Title:   Mean-field limits by the modulated energy method (Part 2)
Speaker:   Sylvia Serfaty
Abstract:   We will discuss joint works with Matt Rosenzweig, Antonin Chodron de Courcel and Hung Q. Nguyen, in which we study the question of mean-field limits, or deriving effective evolution equations of PDE type for a system of N points in singular interaction, for instance of Coulomb or Riesz, evolving by gradient flow or conservative flow (such as the point vortex system in 2D), with or without noise.
We will describe the convergence to the mean-field limit by the modulated energy method, that relies on a functional inequality of commutator estimate type. We will discuss various ways of proving the commutator estimate, and recent progress that provides sharp and localizable estimates.
We also discuss the question of obtaining global-in-time convergence and its connection with modulated log-Sobolev inequalities.
Hashtag: #workshop

Title:   Instability and non-uniqueness for the Euler equations of an ideal incompressible fluid (Part 2)
Speaker:   Mikhail Vishik
Abstract:   
We plan to discuss the following topics:

Evolution of small perturbations for an incompressible flow of an ideal fluid.
Definitions of instability, (non-selfadjoint) spectral theory, approximation theorem.
Bicharacteristic amplitude system of equations.
Essential spectral radius of the evolution operator.
Instability in an ideal incompressible fluid and Lyapunov exponents. Examples.
Growth bound equals spectral bound?
Discrete spectrum of the generator. Examples.
Instability of radial vortices.  
Non-uniqueness for the forced Euler equation in dimension 2 with a radially 
symmetric force.
Spectrum of the generator.
Construction of a nontrivial solution.
Hashtag: #workshop

Title:   Mean-field limits by the modulated energy method (Part 3)
Speaker:   Sylvia Serfaty
Abstract:   We will discuss joint works with Matt Rosenzweig, Antonin Chodron de Courcel and Hung Q. Nguyen, in which we study the question of mean-field limits, or deriving effective evolution equations of PDE type for a system of N points in singular interaction, for instance of Coulomb or Riesz, evolving by gradient flow or conservative flow (such as the point vortex system in 2D), with or without noise.
We will describe the convergence to the mean-field limit by the modulated energy method, that relies on a functional inequality of commutator estimate type. We will discuss various ways of proving the commutator estimate, and recent progress that provides sharp and localizable estimates.
We also discuss the question of obtaining global-in-time convergence and its connection with modulated log-Sobolev inequalities.
Hashtag: #workshop

Title:   Regularity theory for quasilinear hyperbolic equations (Part 1)
Speaker:   Igor Rodnianski
Abstract:   I will discuss a geometric approach to the local well-posedness problem for quasilinear hyperbolic equations and survey some of the advances made in the last 25 years.
Hashtag: #workshop

Title:   TBA
Speaker:   Yusheng Luo [Cornell University]

Abstract:   TBA
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Title:   Instability and non-uniqueness for the Euler equations of an ideal incompressible fluid (Part 3)
Speaker:   Mikhail Vishik
Abstract:   
We plan to discuss the following topics:

Evolution of small perturbations for an incompressible flow of an ideal fluid.
Definitions of instability, (non-selfadjoint) spectral theory, approximation theorem.
Bicharacteristic amplitude system of equations.
Essential spectral radius of the evolution operator.
Instability in an ideal incompressible fluid and Lyapunov exponents. Examples.
Growth bound equals spectral bound?
Discrete spectrum of the generator. Examples.
Instability of radial vortices.  
Non-uniqueness for the forced Euler equation in dimension 2 with a radially 
symmetric force.
Spectrum of the generator.
Construction of a nontrivial solution.
Hashtag: #workshop

Title:   Regularity theory for quasilinear hyperbolic equations (Part 2)
Speaker:   Igor Rodnianski
Abstract:   I will discuss a geometric approach to the local well-posedness problem for quasilinear hyperbolic equations and survey some of the advances made in the last 25 years.
Hashtag: #workshop

Title:   What would be a mathematical theory of turbulence? (Part 1)
Speaker:   Jacob Bedrossian
Abstract:   Statistical theories of hydrodynamic turbulence are of fundamental importance in many applications from aerospace, nautical, and civil engineering to weather and climate prediction. However, there is currently no predictive theory which starts only from the Navier-Stokes equations and accurately matches the observations, and certainly nothing of this kind which is mathematically rigorous. In this series I will introduce the basics of the Navier-Stokes equations at high Reynolds number and the idea of hydrodynamic turbulence. Then I will explain how to phrase the basic predictions of the statistical theories of turbulence such as K41 theory as concise, mathematically rigorous conjectures for statistically stationary solutions of the Navier-Stokes equations subjected to stochastic forcing in a periodic box. These remain far out of reach, so I will then discuss some work done by my collaborators and I on related problems, such as Batchelor's law of passive scalar turbulence, estimating Lyapunov exponents, proving non-uniqueness of stationary measures in degenerately forced SDEs, and dissipation in SDEs with highly degenerate damping, with the goal of building up tools and understanding related phenomena in simpler systems.
Hashtag: #workshop

Title:   What would be a mathematical theory of turbulence? (Part 2)
Speaker:   Jacob Bedrossian
Abstract:   Statistical theories of hydrodynamic turbulence are of fundamental importance in many applications from aerospace, nautical, and civil engineering to weather and climate prediction. However, there is currently no predictive theory which starts only from the Navier-Stokes equations and accurately matches the observations, and certainly nothing of this kind which is mathematically rigorous. In this series I will introduce the basics of the Navier-Stokes equations at high Reynolds number and the idea of hydrodynamic turbulence. Then I will explain how to phrase the basic predictions of the statistical theories of turbulence such as K41 theory as concise, mathematically rigorous conjectures for statistically stationary solutions of the Navier-Stokes equations subjected to stochastic forcing in a periodic box. These remain far out of reach, so I will then discuss some work done by my collaborators and I on related problems, such as Batchelor's law of passive scalar turbulence, estimating Lyapunov exponents, proving non-uniqueness of stationary measures in degenerately forced SDEs, and dissipation in SDEs with highly degenerate damping, with the goal of building up tools and understanding related phenomena in simpler systems.
Hashtag: #workshop

Title:   Regularity theory for quasilinear hyperbolic equations (Part 3)
Speaker:   Igor Rodnianski
Abstract:   I will discuss a geometric approach to the local well-posedness problem for quasilinear hyperbolic equations and survey some of the advances made in the last 25 years. 
Hashtag: #workshop