

Fluid mechanics has always been the source of challenging mathematical problems. This workshop 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 decades.
The purpose of this workshop is to bring together recognized experts and young researchers working in this field and to foster scientific interaction among the participants. The minicourse of Prof. Cordoba and the invited talks will reflect the variety of mathematical tools and problems arising in the study of the dynamical aspects of fluids. This meeting will take place at the University of Lyon 1, Institut Camille Jordan (the Mathematics department) in the Braconnier building. Lyon forms the secondlargest 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. Organizing Committee
Minicourse: Diego Córdoba (Instituto de Ciencias Matemáticas, Madrid): Interface dynamics for incompressible flows in 2D: SQG, Muskat and Water Waves Abstract: The evolution of an interface between two immiscible incompressible fluids can develop singularities in finite time. In particular those contour dynamics that are given by basic fluid mechanics systems; Euler´s equation, Darcy´s law and the Quasigeostrophic equation. These give rise to problems such as water wave, Muskat, and the evolution of sharp fronts of temperature. We will present the main ideas and arguments of the corresponding singularities for these models. Main Lectures: Christophe Cheverry (Université de Rennes 1): Long time gyrokinetic equations. Application to whistler waves Abstract: In this talk, I will present two results. The first provides a new approach allowing to extend in longer times the classical insights on fast rotating fluids. This will be applied to show that a plasma can be confined by a magnetic field. The second is based on a study of oscillatory integrals implying special phases. This will be applied to give a better understanding of whistlermode chorus emissions in space plasmas. The framework will be relativistic VlasovMaxwell equations, with a penalized skewsymmetric term where the inhomogeneity of the magnetic field plays an essential part. Gianluca Crippa (Universität Basel): Mixing and loss of regularity for twodimensional flows Abstract: Consider a passive scalar which is advected by a smooth incompressible twodimensional velocity field. The following question is of interest: Starting from a given initial distribution of the passive scalar, and given a certain energy budget, power budget, or palenstrophy budget, what velocity field best mixes the passive scalar? While it is easy to see that under energy bounds perfect mixing can be accomplished in finite time, it has been recently proven that under power bounds the mixing rate is at most exponential in time. In the talk I will present a construction from a work in progress with Giovanni Alberti (Pisa) and Anna Mazzucato (Penn State), illustrating the optimality of the exponential bound on the mixing rate under power bounds. The velocity field can in fact be required to satisfy $W^{1,p}$ bounds uniformly in time, for any value of $1 \leq p \leq \infty$: the Lipschitz case is also included. As a consequence, we deduce the existence of velocity fields in $W^{k,q}$ (where $k$ and $q$ are real numbers, so that such Sobolev space is not embedded in the Lipschitz space) such that any fractional regularity of the initial datum is instantaneously destroyed. Isabelle Gallagher (Université Paris 7): Blow up of critical norms for the NavierStokes equations Abstract: In this lecture we shall present some recent results on the behavior of the solutions of the NavierStokes equations near the (possible) blowup time. In particular we shall be interested in the blowup of scale invariant norms. David GérardVaret (Université Paris 7): The Taylor model in magnetohydrodynamics (MHD) Abstract: We shall discuss a model introduced by J.B. Taylor in 1963, that comes from a formal asymptotic limit of MHD equations with rotation. This asymptotic model, relevant to the Earth's dynamo problem, should in principle allow for easier numerics. However, its simulation has been unsuccessful so far, due to unclear stability properties. The aim of the talk is to present recent mathematical results on this stability issue (joint work with E. Dormy, I. Gallagher, L. SaintRaymond). Taoufik Hmidi (Université de Rennes 1): On the doubly connected Vstates for Euler equations Abstract: We shall discuss some results on the Vstates for the planar Euler equations. They are special patches rotating uniformly around their centers of mass . In the first part, we review some classical results on the simply connected case. What is known is that close to Rankine vortices the Vstates are described by a countable family of bifurcating onedimensional curves. We shall discuss in the