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Lyon, July 15-24, 2010 Institut Camille Jordan - Université Lyon 1  
About the Franco-Brazilian Fluids Summer School
The Summer School is dedicated to the subject of Fluid Dynamics and PDE, focusing on modeling and rigorous analysis of physically relevant problems in fluid dynamics, particularly highly irregular incompressible flows. The research in this field touches upon many different problems with relevant applications, such as problems in aerodynamics, oceanographic flows and geophysical flows. It is a field which has generated intense international research activity over the last two decades. This School is designed to attract young researchers and students to the field and to foster scientific interaction among the participants. Given these goals, the School has been structured so as to have both minicourses on topics of current interest in the field and lectures on recent results. There will also be opportunity for select contributed communications. This endeavor stems from an ongoing collaboration between researchers in France and Brazil and it is a part of a joint project between members of the Scientific Committee. Location The 1st Franco-Brazilian Fluids Summer School took place in Campinas, January 2010. The second Summer School will take place at the University of Lyon 1, Institut Camille Jordan (the Mathematics department) in the Braconnier building. Lyon forms the second-largest metropolitan area in France after that of Paris, with a population of 1,3 million inhabitants. It has a reputation as the French capital of gastronomy. The University of Lyon 1 represents one of the main higher education institutions in the country, both in terms of student population and scientific productivity. July in Lyon is a summer month, with temperatures which, typically, do not exceed 30°C, although higher temperatures cannot be excluded (the mean daily maximum temperature is 27°C). Scientific and Organizing Committee
Didier Bresch (Université de Chambéry): Global weak solutions: From incompressible to compressible flows Abstract: During these lectures, we will discuss the ideas used to prove global existence of weak solutions "à la Leray" starting from the standard homogeneous incompressible Navier-Stokes equations and finishing with the compressible Navier-Stokes equations with heat conductivity. The main idea is to allow to understand at each level the difficulties and the main tools. Helena Nussenzveig Lopes (University of Campinas): On the vortex sheet problem for 2D incompressible fluid flow Abstract: Vortex sheets are an idealized model for flows with strong shears concentrated in a thin region, such as what occurs in the flow trailing an airfoil. In these lectures we will discuss two different approaches to the problem of modeling vortex sheet evolution: the so-called explicit approach, leading to the Birkhoff-Rott equations and the implicit approach, involving weak solutions with vorticity being a measure. We will make a tour of the progress achieved over the past 30 years, including recent results comparing these two approaches. Marius Paicu (Université Paris-Sud): Global wellposedness of some problems in fluid mechanics Abstract: We review some classical results on the wellposedness theory for the Navier-Stokes equations and we present some new results and methods in this field. We also discuss the global existence of solutions for some related models as the Boussinesq system (which describes the heat convection in an incompressible fluid) and the system of nematic liquid crystals. Armen Shirikyan (Université de Cergy-Pontoise): Ergodic theory of random dynamical systems and application to stochastic Navier-Stokes equations Abstract: The aim of this short course is to give a quick introduction to the ergodic theory of randomly perturbed dynamical systems and to show how to apply it to some problems arising in mathematical fluid dynamics. We begin with a general study of a class of Markov random dynamical systems (RDS) in a phase space X. It will be shown, in particular, that such an RDS defines a semigroup in the space of probability measures on X. We then introduce the concept of a stationary distribution and describe the Bogolyubov-Krylov argument for its construction. The core of the course is a result on uniqueness and mixing properties of stationary distribution. Its proof if based on a coupling argument which enables one to show that the above-mentioned semigroup is a contraction in the space of measures endowed with the Kantorovich-Wassertein metric. In conclusion, we prove that the results obtained for Markov RDS apply to 2D Navier-Stokes equations