Special Topic: Incompressible Navier–Stokes Equations and the Le Jan–Sznitman Cascade

Abstract

The three-dimensional incompressible Navier–Stokes equations are nonlinear partial differential equations formulated in an effort to embody the basic physics of fluid flow in accordance with Newton’s laws of motion (See Landau and Lifshitz. Fluid mechanics, course of theoretical physics, vol. 6, 2nd revised edn. Pergamon Press, Oxford (1987)). The equations are involved in models of fluid flow with applications ranging from modeling ocean currents in oceanography to blood flow in medicine, among many others. However, understanding the existence and/or uniqueness of smooth solutions to these equations, when the initial data is smooth, presents one of the great unsolved mathematical challenges of the twentieth and twenty-first centuries (Ladyzhenskaya (Russian Math Surv 58(2):25–286, 2003), Fefferman (Existence and smoothness of the Navier–Stokes equation. In: Millennium prize problems. Clay Mathematics Institute, Cambridge, pp 57–67 (2006)). The main goal of the present chapter is to present a probabilistic cascade model of LeJan and Sznitman (LeJan and Sznitman (Prob Theory Rel Fields 109:343–366, 1997)) and its subsequent extensions, in which solutions may be represented as expected values of a certain vector product over a random tree, provided the expectations exist.