ICMS

CANPDE - Dynamics in infinite-dimensions: ergodic theory and PDEs

May 21, 2012 - May 25, 2012

ICMS, 15 South College Street Edinburgh

Organisers

Name	Institution
Beck, Margaret	Heriot-Watt University and Boston University
Melbourne, Ian	University of Surrey
Ott, William	University of Houston

 

This meeting will focus on the analysis of PDEs using techniques motivated by phenomena that occur in finite dimensions. In particular, the goal will be to develop the ergodic theory of infinite- dimensional dynamical systems. There are many examples, such as invariant manifolds and stability analysis, in which the rigorous analysis of the phenomenon in finite dimensions lead to the understanding of it also in infinite dimensions. One area in which this connection has not been extensively explored is ergodic theory, which in finite dimensions is reasonably mature. The aim of this workshop is to bring this accumulated knowledge to bear on the study of certain PDEs by fostering new collaborations between researchers working in these areas.

 

This meeting is funded by the CANPDE project http://www.maxwell.ac.uk/PDEs

 

Confirmed speakers

Alexandre Boritchev (Ecole Polytechnique)
Alexander Bufetov (Steklov Mathematical Institute & Rice University)
Thierry Gallay (Universite de Grenoble I)
Nathan Glatt-Holtz (Indiana University)
Georg Gottwald (University of Sydney)
Martin Hairer (University of Warwick)
Carlangelo Liverani (Universita di Roma)
Kening Lu (Brigham Young University)
Karsten Matthies (University of Bath)
Dmitry Turaev (Imperial College London)
C. Eugene Wayne (Boston University)
Hendrik Weber (University of Warwick)
Sergey Zelik (University of Surrey)

 

Arrangements

Participation
Speakers have been sent invitations. If you are interested in attending the workshop you are welcome to submit your applications through the public application form (link at top right of screen). Places are limited and the organisers may have to prioritise places.

UK Visas
If you are travelling from overseas you may require an entry visa. A European visa does not guarantee entry to the UK. Please use this link to the UK Visas site to find out if you need a visa and if so how to apply for one.

Venue
The workshop will be held at 15 South College Street, Edinburgh.  You may also find this map useful for the workshop.  All lectures will be held in the Newhaven Lecture Theatre.  To view this room and a list of the visual equipment available click here.  In addition, two blackboards have recently been installed.  The projector and one board may be used simultaneously.  

Wireless Access
The workshop venue, 15 South College Street, has wireless access throughout. On arrival at Registration you will be given instructions and a code for accessing the wireless network. Alternatively, we have EduRoam for those who are already registered for this service at their home university. For those without laptops, there will also be a couple of computers available for you to check your emails.

Travel
Information about travel to the UK and Edinburgh is available here.

Please note that it is your responsibility to have adequate travel insurance to cover medical and other emergencies (such as volcanic ash disruptions) that may occur on your trip.

A taxi directly from the airport will cost approximately 15.00 to 20.00 GBP to the city centre for a one-way journey.     There is also a bus service direct from the airport to the city centre which will cost 3.50 single or 6.00 GBP return - the Airlink 100.  This is a frequent service (every 10 minutes during peak times) and will bring you close to Waverley Railway Station, only a short walk to the accommodation and the workshop venue (see map in 'Venue' section above). 

Lothian buses charge £1.40 for a single, £3.50 for a day ticket. Please note that the exact fare is required and no change is given.

If travelling by train, please note that Edinburgh has two railway stations - Waverley Railway Station being the main station and closest to the workshop venue at 15 South College Street. If you alight at Edinburgh Waverley, the workshop venue is an easy 10 minute walk over North and South Bridge. The second railway station is called Haymarket and is at the West End of the city centre.

 

Programme

Preliminary programme published Friday 4 May 2012. Last update 17 May.

Monday 21 May

09.30 - 11.00	Registration and coffee
11.00 - 12.00	Sergey Zelik (University of Surrey) Infinite-dimensionality in the "finite-dimensional" dissipative dynamics
12.00 - 14.00	Lunch served in the chapterhouse, level 1
14.00 - 15.00	Martin Hairer (University of Warwick) Mini lecture series 1(4): Ergodic theory for SPDEs
15.00 - 15.45	Coffee and tea (served in the Chapterhouse)
15.45 - 16.45	Alexandre Boritchev (Ecole Polytechnique)   1D Burgers turbulence as a model case for the Kolmogorov theory
16.45 - 18.00	Wine reception

Tuesday 22 May

09.30 - 10.30	Kening Lu (Brigham Young University) Lyapunov exponents and chaotic behavior for random dynamical systems in a Banach space
10.30 - 11.00	Coffee and tea (served in the Chapterhouse)
11.00 - 12.00	Martin Hairer  (University of Warwick) Mini lecture series 2 (4): Ergodic theory for SPDEs
12.00 - 14.00	Lunch served in the Chapterhouse
14.00 - 15.00	Martin Hairer  (University of Warwick) Mini lecture series 3 (4): Ergodic theory for SPDEs
15.00 - 15.45	Coffee and tea (served in the Chapterhouse)
15.45 - 16.45	Alexander Bufetov (Steklov Mathematical Institute & Rice University) Ergodic decomposition for actions of infinite-dimensional group