perturbed by an external random force. David Ambrose (Drexel University): Some Existence Problems in Interfacial Fluid Dynamics Abstract: For many problems in interfacial fluid dynamics, such as the water wave or the vortex sheet with surface tension, existence of solutions for initial value problems and existence of traveling solutions has been established in recent years. In this talk, we will look at some other existence problems. First, we will explore the question of existence of weak solutions for interfacial Navier-Stokes flows, in the presence of surface tension. One significant challenge in this problem is finding a weak formulation of the surface tension force, which is supported only at the interface, and is given in terms of the curvature of the interface. As time allows, we will also discuss existence of time-periodic interfacial flows, as well as solutions to elliptic problems in interfacial fluid dynamics such as the unregularized vortex sheet and Boussinesq equations. This includes joint work with Milton Lopes Filho, Timur Milgrom, Helena Nussenzveig Lopes, Walter Strauss, and Jon Wilkening. Christophe Cheverry (Université de Rennes 1): Oscillating solutions with nilpotent Jacobian matrix for three dimensional Burger equations Abstract: In this talk, we will explain how to construct large amplitude oscillating waves which are solutions on a non trivial domain of both the three dimensional Burger equations (without source term) and of the incompressible Euler equations (without pressure). The solutions are mainly characterized by the fact that the associated Jacobian matrices are nilpotent of rank two. We will insist on the special geometrical features of the involved phasis and on the turbulent phenomena which can be exhibited through the method. Darren Crowdy (Imperial College London): A new calculus for two dimensional vortex dynamics Abstract: In classical fluid dynamics, an important problem arising in a variety of applications is to understand how vorticity interacts with solid objects (e.g. aerofoils, obstacles or stirrers). For planar flows, a variety of powerful mathematical results exist (complex variable methods, conformal mapping, Kirchhoff-Routh theory) that have been used to study such problems but the constructions are usually restricted to problems with just one, or perhaps two, objects. Expressed another way, most studies deal only with fluid regions that are simply or doubly connected. There has been a general and longstanding perception that problems involving fluid regions of higher connectivity are too challenging to be tackled analytically.The talk will show that there is a way to formulate the theory so that the relevant fluid dynamical formulae are exactly the same irrespective of the connectivity of the domain. This provides a flexible and unified tool for modelling the fluid dynamical interaction of multiple objects/aerofoils/obstacles/stirrers in ideal flow and their interaction with free vortices. Olivier Glass (Université Paris-Dauphine): On the control of the displacement of a fluid zone Abstract: We consider the 2 dimensional Euler equation for perfect incompressible fluids in a bounded domain, where we can use a part of the boundary conditions as a control, that is, as a way to influence the system which we can choose. We consider the problem of moving a given zone of the fluid to another fixed zone, by choosing a relevant control. This is a joint work with T. Horsin. James Kelliher (University of Riverside): Some recent results on the vanishing viscosity limit for incompressible fluids Abstract: I will give a brief overview of what is known regarding the vanishing viscosity limit of solutions to the Navier-Stokes equations for no-slip boundary conditions. I will then discuss recent work with Roger Temam and Xiaoming Wang in which we establish the corresponding limit for the infinite Prandtl-Darcy number Darcy-Brinkman-Boussinesq system. I will attempt to make clear in which way the problems are very similar, and also why the result can be obtained in the latter case though not (so far) in the former. Christophe Lacave (Université Paris 7): Two Dimensional Incompressible Flow around Several Obstacles Tending to a Curve Abstract: In the Atlantic ocean, the stream waters, when meeting the Brazil coast, create big vortices, which move to the gulf of Mexico. These vortices would be a cause of hurricane apparition which damages USA. A natural question is to know whether the chain of islands in the Caribbean sea could be a wall for these vortices. More precisely, we consider n disjoint obstacles of size ε, uniformly distributed on a ``imaginary'' open curve Γ. We study the