ICMS

Jul 16, 2007 - Jul 20, 2007

David Hume Tower, University of Edinburgh

Organisers

Name	Institution
Cullen, Mike	Meteorological Office

Scientific Organisers
Professor Luis Caffarelli, University of Texas at Austin
Dr Mike Cullen, Meteorological Office
Professor L Craig Evans, University of California, Berkeley
Professor Mikhail Feldman, University of Wisconsin-Madison
Professor Wilfred Gangbo, Georgia Institute of Technology
Professor Robert McCann, University of Toronto
Dr Ian Roulstone, University of Surrey

Short Report

The purpose of the meeting was to bring together experts in two areas of mathematics, namely mass transportation and geometry, with experts in geophysical fluid dynamics. Geophysical dynamics seeks to understand the evolution of the atmosphere and oceans, which is fundamental to weather and climate prediction. It does this by studying systems of equations which accurately model the large-scale behaviour of the solutions, rather than the small-scale turbulent motions. Mass transportation theory was originally developed as a way of solving optimisation problems in the areas of operations research, probability and statistics. It has more recently found a much wider range of applications, such as network design, economics, medical imaging and reflector design. A wide variety of other application areas were discussed at the workshop, such as biological aggregation, plasma physics, aerodynamics and relativistic physics. In particular, it has been shown that it can be applied to fluid dynamical problems, for instance those governing the large-scale behaviour of the atmosphere and oceans. Mass transportation theory can also be given a geometrical interpretation, which has led to important extensions in its applicability.

This links to fluid dynamics through the identification of incompressible flows as geodesics on an appropriate space. Recent work suggests that the classical definition of such flows has to be relaxed in order to ensure that they can be proved to exist. Recent results were described which show that mass transportation methods may be more successful in curved spaces, such as the surface of a sphere, than in at spaces. The meeting contained a significant training element with Rick Salmon giving a mini-course on geophysical fluid dynamics and Felix Otto giving a mini-course on mass transportation methods. Several talks and posters were presented by graduate students and beginning researchers.

Participants list and links to available presentations are further down this page.

Download the pdf file of the full report

Original Details

Optimal transportation is a mathematically rigorous method for solving a wide class of nonlinear optimization problems, and has been applied successfully to problems in economics, statistics, image matching, cosmology and geophysics. It has also been used to prove results in geometry and partial differential equations (PDE). A review of the method and many of the applications is given in the book by Villani (2003).

This workshop has three main themes:

1. To review the links between optimal transportation and geophysical fluid dynamics. These are best developed in the context of the semi-geostrophic model, which describes large-scale flows in the atmosphere and ocean (Cullen (2006)), and in the anelastic model, which is mathematically equivalent to an incompressible model and describes small scale buoyancy-driven flows. There are also a wide variety of other models used to study the different regimes present in geophysical flows, and there are potentially new applications in these areas. (Norbury and Roulstone (2002), Majda (2003))
2. To review the links between optimal transportation theory and geometry. Areas of recent progress include application of optimal transportation on manifolds, which widens the application area considerably, geometrical aspects of the theory of elliptic Monge-Ampere equations, and study of differential equations on spaces of probability measures.
3. A broader survey of problems and developments surrounding optimal transportation and fluid mechanical models.

References

Ambrosio, L., Gigli, N. and Savare, G. (2005) Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lecture in Mathematics, ETH Zurich, Birkhauser.

Cullen, M. J. P. (2006) A Mathematical Theory of Large-Scale Atmospheric Flow. Imperial College Press.

Majda, A. J. (2003) Introduction to PDEs and waves for the atmosphere and ocean. Courant Lecture Notes, 9, American Math. Society.

Norbury, J. and Roulstone, I. eds. (2002) Large-Scale Atmosphere-Ocean Dynamics., vols.1 and 2. Cambridge University Press.

Villani, C. (2003) Topics in optimal transportation. Vol. 58 of Graduate Studies in Mathematics, Amer.Math. Soc., Providence, RI.

Arrangements

Venue
The Workshop will be held in the David Hume Tower, University of Edinburgh. The Conference Rooms are equipped with OHPs, data projectors and white boards.

Registration
There will be a combined Registration and Welcome Buffet from 18.30 to 21.00 on Sunday 15 July, in the St Trinnean's Room, St Leonard's Hall, Pollock Halls. Food will be available throughout that period and you may register at any time between 18.30 and 21.00.
Those who cannot register on Sunday evening may do so at the workshop venue from 08.30-09.00 before the talks start on Monday morning.

