ICMS

Dissipative PDEs in Bounded and Unbounded Domains and Related Attractors

Sep 20, 2010 - Sep 24, 2010

15 South College Street, Edinburgh EH8 9AA

Organisers

Name	Institution
Bartuccelli, Michele	University of Surrey
Grasselli, Maurizio	Politecnico di Milano
Shirikyan, Armen	Université de Cergy-Pontoise
Zelik, Sergey	University of Surrey

 

The workshop gathers together the leading experts working in the area of dissipative PDEs and related attractors theory. We will discuss the recent achievements in the field and  the development of new approaches to tackle the fundamental open problems of the theory.

Some of the central themes of this workshop will be the following:

-- New results on the existence of global and exponential attractors in bounded domains:
finite and infinite-dimensional attractors for singular and degenerate problems, finite-dimensional reduction  for the non-autonomous  PDEs, trajectory   dynamical systems and trajectory attractors for ill-posed problems.

-- Attractors, exponential attractors and invariant measures for the stochastically perturbed dissipative systems in bounded and unbounded domains.

-- Infinite-energy solutions in unbounded domains.

 

Arrangements

Registration for this workshop is now closed.
The workshop will commence at lunchtime on Monday 20 September and close at 13.00 on Friday 24 September.  

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. 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. Follow this link for a map showing the location of 15 South College Street.  You may also find this map useful for the workshop. 

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.  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 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.

Lothian buses charge £1.20 for a single, £3.00 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.

Accommodation
ICMS has arranged en-suite rooms in accommodation nearby for those who requested this. Accommodation is typically no more than 5-10 minutes walk from the workshop venue. Participants making their own arrangements may claim back the cost, with original receipts, up to a maximum of £50.00 per night bed and breakfast for a maximum of 5 or 6 nights.  A list of Edinburgh accommodation of various sorts and prices is available here (Section 4 is the relevant section for accommodation near to the workshop venue at 15 south College Street).

Catering
Refreshments will be provided during the coffee breaks for the duration of the event.  On Thursday 23 September there will be a workshop dinner at the Magnum Restaurant, 1 Albany Street, Edinburgh, EH1 3PY.   I am pleased to inform you that the workshop dinner will now be free of charge to all participants.  

Wednesday Afternoon Walking Tour
On Wednesday afternoon at 14.00, a group walking tour of Edinburgh’s 'Old Town' has been arranged.  Anyone wishing to participate should gather in the Chapterhouse of 15 South College Street at 13.50.  The tour will leave at 14.00.   Two local guides will show groups various highlights of the 'Old  Town', pointing out places of interest and giving information about places you may wish to explore.

Financial Matters
The workshop grant will cover the cost of bed and breakfast accommodation for all participants and refreshments during the coffee breaks for the duration of the workshop, lunch on Monday only, and a workshop dinner at the Magnum Restaurant on Thursday evening.  

If we have agreed to pay some of your travel costs, you will be informed in the 'Final Information' email.  Reimbursement will take place after the workshop and will involve payment directly into your bank account.  At Registration you will be given an expenses claim form and this should be submitted to ICMS, with original receipts. It would be helpful if you could bring your bank details to the workshop. In addition to the bank account number, participants from the USA and Canada will require their bank’s routing number, those from the UK will be asked for the bank sort code, and those from  the rest of the world, an IBAN and SWIFT/BIC code. We cannot reimburse any item without a signed claim form and original receipts.

Please note that there is NO registration fee payable for this workshop. 

