Collaborative Research: Stochastic and Multiscale Structure Associated with the Navier Stokes Equations

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This project will develop the theory of the motion of fluids as embodied by the Navier-Stokes equations using new probabilistic methods that exploit the power of stochastic calculus and probabilistic limit theory. Although the Navier-Stokes equations are essentially deterministic, the approach used in this work will build on a representation of the equations as a functional of an underlying branching random walk. This representation, which was recently discovered by LeJan and Sznitman in France, is clearly intrinsic to the structure of the Navier-Stokes equations. While this is not the first attempt to use stochastic methods in connection with the flows associated with the Navier-Stokes equations, it does represent an entirely new direction which has the potential to transcend much of existing theory. Specific problems considered in this proposal seek to provide a better understanding of the role of spatial dimensions, boundary conditions, multi-scaling exponents and singularities, viscosity, homogeneity, isotropy and rotational accelerations, stationary flows and long-time evolution. The Navier-Stokes equations describe the basic physics governing the motion of fluid in its various forms of air, water, oil, etc. As such these equations play a fundamental role in science and engineering through the modeling of all varieties of fluid flow, from atmospheric and oceanic circulation to the flow of water beneath the earth's surface. Improved understanding of these equations and their solutions is essential to applications which range from tracking climate change and dispersion of contaminants in the Earth's environment, to more stable aerospace and sea vessel designs. The nonlinearity inherent in these equations makes explicit solutions possible only for the simplest of flows. Consequently the development of a more complete understanding of these equations at all physical length scales ranks among the most important outstanding problems of contemporary mathematical physics