Accommodation
ICMS has arranged en suite rooms in university accommodation in Pollock Halls for senior participants who require it, with standard accommodation for students. Participants who are being funded are also free to make their own arrangements and may claim back the cost, with receipts, up to a maximum of £45 per night for bed and breakfast. A list of Edinburgh accommodation of various sorts and prices can be found on the Accommodation Page on the ICMS web site. Section 4 is particularly relevant. July is very busy in Edinburgh and early booking is recommended.

Meals and Refreshments
There will be a Registration buffet in the St Trinneans Room in St Leonards, Pollock Halls, (near the Reception Centre – see map) on Sunday evening and a Workshop dinner on Thursday evening in the St Trinnean's Room. Morning and afternoon refreshments will be provided at the workshop venue.

Financial support available
We expect to be able to contribute about £100 to the travel costs of UK invited attendees, £250 for those based in Northern Europe, £350 for those based in southern Europe, £600 for those based in the USA and £750 for those based in the rest of the world.

Further information will be provided in the Invitation Letter.

Programme

Sunday 15 July

18.30-19.00

Registration & buffet, St Trinnean's Room, St Leonard's, Pollock Halls

Monday 16 July

08.30-09.00	Registration
09.00-10.00	Robert Douglas (University of Wales Aberystwyth) Introduction to mass transportation and applications to geophysics
10.05-11.05	Rick Salmon (Scripps Institution of Oceanography) Mini-course on geophysical fluid dynamics (1)
11.05-11.35	Coffee Break
11.35-12.20	Guiseppe Buttazzo (University of Pisa) Long-term planning versus short-term planning in location problems Presentation
12.20-14.30	Lunch Break
14.30-14.45	Marjolaine Puel (Université Paul Sabatier) A mass transport approach for the relativistic heat equation Presentation
14.50-15.05	Yann Brenier (CNRS Nice/Paris VI) Polymers and optimal transportation
15.10-15.25	Gordon Blower (University of Lancaster) Young diagrams and concentration Presentation
15.25-16.05	Coffee Break
16.05-16.50	Wilfred Gangbo (Georgia Institute of Technology) Euler-Poisson systems as action-minimizing paths in the Wasserstein space
16.55-17.40	Eric Carlen (Georgia Institute of Technology) Some new displacement convex functionals and some variations on displacement convexity

Tuesday 17 July

09.00-10.00	Rick Salmon (Scripps Institution of Oceanography) Mini-course on geophysical fluid dynamics (2)
10.05-10.50	José Carrillo (Universitat Autònoma de Barcelona) The Keller-Segel model: entropy, geometric inequalities and mass transport
10.50-11.20	Coffee Break
11.20-12.05	Aldo Pratelli (University of Pavia) A quantitative version of the Sobolev inequality
12.10-12.25	Adrian Tudorascu (Georgia Institute of Technology) Pressureless Euler/Euler-Poisson systems via adhesion dynamics and scalar conservation laws Presentation
12.25-14.00	Lunch Break
14.00-14.30	Poster Session
14.30-14.45	Bob Peeters (University of Twente) Hamiltonian-based numerical methods for forced-dissipative climate prediction Presentation
14.50-15.05	Francisco Gancedo (NRC Madrid) Two contour dynamics problems in incompressible flows: the Muskat problem and the QG sharp front Presentation
15.10-15.25	Maria del Mar Gonzalez (Universidad Politecnica de Catalunya) Fractional powers of the Laplacian and the geometry of phase transition models Presentation
15.30-15.45	Dorian Goldman (University of Toronto) Chaotic response of the 2D semi-geostrophic equations to gentle periodic forcing Presentation
15.50-16.05	Paul Lee (University of Toronto) Hamiltonian Reductions relating to optimal mass transport Presentation
16.05-16.35	Coffee Break and Poster Session
16.35-17.20	Alexander Plakhov (University of Aveiro) Shapes of optimal aerodynamic resistance and optimal mass transportation Presentation
17.25-18.10	Dejan Slepcev (Carnegie Mellon University) Interfacial aspects of biological aggregation Presentation