Programme

Monday 20 September 2010

13.00 - 13.55	Registration and lunch in the Chapterhouse, 15 South College Street
13.55 - 14.00	Welcome & introduction
14.00 - 14.40	Anatoli Babine (University of California, Irvine) Derivation of classical and quantum mechanical effects for charged particles from dynamics of PDE    pdf of presentation
14.45 - 15.25	Edriss Titi (University of California, Irvine & Weizmann Institute of Science) Computing slowly advancing features in fast-slow systems without scale separation - a young measure approach   pdf of presentation
15.25 - 15.45	Coffee/Tea in the Chapterhouse (ground floor)
15.45 - 16.25	Eduard Feireisl (Academy of Sciences of the Czech Republic) Asymptotic properties of complete fluid systems  pdf of presentation
16.30 - 17.10	Genevieve Raugel (CNRS and Université Paris-Sud) Genericity of Kupka-Smale and Morse-Smale properties for scalar parabolic equations pdf of presentation
17.10 - 17.55	Mark Groves (Universität des Saarlandes) Existence and stability of fully localised three-dimensional gravity-capillary solitary water waves  pdf of presentation

 

Tuesday 21 September 2010

09.00 - 09.40	John Gibbon (Imperial College London) The dynamics of the gradient of potential vorticity  pdf of presentation
09.45 - 10.25	Koji Ohkitani (University of Sheffield) Dissipative and ideal surface quasi-geostrophic equations
10.25 - 10.45	Coffee/Tea in the Chapterhouse (ground floor)
10.45 - 11.25	Guido Gentile (Università di Roma Tre) Periodic solutions for nonlinear PDEs in higher dimension   pdf of presentation
11.30 - 12.10	James Robinson (University of Warwick) Dimensions, embeddings, and attractors pdf of presentation
12.15 - 12.55	Gregory Seregin (University of Oxford) How does L_3-norm approach potential blow-up?
12.55 - 14.30	Lunch break
14.30 - 15.10	Peter Constantin (University of Chicago) Oldroyd-B systems   pdf of presentation
15.15 - 15.55	Ian Roulstone (University of Surrey) A geometric description of Navier-Stokes flows
15.55 - 16.15	Coffee/Tea in the Chapterhouse (ground floor)
16.15 - 16.55	Giovanni Gallavotti (University of Roma) Thermostats models and related PDE's   pdf of presentation
17.00 - 17.40	Marcel Oliver (Jacobs University Bremen) Non-dissipative regularization of first order balance models   pdf of presentation

 

Wednesday 22 September 2010

09.00 - 09.40	Sergei Kuksin (CMLS, École Polytechnique) Damped and driven KdV equation
09.45 - 10.25	Marco Romito (Università di Firenze) Analysis of a model for amorphous surface growth  pdf of presentation
10.25 - 10.45	Coffee/Tea in the Chapterhouse (ground floor)
10.45 - 11.25	Arghir Zarnescu (University of Oxford) Energy dissipation, regularity and statistical dynamics for a nematic liquid crystal flow pdf of presentation
11.30 - 12.10	Ludovic Goudenege (ENS de Cachan - Antenne de Bretagne) Stochastic Cahn-Hilliard equation with singularities  pdf of presentation
12.15 - 12.55	Lihu Xu (EURANDOM) Exponentially mixing of stochastic 3D Navier-Stokes equations driven by mildy degenerate noises   pdf of presentation
12.55	Lunch break and discussions
13.50 - 16.00 approximately	Walking Tour of 'Edinburgh's 'Old Town' This tour is optional.  Please gather in the Chapterhouse around 13.50 if you wish to attend.  Guides will show you highlights of the 'Old Town'.

