Applied Mathematics Perspectives

July 12-15, 2011

Conference Manual

2011 ICIAM Satellite Meetings

Applied Analysis and Applied PDEsUniversity of Victoria

Welcome to Applied Mathematics Perspectives, a sequence of workshops supporting the flagship ICIAM 2011 meeting of the Applied Mathematics community. Although the historical roots of Applied Mathematics lie in applied analysis and the development of Newtonian physics, today Applied Mathematics is a far reaching set of interwoven disciplines and skills, which are applied to understanding every facet of nature, science, industry and modern life. This dynamic and ever-changing field includes classical mathematical areas (differential equations, numerical analysis, asymptotic and variational methods, applied probability), all pursued from the application perspective, as well as areas such as applied mechanics and fluid dynamics. Each of these areas advances apace, embracing in particular high speed computation, but newer areas are also intertwined into the applied mathematics fabric. The workshops of Applied Mathematic Per-spectives 2011 reflect the diversity and strengths of applied mathematics today.

This sequence of workshops has been supported by PIMS, MITACS, BIRS, CAIMS/SCMAI. Substantial travel funds have been awarded to US participants by the NSF. Local support at UBC has come from PIMS, the UBC Department of Mathematics and the Institute of Applied Mathematics. We thank these organisations for their generosity and help in making this a successful scientific event. We also acknowledge with gratitude the contributions of members of the local organising committee and scientific committee.

Visit the official website: http://www.mitacs.ca/goto/amp2011

Local Organising Committee:I. Frigaard (University of British Columbia)T. Hillen (University of Alberta)B. Khouider (University of Victoria)M. Lamoureux (University of Calgary)R. LeVeque (University of Washington)N. Nigam (Simon Fraser University)R. Russell (Simon Fraser University)R. Spiteri (University of Saskatchewan)M. Ward (University of British Columbia)

Scientific CommitteeR. Craster (Imperial College, UK) M. Davidson (University of Western Ontario) I. Frigaard (University of British Columbia)G. Homsy (University of British Columbia) S. Howison (Oxford University, UK) H. Othmer (University of Minnesota, US) M. Overton (Courant Institute, US) O. Scherzer (Vienna, Austria) R. Spiteri (University of Saskatchewan)

Applied Analysis and Applied PDEs :::PDEs primarily arise from physical models in fluid dynamics, mathematical physics, optimal mass transpor-tation, etc. Their analysis is a broad subject. Some applied mathematicians tend to study nonlinear PDEs rather as purely mathematical objects, often modifying them for technical reasons, while others prefer to focus on the relevant aspects of the equations and the phenomena underlying them, without much inter-est on their mathematical deepness. While these two aspects of PDEs are equally important, it was some-how reasonable to concentrate on either side of them, with little or no interest on the other. However, after the past few decades of rapid progress in the nonlinear analysis, it is getting more and more realistic to put these areas together as a vast but unified field. Consequently, a cross-communication among re-searchers in these closely related sub-fields of PDEs is necessary for the progress in nonlinear PDEs.

The aim of this workshop is to bring together researchers in applied PDEs and optimal transportation, with special focus on applications in fluid dynamics and waves in atmospheric science.

There will be three mini-courses on selected topics from optimal transportation theory, Navier-Stokes type equations, and PDEs for waves in atmospheric science. The mini-courses will be given by worldwide experts on the above subjects. In addition, there will be a

moderate number of talks highlighting the most recent progress in these directions.

This event is sponsored by PIMS, Mitacs, and the University of Victoria.

Organizers:::Martial Agueh & Slim Ibrahim, Department of Mathematics and Statistics, University of Victoria

Samuel Stechmann, Department of Mathematics, University of Wisconsin-Madison

Locations :::Workshops - David Strong Building C103. See UVic map for location

SCHEDULE :::

TUESDAY, JULY 12, 2011

MORNING9:00 – 9:10 Welcome Address

9:10 – 10:00 Leslie Smith Minicourse: A Hierarchy of PDE Reduced Models for Rotating Stratified Flows

10:10 – 11:00 Yoshi Giga Minicourse: Blow-up arguments and the Navier-Stokes equations

11:00 – 11:30 Coffee Break

11:30 - 12:20 Martial Agueh Minicourse: The Optimal transport problem: overview, existence of solutions, and the Monge- Ampère equationAFTERNOON12:20 – 2:00 Lunch (on your own)

2:00 – 2:25 Young-Heon Kim Regularity of optimal transport maps.

2:30 – 2:55 Adam Oberman Fast finite difference solvers for singular solutions of the elliptic Monge-Ampère equation

3:00 – 3:25 Brittany Froese Numerical Methods for the Monge- Ampère Equation with Transport Boundary Conditions

3:30 – 4:00 Coffee Break

4:00 – 4:25 Edriss Titi On the Loss of Regularity for the Three-Dimensional Euler Equations

4:30 – 4:55 Hakima Bessaih Invariant Gibbs measures of the energy for shell models of turbulence

5:00 – 5:25 Makram Hamouda Boundary layers for the primitive equations

WEDNESDAY, JULY 13, 2011

MORNING9:10 – 10:00 Yoshi Giga Minicourse: Blow-up arguments and the Navier-Stokes equations

10:10 – 11:00 Martial Agueh Minicourse: Optimal transport and geometric functional inequalities.