behavior of an ideal incompressible flow around these obstacles, when the size tends to zero and the number of obstacles tends to infinity. The main goal is to determine the limit flow and to compare it with the flow around a curve. Is there a part of the flow which crosses the curve? This is work in collaboration with M.C. Lopes Filho and H.J. Nussenzveig Lopes. Marcel Lesieur (Institut Polytechnique de Grenoble et Académie des Sciences): Recent developments in large-eddy simulations of inhomogeneous turbulence Abstract: After having discussed the limits of turbulence direct-numerical simulations, one presents large-eddy simulations methods, where small scales of velocity and passive scalars are filtered out and modelled by appropriate eddy coefficients in the evolution of large scales. We show how to visualize vortices thanks to the Q-criterion. We concentrate on models developed originally in Fourier space, and leading in real space to the structure-function based models. We first study the incompressible or weakly-compressible plane channel, and discuss drag reduction by longitudinal riblets. Afterwards, the channel with spanwise rotation is looked at. We show a universality with anticyclonic rotating mixing layers. We consider also mixing in incompressible oaxial jets. Finally we discuss compressible turbulence LES with applications to a subsonic (Mach 0.7) and supersonic (Mach 1.4) round jet. Milton Lopes Filho (University of Campinas): Incompressible flow around small obstacles Abstract: The purpose of this talk is to briefly survey what is known about the asymptotic behavior of incompressible flow around small obstacles, the techniques of proof that have been used and some of the problems that remain open. Andro Mikelic (Université Lyon 1): Global-in-time solutions for the isothermal Matovich-Pearson equations Abstract: In this talk we present the mathematical study of the equations of Matovich and Pearson describing the process of glass fiber drawing. These equations may be viewed as a 1D-reduction of the incompressible Navier-Stokes equations including free boundary, valid for the drawing of a long and thin glass fiber. We concentrate on the isothermal case without surface tension. Then the equations of Matovich and Pearson represent a nonlinearly coupled system of an elliptic equation for the axial velocity and a hyperbolic transport equation for the fluid cross-sectional area. We first prove existence of a local solution, and, after constructing appropriate barrier functions, we deduce that the fluid radius is always strictly positive and that the local solution remains in the same regularity class. To the best of our knowledge, this is the first global existence and uniqueness result for this important system of equations. This is a joint work with Eduard Feireisl (Institute of Mathematics of the Academy of Sciences of the Czech Republic, Prague) and Philippe Laurençot (Institut de Mathématiques de Toulouse). Evelyne Miot (Università di Roma La Sapienza): The Cauchy problem for the 3-D Vlasov-Poisson system with point charges. Abstract: In this talk, I will consider a modified Vlasov-Poisson system describing the evolution of a three-dimensional bounded distribution of electrical particles, a plasma, interacting with a finite number of charged point particles. I will present a global existence and uniqueness result in the case of repulsive interaction. This is joint work with Carlo Marchioro and Mario Pulvirenti. Gabriela Planas (University of Campinas): Some problems related to irreversible phase change with fluid flow Abstract: In this talk we present some models for the evolution of the process of irreversible phase change of certain pure materials by taking into account the effects of fluid flow in the molten regions. These models consist of systems of highly non-linear free-boundary equations which include a heat equation, a doubly nonlinear inclusion for the phase change and Navier-Stokes type equations singularly perturbed. This is a joint work with J.L. Boldrini and L.H. de Miranda. Fabio Ramos (University of Bonn and Universidade Federal do Rio de Janeiro): Scaling in turbulent flows: Heuristics and rigorous results Abstract: In this talk I will present some classical heuristics and related recent rigorous results for the Kolmogorov-Obukhov spectral scaling for fluid flows at high Reynolds number. In particular, I will describe the main ideas introduced in the recent work: "Universal bounds for the Littlewood-Paley first-order moments of the 3D NSE", joint with Felix Otto. Geneviève Raugel (Université Paris-Sud): Dynamics of two-dimensional second grade fluids Abstract: In this talk, we consider the