Filtered Viscous Fluid Equations

The Eulerian description of an incompressible viscous fluid of constant density and temperature is concerned with the fluid velocity u(x, t) and pressure p(x, t) recorded at fixed positions x ∈ R, n = 3 as functions of time t. The solutions of the incompressible, inviscid Euler equations can be thought of as geodesic paths on an infinite dimensional group of transformations ([1]). This is done using the Lagrangian description. In this description the basic object is a transformation a 7→ X(a, t) that represents the position x = X(a, t) at time t of the fluid particle that started at t = 0 from a. At time t = 0 the transformation is the identity, X(a, 0) = a. An Eulerian-Lagrangian formulation of the Euler equations ([8]) can be written in terms of the inverse map A(x, t) = X−1(x, t) as (∂t + u · ∇)A = 0, u = W [A]