second part of the talk how to extend this result to the Vstates with only one hole. Christophe Lacave (Université Paris 7): The Vortex Method in the Exterior of a Disk Abstract: The vortex method is a common numerical and theoretical approach used to implement the motion of an ideal flow, in which the vorticity is approximated by a sum of point vortices, so that the Euler equations read as a system of ordinary differential equations. Such a method is well justified in the full plane, thanks to the explicit representation formulas of Biot and Savart. In an exterior domain, we also replace the impermeable boundary by a collection of point vortices generating the circulation around the obstacle. The density of these point vortices is chosen in order that the flow remains tangent at midpoints between adjacent vortices. In this presentation, we provide a rigorous justification for this method in exterior domains. One of the main mathematical difficulties being that the BiotSavart kernel defines a singular integral operator when restricted to a curve. For simplicity and clarity, we only treat the case of the unit disk in the plane approximated by a uniformly distributed mesh of point vortices. This work is in collaboration with D. Arsénio (Paris Diderot) and E. Dormy (ENSParis). Milton Lopes Filho (Universidade Federal do Rio de Janeiro): The limit of small viscosity and small elastic response for the secondgrade fluid equations Abstract: In this talk we consider the secondgrade fluid system with viscosity $\nu$ and elastic response parameter $\alpha$ in a twodimensional, smooth bounded domain with smooth initial data and noslip boundary conditions. For a suitably defined converging family of initial data we consider a corresponding Family of solutions to this system, depending on these parameters. We examine the limits of vanishing $\alpha$ and $\nu$ obtaining, for certain regimes, convergence to solutions of the incompressible Euler equations, while in other regimes we obtain sharp criteria for convergence, analogous to Kato's criterion. This talk is based in joint work with H. Nussenzveig Lopes, E. Titi and A. Zang. Carlo Marchioro (Università di Roma "La Sapienza"): On the VlasovPoisson fluid with unbounded mass Abstract: We consider a onespecies plasma moving in an infinite cylinder, in which it is confined by means of a magnetic field diverging on the walls of the cylinder. It is assumed that initially the particles have bounded velocities and are distributed according to a density which is bounded, without any decreasing at infinity. Hence the total mass cound be infinite. The mutual interaction is of Yukawa type, i.e. Coulomb at short distance and exponentially decreasing at infinity. We prove the global in time existence and uniqueness of the time evolution of the plasma and its confinement. We discuss also the case of a plasma mouving in the whole space. Takayoshi Ogawa (Tohoku University): Threshold for the large time behavior of weak solutions to degenerate driftdiffusion system in between two critical exponents Abstract: We consider a degenerate driftdiffusion system which can be derived from the damped compressible Euler or NavierStokesPoisson system. Observing that there are two important critical exponents related with adiabatic constant, we introduce the threshold of initial mass for the large time behavior of weak solutions. We show that the threshold separating the global existence of the solution and finite time blowup of solutions is given by the best possible constant of the modified HardyLittlewoodSobolev inequality. Marius Paicu (Université Bordeaux 1): Global wellposedness for inhomogeneous fluids Abstract: In this talk I will present some results about the global wellposedness of inhomogeneous fluids with bounded density and some large initial data. In the twodimensional space we obtain the global solution for rough density and general velocity. In the three dimensional space we obtain the global wellposedness for a small inhomogeneity and an initial data verifying a nonlinear smallness condition. These results are in collaboration with J. Huang, P. Zhang and Z. Zhang. Geneviève Raugel (Université ParisSud): The "hyperbolic perturbed" NavierStokes equations revisited Abstract: In this talk, we consider the hyperbolic NavierStokes equations in the whole plane. We improve the known existence results of solutions. We show that these hyperbolic equations are a regular perturbation of the NavierStokes equations, which has interesting consequences in the qualitative study of the dynamics (partly joint work with M. Paicu). José L. Rodrigo (University of Warwick): On nonresistive MHD systems connected to magnetic relaxation. Abstract: In this talk I will present several results connected with the