Abstract
We present a stochastic Lagrangian view of fluid dynamics. The velocity solving the deterministic Navier–Stokes equation is regarded as a mean time derivative taken over stochastic Lagrangian paths and the equations of motion are critical points of an associated stochastic action functional involving the kinetic energy computed over random paths. Thus the deterministic Navier–Stokes equation is obtained via a variational principle. The pressure can be regarded as a Lagrange multiplier. The approach is based on Itô’s stochastic calculus. Different related probabilistic methods to study the Navier–Stokes equation are discussed. We also consider Navier–Stokes equations perturbed by random terms, which we derive by means of a variational principle.
Highlights
We consider Navier–Stokes equations perturbed by random terms, which we derive by means of a variational principle
The dynamics of an incompressible viscous fluid is modeled by the Navier–Stokes equation, a second order, nonlinear, partial differential equation describing the balance of mass and momentum of the fluid flow
The goal of this review is to show that probabilistic methods play a central role in the study of the deterministic Navier–Stokes equation, both from a conceptual and a practical point of view
Summary
The dynamics of an incompressible viscous fluid is modeled by the Navier–Stokes equation, a second order, nonlinear, partial differential equation describing the balance of mass and momentum of the fluid flow. In the absence of external forces and considering a perfectly incompressible fluid, the Navier–Stokes equation reads ∂. Where the velocity field u is required to satisfy the incompressibility condition divu = 0, the fluid density being equal to 1. The Navier–Stokes equation is deterministic, but it is well known that some of its solutions seem to exhibit random behavior, which might eventually provide an insight into the onset of turbulence from a purely mathematical point of view, the very important problem of existence and smoothness of the solutions to Equation (1) remains largely unsolved to this day. In this article we will review various ways of introducing randomness into the analysis of the solutions to Equation (1)
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