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.
Organisations
- University of Warwick (Lead Research Organisation)
- Universidade de São Paulo (Collaboration)
- University of Augsburg (Collaboration)
- Complutense University of Madrid (Collaboration)
- University of Manchester (Collaboration)
- University of Zurich (Collaboration)
- Polish Academy of Sciences (Collaboration)
- Autonomous University of Madrid (Collaboration)
- Polytechnic University of Milan (Collaboration)
- University of Nevada (Collaboration)
- University of Sussex (Collaboration)
- Xi'an Jiaotong Liverpool University (Collaboration)
- University of Seville (Collaboration)
- University of Warsaw (Collaboration)
- State University of Campinas (Collaboration)
- University of Chicago (Project Partner)
- Complutense University of Madrid (Project Partner)
- Weizmann Institute of Science (Project Partner)
People |
ORCID iD |
James Robinson (Principal Investigator) |
Publications
Vidal-López A
(2016)
A complete characterisation of local existence for semilinear heat equations in Lebesgue spaces
in Annales de l'Institut Henri Poincaré C, Analyse non linéaire
Vidal-López A
(2013)
A note on the existence of global solutions for reaction-diffusion equations with almost-monotonic nonlinearities
in Communications on Pure and Applied Analysis
Sánchez-Gabites J
(2011)
How strange can an attractor for a dynamical system in a 3-manifold look?
in Nonlinear Analysis: Theory, Methods & Applications
Sadowski W
(2013)
A remark on the box-counting dimension of the singular set for the Navier-Stokes equations
in Communications in Mathematical Sciences
Sadowski W
(2012)
Detecting local time singularities of the micropolar fluid flow
in Physica D: Nonlinear Phenomena
Robinson, James C.; Rodrigo, Jose L.; Sadowski, Witold
(2012)
Mathematical Aspects of Fluid Mechanics
Robinson, James C. (University Of Warwick); Rodrigo, Jose L.
(2009)
Partial Differential Equations and Fluid Mechanics
Robinson JC & Sadowski W
(2009)
Partial Differential Equations and Fluid Mechniacs
Robinson JC
(2012)
Strict inequality in the box-counting dimension product formula
in Real Analysis Exchange
Robinson J
(2012)
Supersolutions for a class of semilinear heat equations
in Revista Matemática Complutense
Robinson J
(2013)
On the Regularity of Lagrangian Trajectories Corresponding to Suitable Weak Solutions of the Navier-Stokes Equations
in Procedia IUTAM
Robinson J
(2009)
Linear embeddings of finite-dimensional subsets of Banach spaces into Euclidean spaces
in Nonlinearity
Robinson J
(2009)
Almost-everywhere uniqueness of Lagrangian trajectories for suitable weak solutions of the three-dimensional Navier-Stokes equations
in Nonlinearity
Robinson J
(2013)
Attractors and Finite-Dimensional Behaviour in the 2D Navier-Stokes Equations
in ISRN Mathematical Analysis
Robinson J
(2014)
A local smoothness criterion for solutions of the 3D Navier-Stokes equations
in Rendiconti del Seminario Matematico della Università di Padova
Robinson J
(2016)
Recent Progress in the Theory of the Euler and Navier-Stokes Equations
Robinson J
(2013)
Dimension prints and the avoidance of sets for flow solutions of non-autonomous ordinary differential equations
in Journal of Differential Equations
Robinson J
(2012)
A note on well-posedness of semilinear reaction-diffusion problem with singular initial data
in Journal of Mathematical Analysis and Applications
Robinson J
(2009)
A Criterion for Uniqueness of Lagrangian Trajectories for Weak Solutions of the 3D Navier-Stokes Equations
in Communications in Mathematical Physics
Robinson J
(2011)
On the Dimension of the Singular Set of Solutions to the Navier-Stokes Equations
in Communications in Mathematical Physics
Robinson J
(2012)