 

Wednesday 23 May

09.30 - 10.30	Georg Gottwald (University of Sydney) Diffusive behaviours in spatially extended systems with symmetry
10.30 - 11.00	Coffee and tea (served in the Chapterhouse)
11.00 - 12.00	Martin Hairer  (University of Warwick) Mini lecture series 4 (4): Ergodic theory for SPDEs
12.00 - 13.00	Lunch served in the Chapterhouse

Thursday 24 May

09.30 - 10.30	Eugene Wayne (Boston University) Metastability in the Navier-Stokes equations
10.30 - 11.00	Coffee and tea (served in the Chapterhouse)
11.00 - 12.00	Dimitry Turaev (Imperial College London) Infinite-dimensional hyperbolic attractor in a driven Swift-Hohenberg equation
12.00 - 14.00	Lunch served in the Chapterhouse
14.00 - 15.00	Nathan Glatt-Holtz (University of Indiana) Invariant measures for dissipative dynamical systems: abstract results and applications
15.00 - 15.45	Coffee and tea (served in the Chapterhouse)
15.45 - 16.45	Thierry Gallay (Universite de Grenoble I)   Quantitative recurrence estimates for extended dissipative systems
19.00	Workshop dinner at The Magnum Restaurant

 

Friday 25 May

09.30 - 10.30	Karsten Matthies  (University of Bath) A semigroup approach to the justification of kinetic PDE
10.30 - 11.00	Coffee and tea (served in the Chapterhouse)
11.00 - 12.00	Hendrik Weber (University of Warwick) Approximations to rough Burgers like equations

 

 

Presentations:

Presentation Details
Boritchev, Alexandre
1D Burgers Turbulence as a Model Case for the Kolmogorov Theory
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The Kolmogorov 1941 theory (K41) is, in a way, the starting point for all models of turbulence. In particular, K41 and corrections to it provide estimates of small-scale quantities such as increments and energy spectrum for a 3D turbulent flow. However, because of the well-known difficulties involved in studying 3D turbulent flow, there are no rigorous results confirming or infirming those predictions. Here, we consider a well-known simplified model for 3D turbulence: Burgulence, or turbulence for the 1D Burgers equation. In the space-periodic case with a stochastic white in time and smooth in space forcing term, we give sharp estimates for small-scale quantities such as increments and energy spectrum.
Bufetov, Alexander
Ergodic decomposition for actions of infinite-dimensional groups
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The talk is devoted to infinite-dimensional dynamical systems arising in representation theory. For example, consider the space of infinite matrices with complex entries, the projective limit of usual finite-dimensional matrix spaces. The infinite unitary group, the inductive limit of finite-dimensional unitary groups, naturally acts on our space of infinite matrices both on the left and on the right. Ergodic probability measures of the resulting action of the Cartesian square of the infinite unitary group form an infinite-dimensional family that admits an explicit classification due to Pickrell. The main result of the talk is that, under some natural assumptions, there are no infinite ergodic measures: every ergodic measure admitting well-defined projections on corners of finite size must itself be finite. This finiteness result is then applied to the problem of ergodic decomposition of unitarily-invariant measures on spaces of infinite matrices. The talk is based on the preprints arXiv:1105.0664 and arXiv:1108.2737.
Gallay, Thierry
Quantitative recurrence estimates for extended dissipative systems
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We study the long-time behavior of solutions to nonlinear dissipative PDE's on unbounded domains. As a consequence of a few simple hypotheses on the local energy dissipation in the system, we obtain general bounds on the integrated energy flux through a given hypersurface which allow us to draw interesting conclusions on the recurrence properties of such a system. For instance, if the space dimension is one or two, we can bound the average time spent by a solution in a neighborhood of any nonequilibrium point. This is a joint work with S. Slijepcevic (Zagreb).
Glatt-Holtz, Nathan
Invariant measures for dissipative dynamical systems: abstract results and applications
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Please see http://www.icms.org.uk/assets/files/downloads/canpde/Glatt-Holtz.pdf
Gottwald, Georg
Diffusive behaviours in spatially extended systems with symmetry
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We present a universal view on diffusive behaviour in chaotic spatially extended systems for anisotropic and isotropic media. For anisotropic systems, strong chaos leads to diffusive behaviour (Brownian motion with drift) and weak chaos leads to superdiffusive behaviour (Levy processes with drift). For isotropic systems, the drift term vanishes and strong chaos again leads to Brownian motion. We establish the existence of a nonlinear Huygens principle for weakly chaotic systems in isotropic media whereby the dynamics behaves diffusively in even space dimension and exhibits superdiffusive behaviour in odd space dimensions.
Hairer, Martin
Ergodic theory for SPDEs
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In this minicourse, we will survey recent results in the ergodic theory for infinite-dimensional Markov processes in general and stochastic PDEs in particular. The outline of the course is as follows. In the first lecture, we provide an introduction to the theory of stochastic PDEs, aimed mostly at an audience with some background in the theory of deterministic PDEs, but little probabilistic background. In the second lecture, we will then review the basics of general ergodic theory for Markov processes and we will introduce some abstract criteria ensuring ergodicity and / or convergence to the invariant measure. The last two lectures will be devoted to the application of the general theory to a class of semilinear parabolic stochastic PDEs including the stochastic Navier-Stokes equations. This will be based on joint work with J. Mattingly.
Lu, Kening
Lyapunov exponents and chaotic behavior for random dynamical systems in a Banach space
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We study the Lyapunov exponents and their associated invariant subspaces for infinite dimensional random dynamical systems in a Banach space, which are generated by, for example, stochastic or random partial differential equations. We prove a multiplicative ergodic theorem. We also prove that for an infinite dimensional random dynamical system with a random invariant set such as random attractor, if it’s topological entropy is positive, then the dynamics on the random invariant set is chaotic. This is based on joint works with Wen Huang and Zeng Lian.
Matthies, Karsten
A Semigroup Approach to the Justification of Kinetic PDE
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A method is presented to show the validity of continuum description for the deterministic dynamics of many interacting particles with random initial data. Considering a simplified case of hard-sphere dynamics, where particles are removed after the first collision, we characterize the many particle flow by collision trees which encode possible collisions. The convergence of the many-particle dynamics in Boltzmann-Grad scaling to the Boltzmann dynamics is achieved via the convergence of associated probability measures on collision trees. These probability measures satisfy nonlinear Kolmogorov equations, which are shown to be well-posed by semigroup methods. We will sketch an extension to full hard-sphere dynamics. Joint work with Florian Theil.
Turaev, Dmitry
Infinite-dimensional hyperbolic attractor in a driven Swift-Hohenberg equation
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We show that the Swift-Hohenberg equation with one spatial variable driven by a small perturbation, periodic in space and time, may possess an infinite-dimensional uniformly-hyperbolic attractor. This is a joint work with S.Zelik.
Wayne, C. Eugene
Metastability in the Navier-Stokes Equations
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Quasi-stationary, or metastable, states play an important role in two-dimensional turbulent fluid flows where they often emerge on time-scales much shorter than the viscous time scale, and then dominate the dynamics for very long time intervals. We give an explanation of the emergence of such time scales for a family of explicit solutions of the two-dimensional Navier-Stokes equations by using Villani's theory of hypercoercivity.
Weber, Hendrik
Approximations to rough Burgers like equations
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We study approximations to a class of vector-valued equations of Burgers type driven by a multiplicative space-time white noise. A solution theory for this class of equations has been developed recently. The key idea was to use the theory of controlled rough paths to give definitions of weak / mild solutions and to set up a Picard iteration argument. In this talk the limiting behaviour of a rather large class of (spatial) approximations to these equations is discussed. These approximations are shown to converge and convergence rates are given, but the limit may depend on the particular choice of approximation. This effect is a spatial analogue to the It^o-Stratonovich correction in the theory of stochastic ordinary differential equations, where it is well known that different approximation schemes may converge to different solutions.
Zelik, Sergey
Infinite-dimensionality in the "finite-dimensional" dissipative dynamics
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We discuss the regularity of the finite-dimensional reduction for dissipative PDEs with finite-dimensional global attractors provided by the so-called Mane' projections. In particular, the examples where these projections are not bi-Lipschitz and even not log-Lipschitz will be presented in the class of abstract semilinear parabolic equations with smooth bounded non-linearities. We also show that, in absence of the spectral gap which guarantees the existence of an inertial manifold, only the Holder continuity of such projections can be expected in general.

Participants

Name	Institution
Beck, Margaret	Heriot-Watt University and Boston University
Boritchev, Alexandre	École Polytechnique
Bufetov, Alexander	Steklov Mathematical Institute & Rice University
Daon, Yair	Weizmann Institute of Science
Gallay, Thierry	Université de Grenoble I
Giles, Will	Imperial College London
Glatt-Holtz, Nathan	University of Indiana
Gottwald, Georg	University of Sydney
Hairer, Martin	University of Warwick
Lu, Kening	Brigham Young University
Matthies, Karsten	University of Bath
Melbourne, Ian	University of Surrey
Mobasheri, Raouf	University of Sussex
Ott, William	University of Houston
Rohani, Saeid	University of Sussex
Savostianov, Anton	University of Surrey
Todd, Mike	University of St Andrews
Turaev, Dmitry	Imperial College London
Wayne, C. Eugene	Boston University
Weber, Hendrik	University of Warwick
Zelik, Sergey	University of Surrey
Zhlobich, Pavel	University of Edinburgh