idea of magnetic relaxation for MHD, including some new commutator estimates (and a counterexample to the estimate in the critical case). This is joint work with various subsets of D. McCormick, J. Robinson, C. Fefferman and JY. Chemin. Miguel Rodrigues (Université Lyon 1): Asymptotic stability and modulation behavior near periodic waves of parabolic and Hamiltonian systems Abstract: Recently, mostly motivated by applications to surface waves, rapid progresses on the stability theory for periodic traveling waves have been obtained. We will review the essentially complete theory available for parabolic systems, that applies to the shallow water description of viscous rollwaves, and shall discuss properties of the linearized evolution about cnoidal waves of the Kortewegde Vries equation, that in particular model long water waves in the small amplitude regime. Frédéric Rousset (Université ParisSud): Landau damping in Sobolev spaces for a simple model Abstract: We shall study the dynamics in the vicinity of spatially homogeneous equilibria of a simple model of interacting particles, the VlasovHMF model. Joint work with E. Faou (Rennes). Franck Sueur (Université Paris 6): Dynamics of a point vortex as limit of a shrinking solid in a 2d perfect incompressible fluid Abstract: In this talk I will present some recent works with Olivier Glass, Christophe Lacave and Alexandre Munnier where we recover the dynamics of a vortex point as the limit of the motion of a rigid body immersed in a twodimensional perfect incompressible fluid when the size of the body vanishes and its mass as well. Contributed talks: Anna Bohun (Universität Basel): Lagrangian solutions to the VlasovPoisson equation with L^{1} density Abstract: In this talk I will review some quantitative stability estimates for Lagrangian flows related to vector fields whose gradient is given by the singular integral of an L^{1} function, and discuss how this method is used to prove existence of Lagrangian solutions of the Vlasov Poisson equation with density in L^{1}. This is a joint work with Gianluca Crippa (Basel) and Francois Bouchut (CNRS). Hugo Decaster (Université Lyon 1): Asymptotic behavior of solutions to the stationary NavierStokes equations Abstract: In this talk, I will address the problem of determining the asymptotic behavior of the solutions of the incompressible stationary NavierStokes system in R^{n}, in dimensions 2 and 3. In dimension 3, if the forcing term behaves at infinity as a field homogeneous of degree 3, we show that the solution behaves at infinity as a homogeneous of degree 1 velocity field which solves the NavierStokes equations with an additional Dirac mass in the forcing. This applies in particular to the case of an exterior domain or a rapidly decaying forcing term with non zero integral. In dimension 2, under some assumptions of symmetry for the forcing term, we show that the asymptotic behavior of the solution at infinity is homogeneous of degree 4 with an explicit formula. Omar Lazar (Instituto de Ciencias Matemáticas, Madrid): Global existence results for a 1D transport equation with nonlocal velocity Abstract: In this talk, we shall study a one dimensional equation introduced by Córdoba, Córdoba and Fontelos. This 1D transport equation can be viewed as a toy model for the 2D SQG equation. It also has some similarities with the BirkhoffRott equation modeling the evolution of a vortex sheet. We will establish some global existence results for this model when the data belong to some weighted spaces. This talk is based on a joint work with PierreGilles LemariéRieusset. Eleonora Pinto de Moura (Universidade Federal do Rio de Janeiro): The vortexwave system with a finite number of vortices as the limit of the Euleralpha model Abstract: In this talk I will discuss solutions of the twodimensional Euleralpha model when the initial vorticity is the superposition of a finite number of point vortices and a bounded background vorticity. Finally I will show that these solutions converge, as alpha tends to zero, to a weak solution of the vortexwave system, introduced by Marchioro and Pulvirenti (1991). This is a joint work with Helena J. Nussenzveig Lopes (UFRJ) and Milton C. Lopes Filho (UFRJ). Eugénie Poulon (Université Paris 6): Profile Decomposition and some applications to Incompressible NavierStokes Equation in R^{3} Abstract: I will present profile theory of P. Gérard and some applications to the Incompressible NavierStokes Equations in the whole space R^{3}. More precisely, I will explain how such a theory has played a key role in the proof of existence of critical elements (or minimal blow up solutions) in the case of data belonging to invariant spaces (under the natural scaling of the NavierStokes equations) and noninvariant spaces.