11:00 – 11:30 Coffee Break

11:30 - 12:20 Leslie Smith Minicourse: Simplified Moist 3D Boussinesq Dynamics for the Hurricane EmbryoAFTERNOON12:20 – 2:00 Lunch (on your own)

2:00 – 2:25 Dongho Chae On the blow-up problem for the Euler equations and the Liouville type results for the fluid equations

2:30 – 2:55 Tsuyoshi Yoneda Long time solvability of the Navier-Stokes-Boussinesq Equations and Related Topics

3:00 – 3:25 Chi Hin Chan Nonuniqueness of Leray-Hopf type solutions to the Navier-Stokes equation on 2 dimensional hyperbolic manifolds

3:30 – 4:00 Coffee Break

4:00 – 4:25 Adrian Tudorascu Weak Lagrangian solutions for the Semi-Geostrophic system in physical space

4:30 – 4:55 Jose M. Mazon A Monge-Kantorovich mass transport problem for a discrete distance

5:00 – 5:25 Mike Waite Buoyancy scale transitions in stratified turbulence

THURSDAY, JULY 14, 2011

MORNING9:10 – 10:00 Martial Agueh Minicourse: Optimal transport and gradient flows: the Riemann and Finsler geometry in the Wasserstein spaces.

10:10 – 11:00 Andy Majda Minicourse: Climate Science, Waves, and PDE’s for the Tropics: Observations, Theory, and Numerics

11:00 – 11:30 Coffee Break

11:30 - 12:20 Yoshi Giga Minicourse: Blow-up arguments and the Navier-Stokes equations

AFTERNOON12:20 – 2:00 Lunch (on your own)

2:00 – 2:25 Boualem Khouider Momentum transport and the multi-scale interactions of convectively coupled tropical waves

2:30 – 2:55 Paul Milewski Stability of Large Amplitude Interfacial Waves

3:00 – 3:25 Georgy Kitavtsev Existence and qualitative behaviour of solutions to lubrication equations in the presence of strong slippage.

3:30 – 4:00 Coffee Break

4:00 – 4:25 Stefan Gustafson Global solutions of the Landau-Lifshitz system

4:30 – 4:55 Natasa Pavlovic On the Cauchy problem for Gross-Pitaevskii hierarchies

5:00 – 5:25 Omar Lazar Global existence for the dissipative critical surface quasi-geostrophic equation

Friday, July 15, 2011

MORNING9:10 – 9:35 Sam Stechmann Nonlinear dynamics of gravity waves: dry and convectively coupled

9:40 – 10:05 Slim Ibrahim On the wellposedness of a full Magneto-Hydro-Dynamic equations 10:40 Closing remarks & Refreshment

karenCalloutBus loop

karenCalloutPIMS office SSMB A425

karenCalloutStudent Union Building

karenCalloutCluster Housing

karenCalloutCraigdarroch Residences

karenCalloutDavid Strong Bldg. C103

List of Abstracts

Applied Analysis and Applied PDEs Workshop

Victoria, July 12-15, 2011

Agueh, Martial (University of Victoria, Canada)

Title: The optimal transport problem: overview, existence of solutions, and the Monge-Ampèreequation. (Mini-course on Optimal transport, Lecture 1)Abstract: The optimal transport problem (OT), also known as the Monge-Kantorovich prob-lem, is a problem which basically deals with the optimal way of allocating resources from onelocation to another while keeping the cost of transportation minimal. It has numerous appli-cations in various areas of science and engineering such as, fluid mechanics, image processing,mathematical economics, meteorology and plasma physics.

In this talk, we will give a brief overview of the problem (its description, origin, etc), and wewill explain the existence of solutions when the cost function is the square distance. We will alsoshow the connection between the OT problem and the Monge-Ampère equation. Finally, we willderive, in one dimension, the explicit solution to this problem. The regularity of the solution aswell as its numerical approximation in higher dimensions will be discussed in some of the talksof the workshop.

————————————-Title: Optimal transport and geometric functional inequalities. (Mini-course on Optimaltransport, Lecture 2)Abstract: We propose a basic framework to prove a wide range of geometric functional in-equalities such as the Brunn-Minkowski, the Sobolev, the logarithmic-Sobolev, the Poincaré anda class of Gagliardo-Nirenberg inequalities. What is apparent in this proof is that all theseinequalities follow from a single and general comparison principle in the motion of a particlesystem which relates the free energy of two states of the system, the energy dissipation, andthe Wasserstein distance between the two states (i.e. the distance obtained from the optimaltransport problem). This framework is widely encompassing as it allows a direct and unifyingway for computing the best constants of these inequalities as well as all their optimal functions.The main idea underlying our proof is a notion of convexity in the space of probability measuresequipped with the Wasserstein metric, i.e. the “displacement convexity”.

————————————-Title:: Optimal transport and gradient flows: the Riemann and Finsler geometry in the Wasser-stein spaces. (Mini-course on Optimal transport, Lecture 3)Abstract: Starting from the pioneering work of Jordan, Kinderlehrer and Otto (1998), it is well-known nowadays that many parabolic diffusion PDEs (such as the heat equation, the porousmedium equation, the Fokker-Planck equation) can be viewed as gradient flows of some energyfunctionals in the 2-Wasserstein space (i.e. the space of probability densities equipped with thequadratic Wasserstein metric).

