Description Flow around obstacles 
Organisation University of Zurich
Country Switzerland 
Sector Academic/University 
PI Contribution Joint research
Collaborator Contribution Joint research
Impact One published paper, one submitted, one in preparation.
Start Year 2009
 
Description Navier-Stokes equations and dynamical systems 
Organisation Polytechnic University of Milan
Country Italy 
Sector Academic/University 
PI Contribution Joint research
Collaborator Contribution Joint research
Impact One published paper, one in preparation.
Start Year 2012
 
Description Navier-Stokes equations and related models 
Organisation Polish Academy of Sciences
Country Poland 
Sector Public 
PI Contribution Joint research
Collaborator Contribution Joint research
Impact 9 published papers. One submitted for publication. Navier-Stokes textbook due to be finished by the end of the year. Conference proceedings in preparation.
Start Year 2013
 
Description Navier-Stokes equations and related models 
Organisation Xi'an Jiaotong Liverpool University 
Country China 
Sector Academic/University 
PI Contribution Joint research
Collaborator Contribution Joint research
Impact 9 published papers. One submitted for publication. Navier-Stokes textbook due to be finished by the end of the year. Conference proceedings in preparation.
Start Year 2013
 
Description Non-autonomous dynamical systems 
Organisation University of Sao Paulo
Country Brazil 
Sector Academic/University 
PI Contribution Joint research
Collaborator Contribution Joint research
Impact Published papers as detailed in main submission. One research-based book as detailed in submission (Carvalho-Langa-Robinson). Edited journal issue currently in press.
Start Year 2006
 
Description Non-autonomous dynamical systems 
Organisation University of Seville
Country Spain 
Sector Academic/University 
PI Contribution Joint research
Collaborator Contribution Joint research
Impact Published papers as detailed in main submission. One research-based book as detailed in submission (Carvalho-Langa-Robinson). Edited journal issue currently in press.
Start Year 2006
 
Description Regularity and singularity in a model of surface growth 
Organisation University of Augsburg
Country Germany 
Sector Academic/University 
PI Contribution Joint research
Collaborator Contribution Joint research
Impact One paper under review, one shortly to be completed
Start Year 2012
 
Description Semilinear heat equations in critical spaces 
Organisation University of Warsaw
Country Poland 
Sector Academic/University 
PI Contribution Joint research
Collaborator Contribution Joint research
Impact Paper submitted for review
Start Year 2014
 
Description Topology and dynamical systems 
Organisation Autonomous University of Madrid
Country Spain 
Sector Academic/University 
PI Contribution Joint research
Collaborator Contribution Joint research
Impact Two joint papers, one research conference
Start Year 2010
 
Description Topology and dynamical systems 
Organisation Complutense University of Madrid
Country Spain 
Sector Academic/University 
PI Contribution Joint research
Collaborator Contribution Joint research
Impact Two joint papers, one research conference
Start Year 2010