Some open problems and research directions in the mathematical study of dynamics

This is an essay in the literal sense: an attempt. As such, it does not con-form to the norm of scientific objectivity but attempts to describe a point of view. In it I describe a number of questions concerning the mathematics of fluids. They range from rather broad issues to technical problems that serve a specific, limited purpose. Some of these questions can be phrased with great precision; others I will have to leave in a form that calls for further devel-opment. The questions and directions discussed here make up an incomplete and personal wish-list; it is my hope that some of them will serve in the development of the field. In an editorial of the Notices of the AMS (vol. 47, Number 3, March 2000), Felix Browder, President of the AMS, refers to “... some of the ma-jor classical problems: the Riemann Hypothesis, the Poincare ́ Conjecture, and the regularity of three-dimensional fluid flows”. I imagine that many beginning graduate students in Mathematics have heard of the first two of these problems, but maybe not so many know about the third. I would like to describe here this third problem in a broader context, involving not only PDE questions of existence, uniqueness and regularity of solutions, but also dynamical issues concerning stability and statistical questions raised by instability. Ordinary incompressible Newtonian fluids are described by the Navier-Stokes equations. These equations have been used by engineers and physi-cists with a great deal of success. The range of their validity and appli-cability is well established. Together with other fundamental systems like the Schrödinger and Maxwell equations, these equations are among the most important equations of mathematical physics. 1 1 PDE There are two ways of describing fluids. The Eulerian description is con-cerned with the fluid velocity u(x, t), density ρ(x, t), and pressure p(x, t) recorded at fixed positions x ∈ Rn, n = 3 as functions of time t. The Navier-Stokes equations relating these quantities to each other is an expression of the balance of forces according to Newton’s second law, F = ma. I will take the density ρ to be constant in order to write the simplest form of the equations. There are n equation