In this talk, we will first review the Riemannian geometry of the 2-Wasserstein space thatjustifies this gradient flow structure. Next, we will show that when the square distance cost is ingeneral replaced by a homogeneous cost of degree p > 1, then the corresponding p-Wassersteinspace can be endowed with a Finsler structure which reduces to a Riemann-Finsler structure

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when p = 2. In particular, we show that, with respect to this Finsler geometry, the parabolicq-Laplacian equation is a gradient flow in the p-Wasserstein space for p = q/(q − 1).

Bessaih, Hakima (Wyoming)

Title: Invariant Gibbs measures of the energy for shell models of turbulenceAbstract: Gaussian measures of Gibbsian type are associated with some shell model of 3Dturbulence; they are constructed by means of the energy, a conserved quantity for the 3D inviscidand unforced shell model. We prove the existence of a unique global flow for a stochastic viscousshell model and of a global flow for the deterministic inviscid shell model, with the property thatthese Gibbs measures are invariant for these flows.

Chae, Dongho (Sungkyunkwan University, Korea)

Title: On the blow-up problem for the Euler equations and the Liouville type results for thefluid equationsAbstract: In the first part of the talk we discuss some new observations on the blow-up problemin the 3D Euler equations. We consider the scenarios of the self-similar blow-ups and theaxisymmetric blow-up. For the self-similar blow-up we prove a Liouville type theorem for theself-similar Euler equations. For the axisymmetric case we show that some uniformity conditionfor the pressure is not consistent with the global regularity. In the second part we presentLiouville type theorems for the steady Navier-Stokes equations for both of the incompressibleand the compressible cases. In the time dependent case we prove that some pressure integralshave definite sign unless the solution is trivial.

Chan, Chi Hin (University of Minnesota, USA)

Title: Nonuniqueness of Leray-Hopf type solutions to the Navier-Stokes equation on 2 dimen-sional hyperbolic manifolds.Abstract: In this talk, I will present a piece of work of Chi Hin Chan and Magdalena Czubak,in which we construct a family of finite energy, finite dissipation solutions to the Naiver-Stokesequation on a 2 dimensional hyperbolic manifold associated to a finite energy initial datumchosen as the gradient of some nontrival bounded harmonic function on a hyperbolic manifold.

Our proof for this interseting but unexpected non-uniqueness result is based on some differ-ential geometric works of D. Sullivan, M. Anderson, and R. Schoen dated back to 1980’s.

In this talk, I will also discuss the significance of our result in regard to the formulation ofthe viscosity operator in the Navier-Stokes equation on a space form due to the important workof D. Ebin and J. Marsden in 1970.

Froese, Brittany (Simon Fraser University, Canada)

Title: Numerical Methods for the Monge-Ampère Equation with Transport Boundary Condi-tions.

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Abstract: The Monge-Ampère equation is a fully nonlinear PDE that arises in optimal trans-port. Solutions may be singular, in which case novel solution techniques are needed to ensureconvergence. We describe a monotone finite difference discretisation, which provably convergesto the weak (viscosity) solution. The resulting nonlinear equations are solved efficiently usingNewton’s method. We also describe a method for implementing the second-type (transport)boundary condition that arises in the context of optimal transport.

Giga, Yoshi ( University of Tokyo, Japan)

Title: Blow-up arguments and the Navier-Stokes equations (Mini-course on Navier-Stokesequations; 3 lectures).Abstract: A blow-up argument is a very strong and flexible tool to prove regularity of solutionsof nonlinear partial differential equations which goes back to the study of the minimal surfacesby E. De Giorgi in 1961. It is also useful to obtain a priori upper bounds for solutions.

In this talk we give two new applications of this method for the Navier-Stokes and Stokesequations. As the first application we give a geometric non-blow-up criterion on the directionof the vorticity for the three dimensional Navier-Stokes flow whose initial data is just boundedand may have infinite energy. We prove that under a restriction on behavior in time (type Icondition) the solution does not blow up if the vorticity direction?is uniformly continuous atplace where vorticity is large. This improves the Lipschitz regularity condition for the vorticitydirection first introduced by P. Constantin and C. Fefferman in 1993 for finite energy weaksolutions.

As a second application we are able to prove that the Stokes operator (with the Dirich-let boundary condition) generates an analytic semigroup in the space of bounded, uniformlycontinuous functions at least when a domain occupied with fluid is a bounded domain in theEuclidean space. This is a longstanding open problem and it was only known when the domainis a half space where an explicit solution formula is available. A conventional method to reducethe problem to a half space does not work since various elliptic and parabolic estimates are notvalid for sup-norms so perturbed terms cannot be controlled. Also, it seems difficult to extendthe method available for the Laplace operator. In our method we reduce the problem to a halfspace problem by a blow-up argument to get a priori estimates.