second grade fluid equations in a two-dimensional torus T2, with initial data in the Hilbert space H3(T2)2 and a forcing term in H1(T2)2. These equations depend on a positive “material coefficient” α. We quickly show that, if the forcing term belongs to Hm(T2)2, where m > 1, then the compact global attractor Aα in H3(T2)2 actually belongs to a more regular space Hs(T2)2, s > 3. If α is sufficiently small, Aα is bounded in Hm+2(T2)2.When α is small, the semiflow Sα(t) generated by the second grade fluid equations can be considered as a non regular perturbation of the semiflow S0(t) generated by the Navier-Stokes equations. Using the previously mentioned regularity result, we are comparing the dynamics of Sα(t) with the dynamics of S0(t), when α goes to zero. Luis Miguel Rodrigues (Université Lyon 1): Vortex-like finite-energy asymptotic profiles for isentropic compressible flows Abstract: Bidimensional incompressible viscous flows with well-localised vorticity are well-known to develop vortex structures. Our purpose is to recover the asymptotic profiles describing these phenomena for homogeneous finite-energy flows as asymptotic profiles for near-equilibrium isentropic compressible flows. This task is performed by extending the sharp description of the asymptotic behaviour of near-equilibrium compressible flows obtained by David Hoff and Kevin Zumbrun to the case of finite-energy vortex-like solutions. Maria Schonbek (University of Santa Cruz): Norm inflation for incompressible magneto-hydrodynamic system Abstract: Based on the construction of Bourgain and Pavlovic , I will show that the solutions to the Cauchy problem for the three dimensional different types of norm inflations in B−1,∞∞. Particularly the magnetic field can develop norm inflation in short time even when the velocity remains small and vice verse.
Lodging
We have secured a limited number of apartments at the Park Avenue Aparthotel untill June 15. We have special prices for the participants at the Summer School: 50 Euros/night for a small 1 bedroom apartment including breakfast (except on Sunday). If you wish to stay at the Park Avenue Aparthotel and take advantage of this special price, please let us know at fluides2010@math.univ-lyon1.fr before June 15. We will make the reservation for you. Please do not wait for the deadline; the reservations will be treated by order of arrival and we only have a limited number of prereserved apartments. The Aparthotel also has one or two apartments with two bedrooms for 60 Euros/night. This is convenient for two participants willing to share (in this case the price is 30 Euros per participant). If you are interested in one of these larger apartments, please let us know as soon as possible (well in advance of the deadline). All lectures will take place at the Institut Camille Jordan (Braconnier building). Instructions to get from the Park Avenue Aparthotel to the Institut Camille Jordan. Take Métro B towards Charpennes and get off at the Terminus (Charpennes Charles Hernu). Next take tram T1 towards IUT-Feyssine and get off at Université Lyon 1. The Braconnier building will be on your left. You can download a basic map of the public transportation in Lyon from here. A detailed map can be found here. Tourism Some tourist activities for the week-end will be proposed later.
Registration
All participants are required to fill out the registration form for the Summer School, for organizational reasons. To register, download the registration form (also available in pdf, odt and rtf) , fill it out and send by email to fluides2010@math.univ-lyon1.fr All registrations will receive a confirmation email. If you did not receive the confirmation email, please resend the registration form. Deadline for sending the registration form: June 15th, 2010 Registration is free but mandatory. Funding The CNRS researchers will have local expenses covered by the organizers (within the limit of the available funding). Some funding is also available to support students and young researchers. If you are interested to obtain funding support from the organizers, please send an application to fluides2010@math.univ-lyon1.fr. The application should include a vita, and for students an additional short letter of recommendation from the PhD advisor. Deadline to request financial support (for CNRS researchers, students and young researchers): June 1st, 2010 Oral presentations and posters There is some limited space for contributed presentations. If you would like to submit a short oral communication or poster then please let us know by email fluides2010@math.univ-lyon1.fr no later than June 1st, 2010.
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