Wednesday 18 July

09.00-10.00	Rick Salmon (Scripps Institution of Oceanography) Mini-course on geophysical fluid dynamics (3)
10.05-10.50	Luigi Ambrosio (Scuola Normale Superiore di Pisa) Variational models for incompressible Euler equations Presentation
10.50-11.20	Coffee Break
11.20-12.05	Jacques Vanneste (University of Edinburgh) Two-dimensional Euler flows in slowly deforming domains: adiabatic invariance and geometric angle Presentation
12.10-12.25	Helena Lopes (University of Campinas) Weak stability in L^1 of Lagrangian solutions to the semigeostrophic equations
12.30-12.45	Milton Lopes-Filho (University of Campinas) On potential vorticity sheets for semigeostrophy
12.45-14.45	Lunch Break
14.45-15.30	Phil Morrison (University of Texas at Austin) Statistical mechanics of fluid and plasma continua Presentation
15.35-16.05	Coffee Break
16.05-16.50	Diogo Gomez (IST Lisbon) Linear programming and homogenization of Hamilton-Jacobi equations
16.55-17.40	Marcel Oliver (Jacobs University Bremen) Transparent boundary conditions as dissipative subgrid closures Presentation

Thursday 19 July

09.00-10.00	Felix Otto (Universität Bonn) Mini-course on some applications of optimal transportation (1)
10.05-10.50	Guiseppe Savare (University of Pavia) Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds Presentation
10.50-11.20	Coffee Break
11.20-11.35	Philippe Delanoë (CNRS) On the "staying away from cut-locus" issue in smooth optimal transport
11.40-12.25	Gregoire Loeper (Université Claude Bernard Lyon 1/BNP Paribas) Continuity of maps solutions of optimal transportation problems
12.30-14.35	Lunch Break
14.30-15.30	Felix Otto (Universität Bonn) Mini-course on some applications of optimal transportation (2)
15.30-16.00	Coffee Break
16.00-16.45	Young-Heon Kim (University of Toronto) Curvature and the continuity of optimal transportation maps Presentation
16.50-17.35	Volodya Roubtsov (Université d'Angers) Geometries and generalized geometries beyond Monge-Ampère Presentation
19.30	Workshop Dinner, St Trinnean's Room, St Leonards

Friday 20 July

09.00-10.00	Felix Otto (Universität Bonn) Mini-course on some applications of optimal transportation (3)
10.05-10.50	Vladimir Zeitlin (École Normale Supérieure) Lagrangian approach to catastrophic adjustment and symmetric instability Presentation
10.50-11.20	Coffee Break
11.20-12.05	Boris Khesin (University of Toronto) Geometry of diffeomorphism groups: shocks of the Burgers equation and non-holonomic mass transport
12.10-12.55	Robert McCann (University of Toronto) Ricci flow, entropy, and optimal transportation

Rick Salmon's Mini-course (NB - very large PDF file)

Felix Otto's Mini-course

Posters
Posters will be displayed all day on Tuesday 17 in the area outside the lecture theatre. The authors will be beside their posters from 14.00-14.30 and 16.05-16.35.

de Oliveira, Josiane
Weak stability in L^1 of Lagrangian solutions to the semigeostrophic equations

Dubinkina, Svetlana
Statistical mechanics of Arakawa's Discretizations

Goldman, Dorian
Chaotic response of the 2D semi-geostrophic equations to gentle periodic forcing

Lanagan, Gareth
Weather forecast error decomposition

Trokhimtchouk, Maxim
Optimal partition of a large labor force into working pairs

Presentations:

Presentation Details
Ambrosio, Luigi
Variational models for incompressible Euler equations
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We study Brenier's variational models for incompressible Euler equations. These models give rise to a relaxation of the Arnold distance in the space of measure-preserving maps and, more generally, measure-preserving plans. We analyze the properties of the relaxed distance, we show a close link between the Lagrangian and the Eulerian model, and we derive necessary and sufficient optimality conditions for minimizers. These conditions take into account a modified Lagrangian induced by the pressure field. Moreover, adapting some ideas of Shnirelman, we show that, even for non-deterministic final conditions, generalized flows can be approximated in energy by flows associated to measure-preserving maps.
Blower, Gordon
Young diagrams and concentration
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This talk contains results about concentration and fluctuations for Plancherel measure on Young diagrams with N boxes as N becomes large.
Brenier, Yann
Polymers and optimal transportation
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We show how optimal transportation based evolution equations (in particular the Vlasov-Monge-Ampere system) occur when modelling polymers by springs interacting with an incompressible ideal fluid.
Buttazzo, Guiseppe
Long-term planning versus short-term planning in location problems
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Planning an economic activity is in general an extremely complex problem, where a high number of parameters often intervene. In addition, the attitude of the planner has to be taken into account: long-term planners take their decision through an optimization process over a large time horizon, while short-term planners behave optimizing day-by-day their strategies. Usually, the first kind of behavior is perceived as more virtuous and efficient, while the second is seen as easier to implement. We analyze the long-term and short-term strategies on a very simple mass transportation problem, the so-called "location problem".
Carrillo, José A
The Keller-Segel model: entropy, geometric inequalities and mass transport
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In this talk, the long-time asymptotics of the Keller-Segel model will be discussed for the critical case using entropy-based arguments. The basic inequality implying these estimates is the so-called logarithmic Hardy-Littlewood-Sobolev inequality. We will show how Log-HLS allows to prove that the JKO scheme is convergent for subcritical masses. One-dimensional numerical simulations will demonstrate that this scheme numerically captures the finite time blow-up of solutions for super-critical masses.
Delanoë, Philippe
On the "staying away from cut-locus" issue in smooth optimal transport
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I will motivate the above issue by discussing a recent estimate of Ma-Trudinger-Wang and treat the case of round spheres (joint work with G. Loeper).
Douglas, Robert
Introduction to mass transportation and applications to geophysics
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In this survey talk we review some developments in optimal mass transportation, and describe an application to geophysics. The archetypal problem is to find the optimal way to transfer mass from a set U to a set V, amongst a set S of allowable strategies, where optimality is measured against a non-negative cost function c=c(x,y). One interprets c(x,y) as being the cost per unit mass of transporting material from x in Uto y in V. We review the formulation of a Monge optimal masstransfer problem, introduce the notation of push-forward measures todescribe S, and consider the relaxed Monge-Kantorovich version of the problem. Since the 1980s there has been considerable progress for a variety of cost functions: a key breakthrough was the discovery of the monotone rearrangement, or Brenier map,the unique optimal mass transfer for a particular cost function. When the Brenier map satisfies a nondegeneracy condition, there is an associated Monge-Ampere problem. We review this and related developments. The Cullen-Norbury-Purser minimum energy principlefor the semigeostrophic equations may be rewritten as an optimal mass transfer problem; it equates to an extra constraint on the system. The following list of references is far from exhaustive but may prove useful to non-experts. References 1. Y. Brenier. Polar factorisation and monotonerearrangement of vector-valued functions. Comm. Pure Appl. Math. 44, 375-417, 1991. 2. M.J.P. Cullen, J. Norbury and R.J. Purser. Generalised Lagrangian solutions for atmospheric and oceanic flows. SIAM J. Appl. Math. 51, 20-31, 1991. 3. R.J. Douglas. Rearrangements of functions with applicationsto meteorology and ideal fluid flow in Large-Scale Atmospheric-OceanDynamics I: analytical methods and numerical models, eds. J. Norbury and I. Roulstone. Cambridge University Press, 288-341, 2002. 4. W. Gangbo and R.J. McCann. The geometry of optimal transportation. Acta Math. 177, 113-161, 1996. 5. C. Villani. Topics in optimal transportation. American Mathematical Society, 2003.
Gancedo, Francisco
Two contour dynamics problems in incompressible flows: the Muskat problem and the QG sharp front
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We will consider two contour dynamics equations given by incompressible fluid flows. A free boundary problem through porous media, which is known in the literature as the Muskat problem, and the evolution of a sharp front for the QG equation.
Gangbo, Wilfred
Euler--Poisson systems as action-minimizing paths in the Wasserstein space
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We use a variational approach to establish existence of solutions $(sigma_t, v_t)$ for the $1$--d Euler-Poisson system by minimizing an action. When $sigma_0,$ $sigma_T$ are prescribed, the minimizer is unique provided that the time interval $[0,T]$ satisfies $T