Thursday 23 September 2010

09.00 - 09.40	Ilia Kamotski (University of Bath) On nonexistence of Baras-Goldstein type
09.45 - 10.25	Igor Kukavica (University of Southern California) Complexity of solutions for parabolic equations with Gevrey coefficients pdf of presentation
10.25 - 10.45	Coffee/Tea in the Chapterhouse (ground floor)
10.45 - 11.25	Irena Lasiecka (University of Virginia) Global Existence and asymptotiic behavior in fully nonlinear thermoelasticity pdf of presentation
11.30 - 12.10	Vittorino Pata (Politecnico di Milano) A new theoretical scheme for the analysis of equations with memory  pdf of presentation
12.15 - 12.55	Vladimir Chepyzhov (IITP, Moscow) Strong and weak trajectory attractors for ill-posed problems  pdf of presentation
12.55 - 14.30	Lunch break
14.30 - 15.10	Alp Eden (Bosphorous University) On Cahn-Hilliard and Convective Cahn-Hilliard equations
15.15 - 15.55	Giulio Schimperna (University of Pavia) On a fourth order degenerate parabolic equation  pdf of presentation
15.55 - 16.15	Coffee/Tea in the Chapterhouse (ground floor)
16.15 - 16.55	Alexei Ilyin (Russian Academy of Sciences) Berezin-Li-Yau bounds with additional term for the Stokes operator and the Dirichlet Laplacian   pdf of presentation
17.00 - 17.40	Stefania Gatti (Università degli Studi di Modena e Reggio Emilia) A globally continuous family of exponential attractors for singularly perturbed problems pdf of presentation
19.00	Workshop Dinner at the Magnum Restaurant, 1 Albany Street, Edinburgh

 

Friday 24 September 2010

09.00 - 09.40	Gautam Iyer (Carnegie Mellon University) Large exit times of diffusions with incompressible drift  pdf of presentation
09.45 - 10.25	Pedro Marin-Rubio (Universidad de Sevilla) On the existence of pullback mathcal{D} -attractors for non-autonomous 2D-Navier-Stokes equations with and without delays
10.25 - 10.45	Coffee/Tea in the Chapterhouse (ground floor)
10.45 - 11.25	Dalibor Prazak (Charles University, Prague) Attractors for nonlinear parabolic problems in unbounded domains  pdf of presentation
11.30 - 12.10	Varga Kalantarov (Koc University) Attractor for the Brinkman-Forchheimer equations   pdf of presentation
12.15 - 12.55	Alain Miranville (Université de Poitiers) The Cahn-Hilliard equation with dynamic boundary conditions  pdf of presentation
12.55	Close of workshop

 

Presentations:

Presentation Details
Babine, Anatoli
Derivation of classical and quantum mechanical effects for charged particles from dynamics of PDE
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There are well-known problems in classical electrodynamics when it is applied to atomic scales. One of ways to solve the problems is through introduction of quantum mechanics. A different approach, which remains in the classical framework, is based on replacing point charges by distributed charge fields with dynamics governed by nonlinear Schrodinger or nonlinear Klein-Gordon equations and coupled with Maxwell equations for electromagnetic fields. In the talk I will explain how Newton's equations with Lorentz forces for point charges are derived from the field equations, how discrete hydrogen spectrum is derived for proton-electron system and how Einstein formula E=mc² can be derived. In conclusion I describe a system of coupled nonlinear Schrodinger and damped wave equations which has interesting mathematical properties and can be used to model transitions between hydrogen states.
Chepyzhov, Vladimir
Strong and weak trajectory attractors for ill-posed problems
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The method of trajectory attractors helps to study effectively the behaviour of solutions of non-linear partial differential equations for which the corresponding boundary-value problem is ill-posed or it well-posedness iis not proved yet. At the same time, as a role, for meaningful equations and problems, we were able to establish the attraction to the corresponding trajectory attractors in a ''maximal'' possible weak topology of the problem we are dealing with. In the present report, for a general dissipative reaction-diffusion system, we prove that its trajectories (global solutions) tend to the trajectory attractor in the ''maximal'' strong topology of the corresponding trajectory phase space. The considered RD-system is ill-posed since it does not satisfy conditions that provide the unique solvability of the corresponding Cauchy problem.
Constantin, Peter
Oldroyd-B systems
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I will discuss global existence, uniqueness and stability for Oldroyd-B systems and related models. The models resemble damped and driven 3D Euler equations with variable density. I will show uniqueness and stability in C^{alpha}cap L^1, a natural space. I will show how the kinetic description allows to introduce a natural non-diffusive regularization. I will give an example of a baby model where a whole cone of large initial data leads to global existence.
Eden, Alp
On Cahn-Hilliard and Convective Cahn-Hilliard equations
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I will present some recent results, joint work with V. Kalantarov and S. Zelik, on the asymptotic properties of the convective Cahn-Hilliard equations on bounded domains and of the Cahn-Hilliard equations on cylindirical domains.
Feireisl, Eduard
Asymptotic properties of complete fluid systems
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We discuss certain asymptotic properties of complete fluid systems. By complete we mean energetically isolated and positive entropy production.
Gallavotti, Giovanni
Thermostats models and related PDE's
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Nonequilibrium staionary states are described in statistical mechanics by a fa mily of PDE's called the BBGKY hierarchy. I plan to illustrate some of their solutions and put them into the context of the microscopic proof of the Fourier's law.
Gatti, Stefania
A globally continuous family of exponential attractors for singularly perturbed problems
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To view abstract follow: http://www.icms.org.uk/downloads/dissipative/Gatti_Abstract.pdf
Gentile, Guido
Periodic solutions for nonlinear PDEs in higher dimension
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I shall discuss recent results on the existence of periodic solutions in a class of nonlinear partial differential equations, including the NLS, the NLW and the NLB equations, in higher dimension. These results include cases of completely resonant equations, where the bifurcation equation is infinite-dimensional, for which there exist periodic solutions which at leading order are wave packets.
Gibbon, John
The dynamics of the gradient of potential vorticity
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In geophysical fluid dynamics (GFD) one of the most important quantities is potential vorticity (PV) defined by $q = mbox{boldmath$omega$} cdotnablatheta$ where $mbox{boldmath$omega$}$ is the vorticity and $theta$ the potential temperature. In non-dissipative flows $q$ is conserved. However, little is understood about the dynamics of $nabla q$ and $nablatheta$ which is essential for an understanding of how PV accumulates in the oceans and the atmosphere. In joint work with Darryl Holm this talk reports on an investigation of the transport of $nabla{q}$ along surfaces of constant potential temperature $theta$ in the stratified Euler, Navier-Stokes and the hydrostatic primitive equations of the oceans and atmosphere, in terms of the divergence-less flux vector $mbox{boldmath$mathcal{B}$} = nabla Q(q)timesnablatheta$, for any smooth function $Q$ of $q$. The flux vector $mbox{boldmath$mathcal{B}$}$ is shown to satisfy a transport equation $$ partial_{t}mbox{boldmath$mathcal{B}$} - mbox{curl}, mbox{boldmath$mathcal{U}$} timesmbox{boldmath$mathcal{B}$}) = - nablabig[qQ'(q),mbox{div},mbox{boldmath$mathcal{U}$}big]timesnablatheta,,$$ where $mbox{boldmath$mathcal{U}$}$ is a formal transport velocity. While the left hand side of this expression is reminiscent of that for magnetic field flux in magnetohydrodynamics, the non-zero right hand side means that $mbox{boldmath$mathcal{B}$}$ is not frozen into the flow when $mbox{div} ,mbox{boldmath$mathcal{U}$} neq 0$. The result may apply to analyzed data for potential vorticity and potential temperature at the tropopause.