The crucial steps of the blow-up arguments consist of the compactness of a blow-up se-quence of rescaled solutions and the uniqueness of the blow-up limit. This general?strategyis well-described in Mi-Ho Giga, Y. Giga and J. Saal, Nonlinear partial differential equations:asymptotic behavior of solutions and self-similar solutions, Birkhauser, 2010. The first part isa joint work with H. Miura (Osaka University) while the second part is a joint work with mystudent Ken Abe (University of Tokyo). The second part is a work in progress while the preprintof the first part is available at http : //eprints3.math.sci.hokudai.ac.jp/2045/

Gustafson, Stephen (University of British Columbia, Canada)

Title: Global solutions of the Landau-Lifshitz system.Abstract: The Landau-Lifshitz (or Schroedinger map) equation is a nonlinear Schroedingerequation of geometric and physical (ferromagnetism) origin. I will describe this system, andpresent some results on singularity (non-)formation and stability in the energy-critical 2D setting,

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joint work with Eva Koo, and with Kenji Nakanishi and Tai-Peng Tsai.

Hamouda, Makram (Indiana University, USA)

Title: Boundary layers for the primitive equations.Abstract: We present in this lecture some convergence results related to the Linearized Prim-iitive Equations (LPEs) as the viscosities go to zero. The (full nonlinear) Primitive Equationsread:

∂ṽ∂t

+ (ṽ · ∇)ṽ + w̃ ∂ṽ∂z

+ fk × ṽ + 1ρ0∇p̃− µṽ∆ṽ − νṽ ∂

2ṽ∂z2

= Fṽ∂p̃∂z

= −ρ̃g,∇ṽ + w̃

∂z= 0,

∂T̃∂t

+ (ṽ · ∇)T̃ + w̃ ∂T̃∂z− µT̃∆T̃ − νT̃ ∂

2T̃∂z2

= QT̃ ,

ρ̃ = ρ0(1− α(T̃ − T0)

).

(1)

One of our aims, among others, is to give the limit solution associated with the linearized systemof (1) that we obtain by dropping the convective terms. How- ever, a diculty for the inviscidLPEs system (that is we set the viscosities to be equal to zero) lies in the fact that no set of localboundary conditions ensures its well-posedness. Several choices of nonlocal boundary conditionsare possible. Hence, in view of the uniqueness, our aim is to give an asymptotic expansion ofthe solution of the LPEs at small viscosities conrming thus, in particular, our choice for theboundary conditions of the limit solution

Ibrahim, Slim (University of Victoria, Canada)

Title: On the wellposedness of a full Magneto-Hydro-Dynamic equations.Abstract: We consider a full Magneto-Hydro-dynamic equations and construct global smallsolutions in ”critical” spaces as well as large local solutions in ”subcritical” spaces. Also, weshow that either the fluid or the magnetic vector field looses regularity as long as the solutionexists. These are joint works with S. Keraani and T. Yoneda

Khouider, Boualem (University of Victoria, Canada)

Title: Momentum transport and the multi-scale interactions of convectively coupled tropicalwaves.Abstract: Recent satellite observations revealed that tropical precipitation is organized intoa hierarchy of wave-like propagating disturbances that are embedded in each other like Rus-sian dolls. They range from mesoscale cloud systems of a few hundred kilometers and a fewhours to synoptic scale convectively coupled waves, also know as cloud superclusters, of a fewthousand kilometers and a few days to planetary scale envelopes with a period of 40 to 60days, know as the Madden-Julian oscillation (MJO). The MJO propagates eastward over theIndian Ocean/Western Pacific as an enhanced region of precipitation, associated with anomalousnear-surface westerly winds, at roughly 5 m/s. Current climate models (or general circulationmodels; GCMs) represent poorly, if at all, the MJO partly due to fact that they do not capturewell the effects of convection on the large scales and the inherent multiscale interactions; thishas an impact on both tropical and extra-topical climate and medium- to long-range weather

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predictability. Convective momentum transport (CMT), which is the mechanism through whichconvective systems deposit momentum on the large scales and accelerate or decelerate the am-bient shear, is believed to play a significant role in the nonlinear interaction between mesoscaleconvective systems, superclusters, and the MJO. In this talk we will present a simple model forthe parameterization of CMT due to mesoscale systems in GCMs and demonstrate its ability tostrengthen and improve the organization of tropical convection on both the synoptic and plane-tary scales, in an idealized climate model?toy GCM. We will then discuss some recent simulationsusing the Weather Forecasting Model (WRF) in an idealized setting to demonstrate the effect ofmesoscale CMT, both on synoptic scale Kelvin waves and on the background vertical shear. Iftime permits we will also discuss some recent results about two way interaction between the MJOenvelope and the embedde d convectively coupled equatorial waves and mesoscale squall linesin a beta plane; the MJO provides the background vertical shear in which equatorial waves ofcertain types can evolve due to large scale advection and in return the waves decelerate the shearthrough CMT thus creating their own demise, consistent with recent detailed observation of afew MJO events in the Western Pacific warm pool region (the TOCA-COARE field campaign).

Kim, Young-Heon (University of British Columbia, Canada)

Title: Regularity of optimal transport maps.Abstract: The theory of optimal transport is concerned with phenomena arising when onematches two mass distributions in a most economic way, minimizing transportation cost ofmoving mass from one location to another. We consider an optimal transportation problemwith costs satisfying certain type of degenerate curvature condition discovered by Ma, Trudingerand Wang. We explain some recent progress in the regularity of optimal maps, especially whenthe mass densities are only bounded. This is joint work with Alessio Figalli and Robert McCann.

