An Eulerian-Lagrangian approach to the Navier-Stokes equations

Details An Eulerian-Lagrangian approach to the Navier-Stokes equations An Eulerian-Lagrangian approach to the Navier-Stokes equations

2000-05-11

arxiv.org/abs/math/0005116v1

Mathematics

Analysis of PDEs

Scientific paper

10.1007/s002200000349

This work presents an approach to the Navier-Stokes equations that is phrased in unbiased Eulerian coordinates, yet describes objects that have Lagrangian significance: particle paths, their dispersion and diffusion. The commutator between Lagrangian and Eulerian derivatives plays an important role in the Navier-Stokes equations: it contributes a singular perturbation to the Euler equations, in addition to the Laplacian. Bounds for the Lagrangian displacements, their first and second derivatives are obtained without assumptions. Some of these rigorous bounds can be interpreted in terms of the heuristic Richardson law of pair dispersion in turbulent flows.

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