Introduced in the mid 19 century, the Navier-Stokes equations are used to analyze fluid flows from laminar to turbulent regimes. Despite their importance and usefulness in engineering and science, a complete theory establishing properties of solutions of these equations continues to be elusive. With applications to physical and biological sciences and aeronautical and naval engineering, the mechanism for energy transfer and dissipation governing fluid motions remain carefully concealed from the current methods to analyze these equations. Â The dramatic improvements in efficiency attained in aircraft, naval and automotive design, serve as testimony of the economic and societal impact of improved control of basic processes modeled by these equations.
In recent years problems in the study of differential equations in general, and in particularly of the Navier-Stokes equations, have given rise to interesting probabilistic structures. The current project aims to elucidate the relation between properties of solutions of the Navier-Stokes equations with properties of a class of branching Markov chains naturally associated to these equations. As well-illustrated by considering self-similar solutions of the Navier-Stokes equations, regularity properties as well as uniqueness of solutions corresponds to properties of a specific branching Markov chain in which the branching nodes have a law determined by the invariance of the Navier-Stokes equations under spatial dilation (with a corresponding time scale change) and rotations. This intrinsic branching structure motivates the formulation of an explosion problem that it is of interest in its own right from the probability point of view. It involves new considerations of the location of the left most particle of the branching process. Furthermore, the branching structure establishes a striking connection between nonlinear PDE's and branching processes that is the object of study in this proposal. A further specific objective of this proposal is to explore the consequences on the branching structure imposed by the incompressible character of the velocity field. Specifically, the Fourier transform of the solution of the Navier-Stokes equations can be represented as an expected value of a multiplicative functional defined on the nodes of the alluded branching Markov chain that reflects the incompressibility of the velocity field. An objective of this proposal is to develop the implications for energy depletion as a consequence of the algebraic structure defined by the indicated multiplication operation. Likewise, regularity and large time behavior of the solutions of the Navier-Stokes equation can also be gleaned from this representation. The proposal also involves methods of graph theory, with the use of semi-algebraic sets in the classification of nodes of random trees. Ultimately, this proposal seeks to elucidate the role of incompressibility in the multiplicative stochastic processes associated with equations of fluid flow.