Kitavtsev, Georgy (Max Planck Institute for Mathematics in the Sciences, Germany)

Title: Existence and qualitative behaviour of solutions to lubrication equations in the presenceof strong slippage.Abstract: In this talk we consider a one-dimensional lubrication model that describes thedewetting process of nanoscopic thin polymer liquid films on hydrophobyzed substrates. Thismodel takes account of intermolecular interactions of the film with the solid substrate due toattractive van der Waals and repulsive Born forces, as well of the effect of large slippage at thepolymer-substrate interface. It is represented by a coupled system of differential equations

Re(∂t(hu) + ∂x(hu

2))

= 4∂x(νh∂xu) + h∂x (σ∂xxh− Π(h))−u

β(2)

∂th = −∂x(hu), (3)

describing evolution in time of the free surface and the averaged lateral velocity of the filmgiven by the functions h(x, t) and u(x, t), respectively. Coefficients Re, ν, σ, β in (2)-(3) de-note Reynolds number, viscosity, capillarity and slip-length, respectively. The system (2)-(3) isconsidered on a bounded domain (0, 1) with the boundary conditions

u = 0 and ∂xh = 0 at x = 0, 1. (4)

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We prove existence of global weak solutions for the system (2)-(3) with (4) and investigate theconvergence of these solutions as either the Reynolds number or the capillarity goes to zero, aswell as their limiting behaviour as the slip length goes to zero or infinity. In particularly, weshow that after an appropriate rescaling they converge to the smooth classical solutions of thelubrication model

∂th = −∂x(h2∂x (σ∂xxh− Π(h))

)which corresponds to the case of the Navier slip condition imposed at the polymer-substrateinterface.

Finally, the late stage of dewetting process, namely coarsening of metastable droplets anddependence of the associated dynamics and coarsening rates on the slip-length will be discussed.

Lazar, Omar (Université Paris-Est Marne-la-Valée, France)

Title: Global existence for the dissipative critical surface quasi-geostrophic equation.Abstract: In this talk, we study the critical dissipative (SGQ) equation. We show the globalexistence for large initial data in a space close to the space of uniformally locally square integrablefunctions. The proof is based on a energy inequality verified by the truncated and regularizedequation.

Majda, Andrew (New York University, USA)

Title: Climate Science, Waves, and PDEs for the Tropics: Observations, Theory, and Numerics.(Mini-course on PDEs in atmospheric science, Lecture 3).Abstract: Geophysical flows are a rich source of novel problems for applied mathematics andthe contemporary theory of partial differential equations. The reason for this is that manyphysically important geophysical flows involve complex nonlinear interaction over multi-scalesin both time and space so developing simplified reduced models which are simpler yet capturekey physical phenomena is of central importance. In mid-latitudes, the fact that the rotationalCoriolis terms are bounded away from zero leads to a strict temporal frequency scale separationbetween slow potential vorticity dynamics and fast gravity waves; this physical fact leads to newtheorems justifying the quasi-geostrophic midlatitude dynamics even with general unbalancedinitial data for both rapidly rotating shallow water equations and completely stratified flows.

At the equator, the tangential projection of the Coriolis force from rotation vanishes iden-tically so that there is no longer a time scale separation between potential vortical flows andgravity waves. This has profound consequences physically that allow the tropics to behave asa waveguide with extremely warm surface temperatures. The resulting behavior profoundlyinfluences longer term mid-latitude weather prediction and climate change through hurricanes,monsoons, El Nino, and global teleconnections with the mid-latitude atmosphere. How this hap-pens through detailed physical mechanisms is one of the most important contemporary problemsin the atmosphere-ocean science community with a central role played by nonlinear interactiveheating involving the interaction of clouds, moisture, and convection. The variable coefficientdegeneracy of the Coriolis term at the equator alluded to earlier leads to both important newphysical effects as well as fascinating new mathematical phenomena and PDEs. In this equatorialcontext, new multi-scale reduced dynamical PDE models are even relatively recent in origin.

After a brief discussion of the observational record as background, this lecturer surveys the

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remarkable new hyperbolic systems that have emerged recently in applications including theirphysical properties, applied mathematical and rigorous mathematical theory. These last topicsinclude novel relaxation limits for climate models with active moisture and new singular limitsfor hyperbolic PDEs with variable coefficients. All of the references in this lecture can be foundat http : //www.math.nyu.edu/faculty/majda/

Mazon, Jose M. (Universitat de Valencia, Spain)

Title: A Monge-Kantorovich mass transport problem for a discrete distance.Abstract: This lecture is concerned with a Monge-Kantorovich mass transport problem inwhich in the transport cost we replace the Euclidean distance with a discrete distance. We fixthe length of a step and the distance that measures the cost of the transport depends of thenumber of steps that is needed to transport the involved mass from its origin to its destination.For this problem we construct special Kantorovich potentials, and optimal transport plans via anonlocal version of the PDE-formulation given by Evans and Gangbo for the classical case withthe Euclidean distance. We also study how this problem, when reescaling the step distance,approximates the classical problem. In particular we obtain, taking limits in the reescalednonlocal formulation, the PDE-formulation given by Evans-Gangbo for the classical problem.Joint work with N. Igbida, J. Rossi and J.J. Toledo.