Registration
All participants are required to register by sending an email to mfd2014@math.univlyon1.fr specifying the name and the affiliation. Deadline for registration: October 1st, 2014. Registration is free but mandatory. Funding Some funding is available to support students and young researchers. If you are interested in obtaining funding support from the organizers, please send an application to mfd2014@math.univlyon1.fr. The application should include a vita, and a short recommendation from the PhD advisor in the case of students. Deadline to request financial support: the funding requests are treated by order of arrival. Please send your application form as soon as possible and in any case before October 1st, 2014. Carolina Azevedo Fernandes (INSA Lyon et Universidade Federal do Rio de Janeiro) Anna Bohun (Universität Basel) Alexandre Boritchev (Université Lyon 1) Lorenzo Brandolese (Université Lyon 1) Renata Bunoiu (Université de Metz) Valentina Busuioc (Université de SaintEtienne) Robert Chahine (École Centrale de Lyon) Abdennasser Chekroun (Université Lyon 1) Christophe Cheverry (Université de Rennes 1) Antoine Choffrut (University of Edinburgh) Sorin Ciuperca (Université Lyon 1) Diego Córdoba (Instituto de Ciencias Matemáticas, Madrid) Fernando Cortez (Université Lyon 1) Gianluca Crippa (Universität Basel) Francesco De Anna (Université Bordeaux 1) Hugo Decaster (Université Lyon 1) Louis Dupaigne (Université Lyon 1) Ennio Fedrizzi (Université Lyon 1) Adrien Fontaine (Université de Rennes 1) Isabelle Gallagher (Université Paris 7) Thierry Gallay (Université Grenoble 1) Ivan Gentil (Université Lyon 1) David GérardVaret (Université Paris 7) Nastasia Grubic (Instituto de Ciencias Matemáticas, Madrid) Taoufik Hmidi (Université de Rennes 1) Dragoş Iftimie (Université Lyon 1) Christophe Lacave (Université Paris 7) Omar Lazar (Instituto de Ciencias Matemáticas, Madrid) Milton Lopes Filho (Universidade Federal do Rio de Janeiro) JeanFrançois Maître (École Centrale de Lyon) Carlo Marchioro (Università di Roma "La Sapienza") Colin Mietka (Université Lyon 1) Petru Mironescu (Université Lyon 1) Evrad Ngom (Université Lyon 1) Takayoshi Ogawa (Tohoku University) Diego Alonso Orán (Instituto de Ciencias Matemáticas, Madrid) Marius Paicu (Université Bordeaux 1) Liviu Iulian Palade (INSA Lyon) Olivier Pierre (Université de Nantes) Eleonora Pinto de Moura (Universidade Federal do Rio de Janeiro) Eugénie Poulon (Université Paris 6) Marco Prestipino (Université Paris 7) Geneviève Raugel (Université ParisSud) GillesAlexis Renaut (École Centrale de Lyon) José L. Rodrigo (University of Warwick) Miguel Rodrigues (Université Lyon 1) Frédéric Rousset (Université ParisSud) Khaled Saleh (Université Lyon 1) Stefano Scrobogna (Université Bordeaux 1) Franck Sueur (Université Paris 6) Benjamin Texier (Université Paris 7) Donato Vallefuoco (École Centrale de Lyon) Mohammad Zafar (Indian Institute of Technology Bombay)
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 yave to let us know at mfd2014@math.univlyon1.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 Apparthotel 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, 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 IUTFeyssine 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 touristic activities for the weekend will be proposed later. 