Goudenege, Ludovic
Stochastic Cahn-Hilliard equation with singularities
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The Cahn-Hilliard equation is a fourth-order PDE (no maximum principle) with a singular potential in -1 and +1. In the deterministic case, there exist compact attractors and various stationary states. Moreover we can study the first bifurcations on various bounded domains. In the stochastic case, there are no more stationary states, but the equation has a unique invariant measure. This measure is ergodic and we show some exponential mixing results.
Groves, Mark
Existence and stability of fully localised three-dimensional gravity-capillary solitary water waves
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A solitary wave of the type advertised in the title is a critical point of the wave energy, which is given in dimensionless coordinates by $$H(eta,xi) = int_{{mathbb R}^2} left{frac{1}{2}xi G(eta)xi + frac{1}{2}eta^2+ beta sqrt{1+eta_x^2+eta_z^2} - beta right},$$ subject to the constraint that the wave impulse $$I(eta,xi) = int_{{mathbb R}^2} eta_x xi$$ is fixed. Here $eta(x,z)$ is the free-surface elevation, $xi$ is the trace of the velocity potential on the free surface, $G(eta)$ is a Dirichlet-Neumann operator and $beta>1/3$ is the Bond number. In this talk I show that there exists a minimiser of $H$ subject to the constraint $I=2mu$, where $0<mu ll 1$. The existence of a solitary wave is thus assured, and since $H$ and $I$ are both conserved quantities its stability follows by a standard argument. `Stability' must however be understood in a qualified sense due to the lack of a global well-posedness theory for three-dimensional water waves.
Ilyin, Alexei
Berezin-Li-Yau bounds with additional term for the Stokes operator and the Dirichlet Laplacian
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We prove Berezin-Li-Yau-type lower bounds with additional term for the eigenvalues of the Stokes operator and improve the previously known estimates for the Dirichlet Laplacian. Generalizations to higher-order operators and applications to the attractors of the Navier-Stokes equations are given.
Iyer, Gautam
Large exit times of diffusions with incompressible drift
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Let Omega be a bounded domain in R^n. For x in Omega, let tau(x) be the expected exit time from Omega of a diffusing particle starting at x and advected by an incompressible flow u. The question I will address is which textit{incompressible} flows maximise tau. That is, which flows u are most efficient in the creation of hot spots. Surprisingly, among all simply connected domains in two dimensions, the discs are the only ones for which the zero flow u = 0 maximises |tau|_{L^infty}. A study of such flows leads very quickly to a peculiar non-linear PDE for which we are unable to prove anything. This is joint work with Alexei Novikov, Lenya Ryzhik, and Andrej Zlatos.
Kalantarov, Varga
Attractor for the Brinkman-Forchheimer equations
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We prove the existence of regular solutions and global attractors for the 3D Brinkmann-Forchheimer equations with the nonlinearity of an arbitrary polynomial growth rate. In order to obtain this result, we prove the maximal regularity estimate for the corresponding semi-linear stationary Stokes problem. The applications of our results to the Brinkmann-Forchheimer equations with the Navier-Stokes inertial term are also considered.
Kamotski, Ilia
On nonexistence of Baras-Goldstein type
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The celebrated result by Baras and Goldstein (1984) established that the heat equation with singular inverse square potential in a smooth bounded domain in the supercritical range does not have a solution for any nontrivial nonnegative initial data. Namely, it was proved that a regular approximation of a possible solution by a sequence of classical solutions of uniformly parabolic equations with bounded truncated potentials diverges. In the present talk, we reveal the connection of this very "singular" evolution with a spectrum of some "limiting" operator. The proposed approach allows us to consider more general higher-order operators and initial data that are not necessarily positive. In particular it is established that, under some natural hypothesis, the divergence result is valid for any $2m$th-order parabolic equation with singular potential and for a wide class of initial data. (This is joint work with V Galaktionov).
Kukavica, Igor
Complexity of solutions for parabolic equations with Gevrey coefficients
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We provide a quantitative estimate of unique continuation for high order parabolic equations with non-analytic Gevrey coefficients. We also provide a new upper bound for the number of spatial oscillations of solutions with a bound depending algebraically on coefficients. (This is a joint work with Mihaela Ignatova).
Kuksin, Sergei
Damped and driven KdV equation
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In my talk I will discuss properties of long-time behaviour of solutions for the KdV eqaution under periodic boundary conditions, perturbed by small dissipation and, maybe, a random force.
Lasiecka, Irena
Global Existence and asymptotiic behavior in fully nonlinear thermoelasticity
