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Analytical and Stochastic Fluid Dynamics October 10, 2005 - October 14, 2005
Registration Deadline: October 01, 2005 over 11 years ago
To apply for Funding you must register by: July 10, 2005 almost 12 years ago
Parent Program: --
Organizers Craig Evans, Susan Friedlander, Boris Rozovsky, Daniel Tataru and David A. Ellwood

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The PDEs known as the Euler and Navier-Stokes equations are important for a number of reasons. They describe the motion of fluids under a wide range of conditions. The Euler equations provide a model for inviscid (i.e., zero frictional) fluid behavior and were presented by Euler in the 1750s. The Navier-Stokes equations include the effects of friction and date to the mid 1800s. Probabilistic versions of the equations provide a model for turbulent behavior. Even though the fluid equations have a long and distinguished history, many of the fundamental mathematical questions associated with them remain an open challenge. For example, the existence and uniqueness of physically reasonable solutions for the Navier-Stokes equation was chosen to be one of the "million dollar" prize problems identified by the Clay Mathematical Institute. Stochastic equations of fluid dynamics is an emerging field bringing together experts from mechanics of fluids, PDEs and stochastic analysis. It has long been suspected that Navier-Stokes and Euler equations with random perturbations might serve as an important mathematical model for the turbulent motion of a fluid with high Reynolds number. For such equations there are many related open problems such as the existence of stochastic flows, Lyapunov exponents and attractors, qualitative properties of invariant measures, large deviations principle. There are also longstanding problems concerning the 'blow-up' of solutions of Navier-Stokes and Euler equations and the detailed connection with the onset of turbulence. Some of the recent developments in stochastic fluids are very promising. These include:

  1. ergodic properties and the structure of invariant measures of Burgers and 2-dimensional Navier-Stokes equations;
  2. stochastic Lagrangian models of fluids;
  3. new vortex filament based models;
  4. development of an analytical theory of Kolmogorov equations for stochastic fluids;
  5. applications of Wiener chaos to de-coupling of the Reynolds equation.

These problems have also been studied in depth by the mathematical physicists. In particular, substantial progress has been made in understanding of Kraichnan's model of turbulence and intermittency effects. Clay Mathematics Institute logoThis workshop is partially supported by the Clay Mathematics Institute. Lecture Schedule with Talk Abstracts      SCHEDULE Monday, October 10 9:30-10:00 Mohammed Ziane, “Remarks on the normal form of the Navier-Stokes equations” 10:00-10:45 Peter Constantin, “Nonlinear Fokker-Planck Navier-Stokes Systems” 10:45-11:15 Morning Tea (6th Floor) 11:15-12:00 Edriss Titi, “Global Regularity for the Three-dimensional Primitive Equations of Ocean and Atmosphere Dynamics” 12:00-2:45 Lunch/Discussion 2:45-3:30 Sergei B. Kuksin, “Asymptotic properties of some SPDE with small dissipation” 3:30-4:00 Afternoon Tea (6th Floor) 4:00-4:45 Franco Flandoli, “Markov selections and their regularity for 3D stochastic Navier-Stokes equations” 4:45-5:15 Roman Shvydkoy, “Spectral problem for the Euler and Navier-Stokes equations” Tuesday, October 11 9:30-10:00 Eric Vanden-Eijnden, "Simple Solvable Models with Cascade of Energy and Anomalous Dissipation" 10:00-10:45 Jonathan Mattingly, “Exponential Mixing for the Degenerately forced Navier Stokes Equations” 10:45-11:15 Morning Tea (6th Floor) 11:15-12:00 Vladimir Sverak, “Regularity of L^{3,\infty} solutions of the Navier-Stokes equations" 12:00-2:45 Lunch/Discussion 2:45-3:30 Charlie Fefferman, “The surface QG-alpha equation” 3:30-4:00 Afternoon Tea (6th Floor) 4:00-4:45 Tom Hou, “The Interplay between Local Geometric Properties and the Global Regularity for 3D Incompressible Flows” 4:45-5:15 Jim Kelliher, “Bounded domain limit for Navier-Stokes and Euler equations” Wednesday, October 12 9:30-10:00 Igor Kukavica, “One direction and one component regularity for the Navier-Stokes equations” 10:00-10:45 Andrea Bertozzi, “Electrowetting in a Hele-Shaw geometry” 10:45-11:15 Morning Tea (6th Floor) 11:15-12:00 Giovanni Gallavotti, “Chaotic motions and developed turbulence: heuristic ideas” 12:00-2:45 Lunch/Discussion 2:45-3:30 Claude Bardos, “Analytic stability and singularities for Kelvin Helmholtz, Rayleigh Taylor, Problems Comparison with the stability of water waves problems” 3:30-4:00 Afternoon Tea (6th Floor) 4:00-4:45 Herbert Koch, “Regularity for a free boundary problem and a conjecture of De Giorgi” 4:45-5:15 Natasa Pavlovic, “Long time behavior of solutions to the 3D Navier-Stokes” Thursday, October 13 9:30-10:00 Anna L. Mazzucato, “On the decay of the energy spectrum for weak solutions to the Navier-Stokes equations” 10:00-10:45 Anatoli Babin, “Linear superposition of nonlinear waves” 10:45-11:15 Morning Tea (6th Floor) 11:15-12:00 Susan Friedlander, “Nonlinear Instability for the Navier Stokes Equations” 12:00-2:45 Lunch/Discussion 2:45-3:30 Poster Session 3:30-4:00 Afternoon Tea (6th Floor) 4:00-4:45 Poster Session 4:45-5:15 Discussion Workshop Friday, October 14 9:30-10:00 Boris Rozovsky, “Passive Scalar Equation in a Turbulent Gaussian Velocity Field” 10:00-10:45 Alexandre Chorin, “Scaling laws in turbulence” 10:45-11:15 Morning Tea (6th Floor) 11:15-12:00 Remigijus Mikulevicius, “On stochastic Euler equation” 12:00-2:45 Lunch/Discussion 2:45-3:30 Xiaoming Wang, “The Emergence of Large Scale Coherent Structure under Small Scale Random Bombardments” 3:30-4:00 Afternoon Tea (6th Floor)

