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.
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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. 
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