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We shall consider fully nonlinear thermoelastic system comprising of nonlinear plate equation coupled with a heat equation. Two cases will be considered: depending whether rotational moments are included or not. In the case ehen these terms are not accounted in teh equation, teh model is of parabolic type with a fully nonlinear terms. Well-posedness and asymptotic behavior are treated by means of abstract parabolic theory developed by Lunardi. In the second case,, which is of hypperbolic nature, new sets of estimates are established that lead to global existence of solutions. The argument depends critically on the presence of thermal dissipation that is propagated into the nonlinear plate.
Marin-Rubio, Pedro
On the existence of pullback mathcal{D} -attractors for non-autonomous 2D-Navier-Stokes equations with and without delays
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In this talk we will present some recent results on the existence and properties of pullback mathcal{D} -attractors for a two-dimesional Navier-Stokes model with non-autonomous force term, where the universe of attracted families mathcal{D} is given by a tempered condition. If possible, we will discuss on some related problems including delay effects. (Partially done in collaboration with Julia Garcia-Luengo and Jose Real)
Miranville, Alain
The Cahn-Hilliard equation with dynamic boundary conditions
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Our aim in this talk is to discuss issues (well-posedness, asymptotic behaviour) related with the Cahn-Hilliard equation with dynamic boundary conditions. Such boundary conditions have recently been proposed in order to account for the wall effects in confined systems. In particular, we are interested in the case of thermodynamically relevant logarithmic potentials.
Ohkitani, Koji
Dissipative and ideal surface quasi-geostrophic equations
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We consider 2D surface quasi-geostrophic (SQG) equations both for dissipative and ideal cases. For the dissipative case, we present some numerical results using hypo-viscosity, that is, a fractional power of Laplacian. Global regularity has been proved if the power is greater than, or equal to 1/2 (Kiselev et al. 2007). We study both super- and sub-critical parameter ranges and have found a long-term damping process with oscillation. They are consistent with global regularity of 2D SQG equation, but may be an obstacle in mathematical analysis. (with T. Sakajo). For the ideal (inviscid) case, we consider a generalised class of equations, where the active scalar is a fractional power Laplacian applied to the stream function. This includes 2D SQG and 2D Euler equations. We present some numerical results of the generalised system and compare them for some different powers. A successive approximation is introduced to treat the SQG equation, which may help to understand the whole family in a systematic manner.
Oliver, Marcel
Non-dissipative regularization of first order balance models
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Non-dissipative regularizations have been used in PDE theory at least since Leray's work on the Navier-Stokes equations. We present a setting in which such terms show up naturally as first order terms in the context of balance models for rotating fluids, and present new analytical and numerical results on this class of models.
Pata, Vittorino
A new theoretical scheme for the analysis of equations with memory
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We discuss a novel approach to the mathematical analysis of equations with memory, based on the notion of state. This is the configuration of the system at the initial time which can be unambiguously determined by the knowledge of the dynamics in the future. As a model, an integro-differential equation arising from viscoelasticity is considered.
Prazak, Dalibor
Attractors for nonlinear parabolic problems in unbounded domains
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We study the long time behavior of solutions to nonlinear parabolic equations in unbounded domains. We propose to show the existence of a global attractor and estimate its Kolmogorov entropy, using the method of trajectories and suitable estimates with Bochner spaces with spatial weights. (Joint work with M Grasselli and G Schimperna)
Raugel, Genevieve
Genericity of Kupka-Smale and Morse-Smale properties for scalar parabolic equations
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In this talk, we first show that, generically with respect to the non-linearity, the semi-flow of a reaction-diffusion equation in S^1 has the Morse-Smale property, that is, all the critical elements (i.e. the equilibria and periodic orbits) are hyperbolic, the stable and unstable manifolds of the critical elements intersect transversally and the non-wandering set is a finite set only composed of critical elements. Afterwards, we show that, generically with respect to the non-linearity, the semi-flow of a reaction-diffusion equation on a bounded domain in {bf R}^n, n geq 1, has the "Kupka-Smale" property.
Robinson, James
Dimensions, embeddings, and attractors
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I will discuss approaches to reproducing the dynamics on finite-dimensional attractors using finite-dimensional dynamical systems. When the attractor has finite box-counting dimension, it has a homemorphic image X in {mathbb R}^k (for k sufficiently large), and one can reproduce the dynamics of the original time T map using a homeomorphism h on {mathbb R}^k, with the property that the attractor of h is arbitrarily close to X. Much stronger assumptions (on the "Assouad dimension" of the attractor) appear to be required if one wishes to do the same with with a flow. The arguments combine powerful embedding theorems with topological results characterising those sets that can be attractors of flows in finite-dimensional spaces. This is joint work with Eleonora Pinto de Moura and Jaime Sánchez-Gabites (both currently at Warwick).
Romito, Marco
Analysis of a model for amorphous surface growth