Goldman, Dorian
Chaotic response of the 2D semi-geostrophic equations to gentle periodic forcing
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Symmetries and Hamiltonian structure are combined with Melnikov's method to show a set of exact soluitons to the 2D semi-geostrophic equations (discovered by McCann and Oberman in 2004) in an elliptical tank respond chaotically to gentle periodic forcing. To leading order, these perturbations correspond to perturbations of domain eccentricity (or potential vorticity for that matter) which are sinusoidal in time for all but countably many periods. The approach simplifies and strengthens the proof by Bertozzi (upon which it is based) concerning the chaotic response of the Kirchoff elliptical vortex patch to gentle shearing in the 2D euler equations. The approach also confirms the chaotic response of the quasi-geostrophic equations to gentle periodic forcing of an external shearing field.
Gomes, Diogo
Linear Programming and Homogenization of Hamilton-Jacobi Equations
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In this talk we will describe the generalized Mather problem and its applications to the homogenization of radom ergodic Hamilton-Jacobi equations.
Khesin, Boris
Geometry of diffeomorphism groups: shocks of the Burgers equation and non-holonomic mass transport
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We describe a relation between curvatures of the group of volume-preserving diffeomorphisms (responsible for Lagrangian instability of ideal fluids) and the generation of shocks for potential solutions of the inviscid Burgers equation (important in mass transport). For this we characterize focal points of the group of volume-preserving diffeomorphism, regarded as a submanifold in all diffeomorphisms and the corresponding conjugate points along geodesics in the Wasserstein space of densities. Further, we consider the non-holonomic optimal transport problem, related to the following non-holonomic version of the classical Moser theorem: given a bracket-generating distribution on a manifold two volume forms of equal total volume can be isotoped by the flow of a vector field tangent to this distribution.
Kim, Young-Heon
Curvature and the continuity of optimal transportation maps
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We will discuss the continuity of optimal transport maps, in view of a semi-Riemannian structure which we have formulated recently. A necessary condition for the continuity is given as some non-negativity condition on the curvature of this semi-Riemannian metric, and this result gives a quite general geometric frame work for the regularity theory of Ma, Trudinger, Wang and Loeper on the potential functions of optimal transport. This is joint work with Robert McCann (University of Toronto).
Lee, Paul
Hamiltonian Reductions relating to optimal mass transport
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In the 60s Arnold showed that the Euler equation can be thought of as the geodesic flow on the group of volume preserving diffeomorphisms. In a similar fashion, Otto showed that the mass transport problem can be consider as the geodesic problem on the space W of all volume forms with the same total volume. In particular, the space W can be regarded as the quotient of the group of all diffeomorphisms by the subgroup of volume preserving ones, while the geodesic flow on the diffeomorphism group, given by the Burgers equation, is closely related to that on the space W. It turns out that this relation between diffeomorphism group and the space W can be understood via Hamiltonian reduction.
Loeper, Gregoire
Continuity of maps solutions of optimal transportation problems
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I will present some results concerning the Holder continuity of maps solutions of optimal transportation problems. Those results hold under some condition on the cost function introduced by Ma, Trudinger and Wang, that I will present from a geometrical point of view.
Lopes, Helena
Weak stability in L^1 of Lagrangian solutions to the semigeostrophic equations
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In this short talk I will describe weak stability, with respect to L^1-norm, of Lagrangian solutions for semigeostrophy. I will show that Lagrangian solutions of L^1-perturbed problems converge, passing to subsequences if needed, to a Lagrangian solution of the limit problem. I will also discuss a singular example.
Lopes-Filho, Milton
On potential vorticity sheets for semigeostrophy
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In this short talk I will report on work in progress regarding a qualitative description of the evolution of vortex sheets for the semigeostrophic equations.
McCann, Robert
Ricci flow, entropy, and optimal transportation
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Let a smooth family of Riemannian metrics $g(tau)$satisfy the backward-time Ricci flow equation on a compact oriented $n$-dimensional manifold $M$. Suppose two families ofnormalized $n$-forms $omega(tau) ge 0$ and $tilde omega(tau) ge 0$ satisfy the forward-time heat equation on $M$ generated by the connection Laplacian $Delta_{g(tau)}$. If these $n$-forms represent two evolving distributions of particles over $M$, the minimum root-mean-square distance $W_2(omega(tau),tilde omega(tau),tau)$ to transport the particles of $omega(tau)$ onto those of $tilde omega(tau)$ is shown to be non-decreasing as a function of $tau$, without sign conditions on the curvature of $(M,g(tau))$. Moreover, this contractivity property is shown to characterise supersolutions to the Ricci flow.
Morrison, Phil
Statistical mechanics of fluid and plasma continua