Milewski, Paul (University of Wisconsin, USA)

Title: Stability of Large Amplitude Interfacial Waves.Abstract: Waves at the interface of two fluids of different densities occur widely in nature.Their mathematical analysis is complicated by the Kelvin-Helmholtz instability which, althoughpresent in the fluid flow, may or may not have significant impact on the waves. In this workwe seek interfacial models for large amplitude waves and study their range of validity from astability perspective. Comparisons of solutions of the models to computations of the primitiveequations will also be shown.

Oberman, Adam (Simon Fraser University, Canada)

Title: Fast finite difference solvers for singular solutions of the elliptic Monge-Ampère equation.Abstract: The elliptic Monge-Ampère equation is a fully nonlinear Partial Differential Equationwhich originated in geometric surface theory, and has been applied in dynamic meteorology,elasticity, geometric optics, image processing and image registration. Solutions can be singular,in which case standard numerical approaches fail.

In this article we build a finite difference solver for the Monge-Ampère equation, whichconverges to the unique viscosity solution of the equation. Regularity results are used to selecta priori between a stable, provably convergent monotone discretization and an accurate finitedifference discretization. The resulting nonlinear equations are then solved by Newton’s method.

Computational results in two and three dimensions validate the claims of accuracy andsolution speed. A computational example is presented which demonstrates the necessity ofthe use of the monotone scheme near singularities.

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Pavlovis, Natasa (University of Texas, USA)

Title: On the Cauchy problem for Gross-Pitaevskii hierarchiesAbstract: The Gross-Pitaevskii (GP) hierarchy is an infinite system of coupled linear non-homogeneous PDEs, which appear in the derivation of the nonlinear Schrödinger equation (NLS).Inspired by the PDE techniques that have turned out to be useful on the level of the NLS, werealized that, in some instances we can introduce analogous techniques at the level of the GP.In this talk we will discuss some of those techniques which we use to study well-posedness forGP hierarchies. The talk is based on joint work with Thomas Chen.

Smith, Leslie (University of Wisconsin, USA)

Title: A Hierarchy of PDE Reduced Models for Rotating Stratified Flows (Mini-course onPDEs in atmospheric science, Lecture 1)Abstract: Starting from the rotating Boussinesq or Shallow-Water equations, we show how toderive a hierarchy of models intermediate between the quasi-geostrophic (QG) approximation andthe full equations. The new PDEs progressively include more effects of inertia-gravity/gravitywaves. We illustrate how the reduced PDEs can be used to identify the nonlinear interactionsprimarily responsible for observed non-QG phenomena, for example (i) anticyclone dominancein rotating shallow water decay and (ii) the growth of horizontal shear flows in strongly stratifiedBoussinesq flow with unbalanced forcing. Pros and cons of our approach will be discussed, aswell as the relation to some previously derived intermediate models.

————————————-Title: Simplified Moist 3D Boussinesq Dynamics for the Hurricane Embryo. (Mini-course onPDEs in atmospheric science, Lecture 2)Abstract: Tropical cyclogenesis is studied in the context of idealized 3D Boussinesq dynamicswith perhaps the simplest possible model for bulk cloud physics. With low-altitude input of watervapor on realistic length and time scales, numerical simulations capture the formation of vorticalhot towers, as well as the eventual merger of low-altitude, small-scale cyclonic activity into alarger-scale tropical depression. From measurements of water vapor, vertical velocity, verticalvorticity and rain, it is demonstrated that the structure, strength and lifetime of the hot towerscompares well to results from models including more detailed cloud microphysics. The favorableagreement was anticipated by the multi-scale analysis of Majda, Xing and Mohammadian (2010).A tropical depression can be generated in the absence of an initial mesoscale cyclonic vortex, butthen the horizontal wind speeds of the depression are relatively low. Vertical shear resulting froman initial zonal velocity profile does not prevent the generation of the hot towers, but inhibitsthe vertical extent of the depression vortex. The results indicate that the simplest condensationand evaporation schemes may be useful for further exploratory numerical simulations aimed atdeeper theoretical understanding.

Stechmann, Samuel (University of Wisconsin, USA)

Title: Nonlinear dynamics of gravity waves: dry and convectively coupled.Abstract: The organization of tropical clouds/convection is intimately linked with nonlineargravity wave dynamics. Nonlinear PDE will be presented to describe these nonlinear dynamics

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and their interactions with wind shear. The first model takes the form of 2-mode shallow waterequations and has many interesting mathematical properties: it is a system of nonconservativePDE with a conserved energy, it is conditionally hyperbolic, and it is neither genuinely nonlinearor linearly degenerate over all of phase space. A simple numerical method was designed andtested in difficult situations to handle the unique mathematical properties of the equations, andits application to the organization of tropical convection will be demonstrated. A second modelwill also be presented for convectively coupled waves, which are coherent structures in the tropicsthat arise from nonlinear interactions between waves, water vapor, and convection. Interactionwith a mean shear leads to convectively coupled wave-mean flow interactions, which are differentfrom classical wave-mean flow interactions in many ways.