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We consider a semilinear fourth order equation arising in surface growth caused by epitaxy or sputtering, for which so far well posedness is an open problem. In the first part of the talk we give a complete analysis of the one dimensional problem forced by space-time white noise in the framework of Markov solutions, proving unique ergodicity and convergence to a unique invariant state. In the second part we analyse the unforced case and give conditions for the emergence of blow up. Finally we briefly introduce the two dimensional problem, which corresponds to the physical case, and give a few preliminary existence results.
Roulstone, Ian
A geometric description of Navier-Stokes flows
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The pressure in the incompressible three-dimensional Navier-Stokes and Euler equations is governed by a Poisson equation and the Laplacian of the pressure is related to the second invariant of the velocity gradient tensor: these relations are studied using the geometry of three-forms in six dimensions. When the Laplacian of the pressure is positive, the three-form associated with the Poisson equation is the real part of a decomposable complex form and an almost-complex structure can be identified. When the Laplacian of the pressure is negative, a real decomposable structure is identified. These results are discussed in the context of coherent structures.
Schimperna, Giulio
On a fourth order degenerate parabolic equation
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I will present some existence, regularity, and long-time behaviour results for a fourth order parabolic equation related to the evolution of thin films. Applications to the Cahn-Hilliard model with degenerate mobility are also discussed. More precisely, after showing existence of at least one (weak) solution by means of an approximation - a priori estimates - passage to the limit argument, I will analyze the generalized dynamical process associated to the equation and prove the existence of a weak trajectory attractor. In case a viscosity term is added and slightly more restrictive conditions on the nonlinearities are assumed, this trajectory attractor can be intended with respect to the strong phase-space topology. Finally, I will present sufficient conditions for having strict positivity of the solution, entailing in particular uniqueness. The result presented in the talk have been obtained in collaboration with Sergey Zelik (university of Surrey).
Seregin, Gregory
How does L_3-norm approach potential blow-up?
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In the talk, there will be discussed the behaviour of the spatial L_3-norm of solutions to the Navier-Stokes equations as time goes to a potential blow-up time.
Titi, Edriss
Computing slowly advancing features in fast-slow systems without scale separation - a Young measure approach
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In the first part of the talk, and in order to set the stage, we will offer a multi-scale and averaging strategy to compute the solution of a singularly perturbed system when the fast dynamics oscillates rapidly; namely, the fast dynamics forms cycle-like limits which advance along with the slow dynamics. We describe the limit as a Young measure with values being supported on the limit cycles, averaging with respect to which induces the equation for the slow dynamics. In particular, computing the tube of the limit cycles establishes a good approximation for arbitrarily small singular parameters. We will demonstrate this by exhibiting concrete numerical examples. In the second part of the talk we will examine singularly perturbed systems which may not possess a natural split into fast and slow state variables. Once again, our approach depicts the limit behaviour as a Young measure with values being invariant measure of the fast contribution to the flow. These invariant measures are drifted by the slow contribution to the value. We keep track of this drift via slowly evolving observables. Averaging equations for the latter lead to computation of characteristic features of the motion and the location the invariant measures. To demonstrate our ideas computationally, we will present some numerical experiments involving a system derived from a spatial discretization of a Korteweg-de Vries-Burgers type equation, with fast dispersion and slow diffusion. (This is a joint work with Z Artstein, W Gear, I Kevrekidis, J Linshiz and M Slemrod).
Xu, Lihu
Exponentially mixing of stochastic 3D Navier-Stokes equations driven by mildy degenerate noises
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We shall study stochastic 3D Navier-Stokes equation driven by mildly degenerate noises and prove that one can select a Markov process from the SPDEs by two methods. By a coupling method, one can also show that the Markov process is exponentially mixing. This is joint work with S Albeverion, A Debussche and M Romito.
Zarnescu, Arghir
Energy dissipation, regularity and statistical dynamics for a nematic liquid crystal flow
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We study a complex non-newtonian fluid that models the flow of nematic liquid crystals. The fluid is described by a system that couples a forced Navier-Stokes system with a parabolic-type system. In the first part of the talk we prove the existence of global weak solutions in dimensions two and three, in the whole space. We show the existence of a Lyapunov functional for the smooth solutions of the coupled system and use the cancellations that allow its existence to prove higher global regularity, in dimension two. We also show the uniqueness in dimension two. In the second part of we consider a simplified version of the system, in which the Navier-Stokes part is neglected, and study statistical dynamics, in particular the validity and limitations of the statistical scaling hypothesis.