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Ideas of statistical mechanics have been applied to describe the turbulence of continuum systems, such as those described by the Vlasov and two-dimensional Euler equations, since the early work of Burgers, Onsager, Lee, Lynden-Bell, Kraichnan, Montgomery, and others. Results using two new approaches for calculating the partition function and obtaining turbulent spectra will be discussed. The first is based on the use of normal coordinates associated with continuum eigenfunctions, which are used in the manner of particle degrees of freedom for finite systems. The second is based on an experimentally verifiable definition of independent subsystems and concomitant independent invariants. Appropriate additive invariants are use in the calculation of the partition function. Regions of validity of the two approaches will be discussed.
Oliver, Marcel
Transparent boundary conditions as dissipative subgrid closures
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We consider the evolution of a passive scalar in a shear flow in its representation as a system of lattice differential equations in wave number space. When the velocity field has small support, the interaction in wave number space is local and can be studied in terms of dispersive linear lattice waves. We close the restriction of the system to a finite set of wave numbers by implementing transparent boundary conditions for lattice waves. This closure is studied numerically in terms of energy dissipation rate and energy spectrum, both for a time-independent velocity field and for a time-dependent synthetic velocity field whose Fourier coefficients follow independent Ornstein-Uhlenbeck stochastic processes. Joint work with Oliver Buehler, Courant Institute.
Peeters, Bob
Hamiltonian-based numerical methods for forced-dissipative climate prediction
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Although our climate is ultimately driven by (nonuniform) solar heating, many aspects of the flow can be understood qualitatively from forcing-free and frictionless dynamics. In the limit of zero forcing and dissipation, our weather system falls under the realm of emph{Hamiltonian fluid dynamics} and the flow preserves several conservation laws such as the conservation of energy and phase space structure. One would like to have a numerical forecasting scheme that can reproduce the correct flow structure for the limit of zero forcing and dissipation, as does the exact system. Many present long-term weather forecast models fail at this point. But the question remains, however: textbf{Is it advantageous to use numerical schemes with a Hamiltonian core for realistic climate modeling?} As a first step, the hydrostatic dynamics for a forcing-free and frictionless atmosphere are formulated within the framework of Hamiltonian fluid dynamics using isentropical coordinates. A Hamiltonian system is completely specified by the emph{Hamiltonian} and its so-called emph{Poisson bracket}. A relative easy way of deriving this Hamiltonian and Poisson bracket for the isentropic atmosphere will be presented, cf. the method of Bokhove & Oliver (2006). As a second step, the Hamiltonian Particle Method is used to discretize the equations of motion, cf. Frank et al. (2002). The method is partially Lagrangian and partially Eulerian. Conserved are mass, energy, circulation and potential vorticity. The method includes a time integrator that preserves the Hamiltonian phase space structure. The algorithm, when applied to our isentropic atmosphere, will be discussed. Preliminary numerical results will be presented. Finally, several future developments will be discussed, including a parameterization of unresolved gravity waves, adequate damping of small scale nonlinear structures and the overturning of isentropes (static instability).
Plakhov, Alexander
Shapes of optimal aerodynamic resistance and optimal mass transportation
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A body moves through a rarefied medium; it is required to find the body’s shape minimizing or maximizing the force of resistance of the medium to the body’s motion. It is assumed that the medium particles do not mutually interact and collide with the body in the absolutely elastic manner. The minimization problem was firstly considered by Newton (1687) in classes of convex axially symmetric bodies. Recently, the problem has been studied by Buttazzo, Kawohl, Lachand-Robert et al in classes of convex (not necessarily symmetric) bodies. We consider the minimization and maximization problems for nonconvex bodies performing both translational and rotational motion. It is shown that that these problems can be reduced to the Monge-Kantorovich mass transfer problem, and the explicit solutions in some special cases are obtained.
Pratelli, Aldo
A quantitative version of the Sobolev inequality
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The Sobolev inequality $S(n,p)|f|_{L^{p^*}} leq |Df|_{L^p}$, valid for any $fin W^{1,p}(R^n)$, $ngeq 2$, $1<p<n$, was established in sharp form via the identification of a family of optimal functions by Aubin and Talenti in the late 1970s. In the case $p=2$, the characterization of these functions as the only optimal functions, as well as a quantitative version of the inequality, were known, respectively, thanks to the works of Gidas-Ni-Nirenberg and Bianchi-Egnell. The uniqueness result for the general case has been established only recently by Cordero-Erasquin, Nazaret and Villani by means of their mass transportation proof of the Sobolev inequality. In a recent joint work with Cianchi, Fusco and Maggi, we prove a quantitative version of the Sobolev inequality, namely, we show that in fact $S(n,p)|f|_{L^{p^*}}(1+kappa d(f)^alpha)leq |Df|_{L^p}$, where $d(f)$ is the $L^{p^*}$ distance of $f$ from the set of the optimal functions, and where $kappa=kappa(n,p)$ is a suitably small positive constant. The proof mixes the mass transportation argument of the above cited work of Cordero-Erasquin et al., with ideas from the theory of rearrangements and sharp trace Sobolev inequalities, and with certain symmetrization techniques that we have developed in a joint paper with Fusco and Maggi to prove a sharp quantitative form of the isoperimetric inequality.