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Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC
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To apply for funding, you must register by the funding application deadline displayed above.

Students, recent Ph.D.'s, women, and members of underrepresented minorities are particularly encouraged to apply. Funding awards are typically made 6 weeks before the workshop begins. Requests received after the funding deadline are considered only if additional funds become available.

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A block of rooms has been reserved at the Hotel Durant. Reservations may be made by calling 1-800-238-7268. When making reservations, guests must request the MSRI preferred rate. If you are making your reservations on line, please go to this link and enter the promo/corporate code MSRI123. Our preferred rate is $129 per night for a Deluxe Queen/King, based on availability.

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Oct 10, 2005
09:30 AM - 10:00 AM
  Remarks on the Normal Form of the Navier-Stokes Equations
Mohammed Ziane
10:00 AM - 10:45 AM
  Nonlinear Fokker-Planck Navier-Stokes Systems
Peter Constantin
11:15 AM - 12:00 PM
  Global Regularity for the Three-dimensional Primitive Equations of Ocean and Atmosphere Dynamics
Edriss Titi
02:45 PM - 03:30 PM
  Asymptotic Properties of Some SPDE with Small Dissipation
Sergej Kuksin
04:00 PM - 04:45 PM
  Markov Selections and Their Regularity for 3D Stochastic Navier-Stokes Equations.
Franco Flandoli
04:45 PM - 05:15 PM
  Spectral Problem for the Euler and Navier-Stokes Equations
Roman Shvydkoy (University of Illinois, Chicago)
Oct 11, 2005
09:30 AM - 10:00 AM
  Simple Solvable Models with Cascade of Energy and Anomalous Dissipation
Eric Vanden-Eijnden (New York University, Courant Institute)
10:00 AM - 10:45 AM
  Exponential Mixing for the Degenerately Forced Navier Stokes Equations
Jonathan Mattingly (Duke University)
11:15 AM - 12:00 PM
  Regularity of L^{3,\infty} Solutions of the Navier-Stokes Equations
Vladimir Sverak (University of Minnesota Twin Cities)
02:45 PM - 03:30 PM
  The Surface QG-alpha Equation
Charles Fefferman (Princeton University)
04:00 PM - 04:45 PM
  The Interplay between Local Geometric Properties and the Global Regularity for 3D Inocompressible Flows.
Thomas Hou
04:45 PM - 05:15 PM
  Bounded Domain Limit for Navier-Stokes and Euler Equations
Jim Kelliher
Oct 12, 2005
09:30 AM - 10:00 AM
  One Direction and One Component Regularity for the Navier-Stokes Equations
Igor Kukavica (University of Southern California)
10:00 AM - 10:45 AM
  Electrowetting in a Hele-Shaw Geometry
Andrea Bertozzi (University of California, Los Angeles)
11:15 AM - 12:00 PM
  Chaotic Motions and Developed Turbulence: Heuristic Ideas
Giovanni Gallavotti
02:45 PM - 03:30 PM
  Analytic Stability and Singularities for Kelvin Helmholtz, Rayleigh Taylor, Problems Comparison with the Stability of Water Waves Problems
Claude Bardos
04:00 PM - 04:45 PM
  Regularity for a Free Boundary Problem and a Conjecture of De Giorgi
Herbert Koch (Rheinische Friedrich-Wilhelms-Universit├Ąt Bonn)
04:45 PM - 05:15 PM
  Long time Behavior of Solutions to the 3D Navier-Stokes
Natasa Pavlovic (University of Texas)
Oct 13, 2005
09:30 AM - 10:00 AM
  On the Decay of the Energy Spectrum for Weak Solutions to the Navier-Stokes Equations
Anna Mazzucato (Pennsylvania State University)
10:00 AM - 10:45 AM
  Linear Superposition of Nonlinear Waves
Anatoli Babine
Oct 14, 2005
09:30 AM - 10:00 AM
  Passive Scalar Equation in a Turbulent Gaussian Velocity Field
Boris Rozovsky (Brown University)
10:00 AM - 10:45 AM
  Scaling Laws in Turbulence
Alexandre Chorin (University of California, Berkeley)
11:15 AM - 12:00 PM
  On Stochastic Euler equation
Remigijus Mikulevicius
02:45 PM - 03:30 PM
  The Emergence of Large Scale Coherent Structure under Small Scale Random Bombardments
Xiaoming Wang