by Jin Feng (University of Kansas)

On March 18, 1999, a small aircraft crashed near St. Louis, and the ensuing FAA investigation concluded that the crash was caused by wake turbulence from a helicopter that had just landed ahead of the plane. One of the FAA recommendations was that the characteristics of rotorcraft vortex descent should be more thoroughly investigated, and, in particular, hazards associated with rotor wash generated by helicopters while hovering or in air taxi operation should be investigated. This is some practical motivation for the mathematical work to model vortex dynamics and to understand the behavior of vortices over extended time periods.

A rigorous mathematical theory of fluid flow in three dimensions is still beyond our understanding, and although the understanding of fluids in two dimensions is much better, there is still much to be done in the mathematical foundations of two-dimensional fluid mechanics. In the middle of the last century the mathematical physicist Lars Onsager proposed an explanation of commonly observed long-time behaviors of 2-D (or nearly 2-D) flows, an explanation that was based on earlier work by C. C. Lin. In Onsager’s work an important principle was codified as the “micro-canonical variational principle” (MVP), which is a relationship between the two fundamental quantities **energy** and **entropy**.

A small research group, of which I am a member, recently met at the American Institute of Mathematics in Palo Alto with the goal of developing a general mathematical framework so that this principle can be derived rigorously from a non-equilibrium probabilistic model of fluid flow. The other group members are Fausto Gozzi (Luiss University in Rome), Tom Kurtz (University of Wisconsin), and Andrzej Swiech (Georgia Tech).

Our approach requires tools and techniques from many areas of mathematics:

- vortex dynamics with stochastic disturbances – highly nontrivial due to singularities hidden in the dynamics
- partial differential equations on spaces of probability measures – highly singular state spaces
- calculus of variations – the micro-canonical variational principle
- theory of large deviations – developed for general metric space valued Markov processes

Here are references that can be consulted to read more about the topics mentioned above:

- L. Ambrosio, N. Gigli, G. Savaré,
*Gradient flows in metric spaces and in the space of probability measures*, Birkhauser, 2005. - G. Eyink, K.R. Sreenivasan,
*Onsager and the theory of hydrodynamic turbulence*, Reviews of Modern Physics,**78**, Jan. 2006. - J. Feng, M. Katsoulakis,
*A Comparison Principle for Hamilton-Jacobi equations related to controlled gradient flows in infinite dimensions*, Archive for Rational Mechanics and Analysis, 2009. - J. Feng, T. G. Kurtz,
*Large deviations for stochastic processes}, Mathematical Surveys and Monographs*, strong>131, American Mathematical Society, Providence, RI, 2006. - J. Feng, A. Swiech, (with Appendix B by A. Stefanov),
*Optimal control for a mixed flow of Hamiltonian and gradient type in space of probability measures*, Transaction of AMS,**365**, 3987-4039. - P. L. Lions,
*On Euler Equations and Statistical Physics*, Scuola Normale Superiore, 1997. - S.R.S. Varadhan,
*Special Invited Paper: Large Deviations*, Annals of Probability, 2008.