Puel, Marjolaine
A mass transport approach for the relativistic heat equation.
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In this talk, we present the Jordan-Kinderleherer-Otto's discrete scheme to construct solutions for a relativistic heat equation. This scheme is based on a mass transport problem involving a discontinuous cost function which is characterisitic for a relativistic phenomenum and the underlying mass transport problem is one of the principal difficulties. The limiting process between the discrete and the continuous equation will also be discussed since it involves a monotonicity argument that is delicate because of the singularities that may appear in the gradient of the solution. This is a joint work with R. McCann.
Salmon, Rick
Mini-course on Geophysical Fluid Dynamics
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An introduction to GFD emphasizing the central importance of potential vorticity.
Savare, Giuseppe
Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds
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We present some new results concerning well posedness of gradient flows generated by semi-convex functionals in a wide class of metric space, including Alexandrov spaces satisfying a lower curvature bound and the corresponding L^2-Wasserstein spaces. Applications to the gradient flow of Entropy functionals in metric-measure spaces with Ricci curvature bounded from below and to the corresponding diffusion semigroup are also considered.
Slepcev, Dejan
Interfacial aspects of biological aggregation
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I will discuss some features of a continuum model of biological aggregation introduced by Topaz, Bertozzi and Lewis. (Similar models have been investigated by Burger, Capasso, and Morale, Burger and di Francesco, and Holm and Putkaradze.) Individuals, described by the model, experience long range attraction to other individuals of their species, but try to avoid overcrowding. In the continuum model for the population density the attraction is described via a nonlocal operator, while the repulsion is modeled by a differential operator. I will describe why such behavior leads to formation of interfaces between a near-constant-density aggregate state and the unpopulated space. The dynamics of the interfaces, and why surface-tension-like effects occur at the interfaces will be discussed. The results rely on the Wasserstein-metric-gradient-flow structure of the problem and asymptotic analysis. On long time scales the interfacial motion leads to coarsening of length scales in the evolution of a typical configuration. The rate of coarsening can be studied using the Kohn-Otto framework. I will describe a geometric viewpoint that unites the coarsening results in a variety of interfacial models. The talk is partially based on joint work with Andrea Bertozzi.
Tudorascu, Adrian
Pressureless Euler/Euler-Poisson systems via adhesion dynamics and scalar conservation laws
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The "sticky particles'' model at the discrete level is employed to obtain global solutions for a class of systems of conservation laws among which lie the pressureless Euler and the pressureless attractive/repulsive Euler-Poisson system with zero background charge. We consider the case of finite, nonnegative initial Borel measures with finite second-order moment, along with continuous initial velocities of at most quadratic growth and finite energy. We prove the time regularity of the solution for the pressureless Euler system and obtain that the velocity satisfies the Oleinik entropy condition, which leads to a partial result on uniqueness. Our approach is motivated by earlier work of Brenier and Grenier who showed that one dimensional conservation laws with special initial conditions and fluxes are appropriate for studying the pressureless Euler system.
Vanneste, Jacques
Two-dimensional Euler flows in slowly deforming domains: adiabatic invariance and geometric angle
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We study the dynamics of a two-dimensional incompressible perfect fluid in a domain that is being deformed slowly. To leading-order, the flow is nearly steady if it is initially so, and it depends instantaneously on the domain shape. It can be obtained by formulating the problem in terms of an (equivalence class of) area-preserving diffeormorphism(s) rearranging the vorticity. The first-order correction to the flow can then be constructed from the time derivative of the diffeomorphisms by solving a linear pseudodifferential problem. The trajectories of fluid particles, including in particular a geometric-angle contribution, follow. The procedure proposed is given a geometric interpretation in terms of lifts in certain principal bundles.
Zeitlin, Vladimir
Lagrangian approach to catastrophic adjustment and symmetric instability
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We develop a systematic Lagrangian approach for symmetric (i.e. independent on one of the horizontal coordinates) configurations in GFD in the framework of two-layer and continuosly stratified models, and study the relaxation (adjustment) of such configurations to equilibrium and their instabilities. The adjusted state is described by the Monge-Ampere equation. The non-equilibrium initial states with non-positive potential vorticity (PV) lead to singularity (catastrophic adjustment), and equilibrium intial states with negative PV are unstable with respect to specific "symmetric" instability. A systematic use of Lagrangian variables allows to describe the process of catastrophic adjustment, and the nonlinear stage of symmetric instability. Nonlinear saturation of this latter and accompanying process of inertia-gravity wave emission may be thus thouroughly studied. A particularly important case of symmetric instability arises in the equatorial region, which allows a systematic Lagrangian treatment as well.

