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:
ergodic properties and the structure of invariant measures of Burgers and 2-dimensional Navier-Stokes equations;
stochastic Lagrangian models of fluids;
new vortex filament based models;
development of an analytical theory of Kolmogorov equations for stochastic fluids;
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. This 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)

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:

- ergodic properties and the structure of invariant measures of Burgers and 2-dimensional Navier-Stokes equations;
- stochastic Lagrangian models of fluids;
- new vortex filament based models;
- development of an analytical theory of Kolmogorov equations for stochastic fluids;
- 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. This 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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