Titi, Edriss (University of California-Irvine, USA)

Title: On the Loss of Regularity for the Three-Dimensional Euler EquationsAbstract: A basic example of shear ow was introduced by DiPerna and Majda to study the weaklimit of oscillatory solutions of the Euler equations of incompressible ideal uids. In particular,they proved by means of this example that weak limit of solutions of Euler equations may, in somecases, fail to be a solution of Euler equations. We use this shear fow example to provide non-generic, yet nontrivial, examples concerning the immediate loss of smoothness and ill-posednessof solutions of the three-dimensional Euler equations, for initial data that do not belong to C1,α

. Moreover, we show by means of this shear fow example the existence of weak solutions for thethree-dimensional Euler equations with vorticity that is having a nontrivial density concentratedon non-smooth surface.

This is very different from what has been proven for the two-dimensional Kelvin-Helmholtzproblem where a minimal regularity implies the real analyticity of the interface. Eventually, weuse this shear fow to provide explicit examples of non-regular solutions of the three-dimensionalEuler equations that conserve the energy, an issue which is related to the Onsager conjecture.

This is a joint work with Claude Bardos.

Tudorascu, Adrian (West Virginia University, USA)

Title: Weak Lagrangian solutions for the Semi-Geostrophic system in physical space.Abstract: Proposed as a simplification for the Boussinesq system in a special regime, theSemi-Geostrophic (SG) system is used by metereologists to model how fronts arise in large scaleweather patterns. In spite of significant progress achieved in the analysis of the SG in dualspace (i.e. the system obtained from the SG by a special change of variables), there are noexistence results on the SG in physical space. We shall argue that weak (Eulerian) solutions forthe Semi-Geostrophic system in physical space exhibiting some mild regularity in time cannotyield point masses in the dual space. However, such solutions are physically relevant to themodel. Thus, we shall discuss a natural generalization of Cullen & Feldman’s weak Lagrangiansolutions in the physical space to include the possibility of singular measures in dual space. Wehave proved existence of such solutions in the case of discrete measures in dual space. Jointwork with Mikhail Feldman.

9

Waite, Michael (University of Waterloo, Canada)

Title: Buoyancy scale transitions in stratified turbulenceAbstract: Geophysical turbulence at intermediate scales is characterized by strong stratifi-cation and weak rotation, a regime with a downscale energy cascade. In recent years, theidealized Boussinesq equations with uniform stratification have been used to reproduce manyof the basic features of this regime. But what happens at smaller scales? The classical pictureis of a transition to isotropic three-dimensional turbulence at the Ozmidov scale. Here, we willpresent numerical evidence for another transition at the (larger) buoyancy scale U/N, where Uis the r.m.s. velocity and N is the buoyancy frequency. This transition appears to result fromKelvin-Helmholtz instability of the large-scale vortices. The implications of using small aspectratio numerical grids - which are common in geophysical fluid simulations but can distort thistransition - will also be discussed.

Yoneda, Tsuyoshi (University of Victoria, Canada)

Title: Long time solvability of the Navier-Stokes-Boussinesq Equations and Related TopicsAbstract: In the talk, we investigate large time existence of solutions of the Navier-Stokes-Boussinesq equations with spatially almost periodic large data when the density stratification issufficiently large.

In 1996, Kimura and Herring examined numerical simulations to show a stabilizing effect dueto the stratification. They observed scattered two-dimensional pancake-shaped vortex patcheslying almost in the horizontal plane. Our result is a mathematical justification of the presenceof such two-dimensional pancakes.

This is joint work with Prof. Ibrahim.

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Workshop Participants :::

Martial Agueh U. Victoria, Canada agueh@math.uvic.caHakima Bessaih U. Wyoming, USA bessaih@uwyo.eduDongho Chae Sungkunkwan U., Korea chae@skku.eduChi Hin Chan U. Minnesota, USA chana002@ima.umn.eduBrittany Froese Simon Fraser U., Canada bdf1@sfu.caYoshi Giga U. Tokyo, Japan labgiga@ms.u-tokyo.ac.jpStephen Gustafson U. British Columbia, Canada gustaf@math.ubc.caSlim Ibrahim U. Victoria, Canada ibrahims@uvic.caBoualem, Khouider U. Victoria, Canada khouider@math.uvic.caYoung-Heon Kim U. British Columbia, Canada yhkim@math.ubc.caGeorgy Kitavtsev Max Planck Institute, GermanyGeorgy.Kitavtsev@mis.mpg.deAndrew Majda New York U., USA jonjon@cims.nyu.eduPaul Milewski U. Wisconsin-Madison, USA milewski@math.wisc.eduAdam Oberman Simon Fraser U., Canada aoberman@math.sfu.caMakram Hamouda U. Indiana, USA Makram.Hamouda@math.u-psud.frOmar Lazar U. Marne-la-Valee, France laz_omar@yahoo.frJose M. Mazon U. Valencia, Spain Jose.M.Mazon@uv.eNatasa Pavlovic U. Texas A&M, USA natasa@math.utexas.eduLeslie Smith U. Wisconsin-Madison, USA lsmith@math.wisc.eduSam Stechmann U. Wisconsin-Madison, USA stechmann@wisc.eduEdriss Titi UC-Irvine, USA etiti@math.uci.eduAdrian Tudorascu West Virginia U., USA adriant@math.wvu.eduMichael Waite U. Waterloo, Canada mwaite@uwaterloo.caTsuyoshi Yoneda U. Victoria, Canada tyoneda@uvic.caKen Abe U. Tokyo, Japan kabe@ms.u-tokyo.ca.jpMichele De La Chevrotiere U. Victoria, Canada delachev@uvic.caMi-Ho Giga U. Tokyo, Japan Nao Hamamuki U. Tokyo, Japan hnao@ms.u-tokyo.ac.jpEva Koo U. British Columbia, Canada evahk@math.ubc.caVenu Kurella U. British Columbia, Canada vkurella@math.ubc.caKody Law U. Warwick, England K.J.H.Law@warwick.ac.ukLouis-Xavier Proulx U. Montreal, Canada roulx@dms.umontreal.caAslin Richardson U. Victoria, Canada ashy@uvic.caLouis-Philippe Saumier U. Victoria, Canada lsaumier@uvic.ca