Participants

Name	Institution
Anthony, Peter	Universxity of Surrey
Babine, Anatoli	University of California, Irvine
Bartuccelli, Michele	University of Surrey
Chepyzhov, Vladimir	IITP, Moscow
Constantin, Peter	University of Chicago
Eden, Alp	Bosphorous University
Feireisl, Eduard	Academy of Sciences of the Czech Republic
Gallavotti, Giovanni	University of Roma
Gatti, Stefania	Università degli Studi di Modena e Reggio Emilia
Gentile, Guido	Università di Roma Tre
Gibbon, John	Imperial College London
Goudenege, Ludovic	ENS de Cachan - Antenne de Bretagne
Grasselli, Maurizio	Politecnico di Milano
Groves, Mark	Universität des Saarlandes
Ilyin, Alexei	Russian Academy of Sciences
Iyer, Gautam	Carnegie Mellon University
Kalantarov, Varga	Koc University
Kamotski, Ilia	University of Bath
Kukavica, Igor	University of Southern California
Kuksin, Sergei	CMLS, École Polytechnique
Lasiecka, Irena	University of Virginia
Marin-Rubio, Pedro	Universidad de Sevilla
Miranville, Alain	Université de Poitiers
Ohkitani, Koji	University of Sheffield
Oliver, Marcel	Jacobs University Bremen
Pata, Vittorino	Politecnico di Milano
Patni, Kavita	University of Surrey
Pennant, Jon	University of Surrey
Prazak, Dalibor	Charles University, Prague
Raugel, Genevieve	CNRS and Université Paris-Sud
Robinson, James	University of Warwick
Romito, Marco	Università di Firenze
Roulstone, Ian	University of Surrey
Schimperna, Giulio	University of Pavia
Sengul, Yasemin	University of Oxford
Seregin, Gregory	University of Oxford
Shirikyan, Armen	Université de Cergy-Pontoise
Titi, Edriss	University of California, Irvine and The Weizmann Institute of Science
Vanneste, Jacques	University of Edinburgh
Watson, Stephen	University of Glasgow
Wilkinson, Mark	University of Oxford
Xu, Lihu	EURANDOM
Zarnescu, Arghir	University of Sussex
Zelik, Sergey	University of Surrey