Lower bounds on blow up solutions of the three-dimensional Navier-Stokes equations in homogeneous Sobolev spaces
in Journal of Mathematical Physics
Robinson J
(2014)
Log-Lipschitz embeddings of homogeneous sets with sharp logarithmic exponents and slicing products of balls
in Proceedings of the American Mathematical Society
Robinson J
(2016)
On finite-dimensional global attractors of homeomorphisms
in Bulletin of the London Mathematical Society
Robinson J
(2013)
Minimal periods of semilinear evolution equations with Lipschitz nonlinearity revisited
in Journal of Differential Equations
Robinson J
(2013)
On finite-dimensional attractors of homeomorphisms
Robinson
(2010)
Dimensions, Embeddings, and Attractors
Robinson
(2016)
The Three-Dimensional Navier-Stokes Equations: Classical Theory
Pooley B
(2016)
An Eulerian-Lagrangian Form for the Euler Equations in Sobolev Spaces
in Journal of Mathematical Fluid Mechanics
Pinto De Moura E
(2011)
Embedding of global attractors and their dynamics
in Proceedings of the American Mathematical Society
Pinto De Moura E
(2010)
Orthogonal sequences and regularity of embeddings into finite-dimensional spaces
in Journal of Mathematical Analysis and Applications
Pinto De Moura E
(2014)
Log-Lipschitz continuity of the vector field on the attractor of certain parabolic equations
in Dynamics of Partial Differential Equations
Pinto De Moura E
(2010)
Lipschitz deviation and embeddings of global attractors
in Nonlinearity
Ozanski W
(2019)
Partial Regularity for a Surface Growth Model
in SIAM Journal on Mathematical Analysis
Ozanski W
(2017)
Partial regularity for a surface growth model
OLSON E
(2015)
Generalised Cantor sets and the dimension of products
in Mathematical Proceedings of the Cambridge Philosophical Society
Olson E
(2009)
Almost bi-Lipschitz embeddings and almost homogeneous sets
in Transactions of the American Mathematical Society
Olson E
(2014)
Generalised Cantor sets and the dimension of products
Nieuwenhuis M
(2014)
Minimal periods for ordinary differential equations in strictly convex Banach spaces and explicit bounds for some L p -spaces
in Journal of Differential Equations
McCormick D
(2014)
Existence and Uniqueness for a Coupled Parabolic-Elliptic Model with Applications to Magnetic Relaxation
in Archive for Rational Mechanics and Analysis
McCormick D
(2016)
Lower Bounds on Blowing-Up Solutions of the Three-Dimensional Navier--Stokes Equations in $\dot H^{3/2}$, $\dot H^{5/2}$, and $\dot B^{5/2}_{2,1}$
in SIAM Journal on Mathematical Analysis
McCormick D
(2013)
Generalised Gagliardo-Nirenberg Inequalities Using Weak Lebesgue Spaces and BMO
in Milan Journal of Mathematics
Marín-Rubio P
(2013)
Solutions of the 3D Navier-Stokes equations for initial data in H ? 1 / 2 : Robustness of regularity and numerical verification of regularity for bounded sets of initial data in H ? 1
in Journal of Mathematical Analysis and Applications
Lukaszewicz G
(2014)
Invariant measures for non-autonomous dissipative dynamical systems
in Discrete and Continuous Dynamical Systems
Lukaszewicz G
(2011)
Invariant Measures for Dissipative Systems and Generalised Banach Limits
in Journal of Dynamics and Differential Equations
Langa J
(2009)
Permanence and Asymptotically Stable Complete Trajectories for Nonautonomous Lotka-Volterra Models with Diffusion
in SIAM Journal on Mathematical Analysis
Laister R
(2013)
Non-existence of local solutions for semilinear heat equations of Osgood type
in Journal of Differential Equations
Laister R
(2014)
Non-existence of local solutions of semilinear heat equations of Osgood type in bounded domains
in Comptes Rendus Mathematique
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 | Universidade de São 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 |