Victoria ::: UVic On Campus

FOOD - University Food Services http://unfs.uvic.ca/ :: in the Cadboro Commons - map http://www.uvic.ca/buildings/com.html : Village Greens (vegetarian and vegan meals) : Caps Bistro (hot calzones and a deli bar) : Village Market (campus grocery store) : in the University Centre - map http://www.uvic.ca/buildings/uvc.html : Centre Caf (full service cafeteria with grill) : Sweet Greens (deli sandwich bar) : in the Student Union Building : Felicitas http://felicitas.ca/

RECREATIONAL ACTIVITIES :: The Stewart Complex (fitness centre, tennis courts, ice rink, and more) : 3964 Golden Head Road :: McKinnon Building (gymnasium, dance studio, pool, and more) : 3800 Finnerty Road (off Gabriola Road) :: Victoria Art Collections : http://uvac.uvic.ca/#section0-9 :: Phoenix Theatre : http://finearts.uvic.ca/theatre/index.shtml :: Jogging Routes : http://www.uvic.ca/maps/joggingmap.html :: Cinecenta : http://www.cinecenta.com/

EMERGENCY/FIRE/SAFTEY ** in an emergency dial 911 for Police, Fire, or Ambulance :: 24 hours emergency/safewalk : 250-721-7599

TRANSPORTATION :: BC Transit http://www.bctransit.com/regions/vic/ or phone 250.382.6161 :: Public Transit : a single fare is valid for 60 minutes (hop on and off as you please) : one ticket - $2.50, day pass - $7.75, monthly pass - $82.50 : http://www.uvic.ca/maps/busroutes.html :: Airport Shuttle/Taxi Link http://www.victoriaairport.com/index.php?pageid=60 :: AKAL shuttle service (250) 386-2525 or 1-877-386-2525 :: Yellow Cab of Victoria (250) 381-2222 or 1-800-808-6881

WEBSITES :: http://www.uvic.ca/buildings/csr.html :: http://www.uvic.ca/visitors/about/

Victoria ::: Off Campus

TOURISM VICTORIA :: visit http://tourismvictoria.com/ or call 1-800-663-3883 :: Butchart Gardens - 55 acres of spectacular beauty : 800 Benvenuto Avenue Brentwood Bay : http://www.butchartgardens.com/index.php?option=com_frontpage&Itemid=1 :: WildPlay - challenging adventure experiences : 1767 Island Highway : http://www.wildplay.com/ :: Carr House - the birthplace and childhood home of Emily Carr : 207 Government Street : http://www.tourismvictoria.com/BusinessProfile.aspx?bID=34050 :: National Geographic Imax Theatre - powerful and immersive movie experience : 675 Belleville Street : http://www.imaxvictoria.com/ :: Centre of the Universe - treat yourself to spectacular views of Victoria and the Cosmos! : 5071 West Saanich Road : http://www.tourismvictoria.com/BusinessProfile.aspx?bID=34053 :: Craigdarroch Castle Historic House Museum - visit this exquisite 1890’s mansion : 1050 Joan Crescent : http://www.thecastle.ca/

:: Maritime Museum of British Columbia - travel back in time to the days of the pirates : 28 Bastion Square : http://mmbc.bc.ca/ :: Miniature World - featuring animated miniature scenes : 649 Humboldt Street : http://www.miniatureworld.com/ :: Pacific Undersea Gardens - largest collection of local marine life : 490 Belleville Street : http://www.pacificunderseagardens.com/ :: Victoria Bug Zoo - enter the amazing world of insects and spiders : 631 Courtney Street : http://www.bugzoo.com/

UVIC ::: Wireless internet access

During the conference you may use the computers at the McPherson Library. Hours are 7:30am-9:00pm on weekdays and 10:00am-6:00pm on weekends. To log on use

Netlink ID: apdespassword: mat4Emax

You will use the same Netlink ID and password for wireless internet, visit http://www.uvic.ca/systems/support/internettelephone/wireless/index.php for more details.