Participants

Name	Institution
Ambrosio, Luigi	Scuola Normale Superiore di Pisa
Barrett, John	Imperial College, London
Bevan, Jonathan	University of Surrey
Blower, Gordon	University of Lancaster
Brenier, Yann	CNRS Nice
Buffoni, Boris	EPF Lausanne
Burton, Geoffrey	University of Bath
Buttazzo, Guiseppe	University of Pisa
Carlen, Eric	Georgia Institute of Technology
Carrillo, José A	Universitat Autònoma de Barcelona
Carvalho, Maria	University of Lisbon
Castro, Angel	CSIC Madrid
Chen, Jun	University of Wisconsin
Cordoba, Diego	NRC Madrid
Cullen, Mike	Meteorological Office
de Oliveira, Josiane Cristina	UNICAMP
Delahaies, Sylvain	University of Surrey
Delanoë, Philippe	CNRS
Douglas, Robert	Aberystwyth University
Dubinkina, Svetlana	CWI Amsterdam
Feldman, Mikhail	University of Wisconsin-Madison
Figalli, Alessio	Scuola Normale Superiore di Pisa
Gancedo, Francisco	NRC Madrid
Gangbo, Wilfred	Georgia Institute of Technology
Goldman, Dorian	University of Toronto
Gomes, Diogo	IST Lisbon
Gonzalez, Maria del Mar	University of Texas at Austin
Karakhanyan, Aram	University of Texas at Austin
Khesin, Boris	University of Toronto
Kim, Hwa Kil	Georgia Institute of Technology
Kim, Young-Heon	University of Toronto
Lanagan, Gareth	University of Aberystwyth
Lee, Paul	University of Toronto
Lisini, Stefano	Universita del Piemonte Orientale
Loeper, Gregoire	Université Claude Bernard Lyon 1/BNP Paribas
Lopes, Helena	University of Campinas
Lopes-Filho, Milton	University of Campinas
Maggi, Francesco	Università degli Studi di Firenze
Markowich, Peter	University of Cambridge
McCann, Robert	University of Toronto
Morrison, Phil	University of Texas at Austin
Norbury, John	University of Oxford
Oliver, Marcel	Jacobs University Bremen
Otto, Felix	Universität Bonn
Pacini, Tommaso	Imperial College London
Panike, Nathan	University of Wisconsin
Peeters, Bob	University of Twente
Plakhov, Alexander	University of Aveiro
Pratelli, Aldo	University of Pavia
Puel, Marjolaine	Université Paul Sabatier
Roubtsov, Volodya	Université d'Angers
Roulstone, Ian	University of Surrey
Salmon, Rick	Scripps Institution of Oceanography
Savare, Giuseppe	University of Pavia
Schonbek, Maria	University of California Santa Cruz
Slepcev, Dejan	Carnegie Mellon University
Sosio, Maria	University of Toronto
Spirn, Dan	University of Minnesota
Thuburn, John	University of Exeter
Trokhimtchouk, Maxim	University of California, Berkeley
Tudorascu, Adrian	Georgia Institute of Technology
Vanneste, Jacques	University of Edinburgh
Wirosoetisno, Djoko	University of Durham
Yolcu, Turkay	Georgia Institute of Technology
Zeitlin, Vladimir	École Normale Supérieure