TY - CHAP

T1 - Diffusion of Lagrangian invariants in the Navier-Stokes equations

AU - Constantin, Peter

PY - 2004

Y1 - 2004

N2 - The incompressible Euler equations can be written as the active vector system (∂tgt; + u. ∇) A = 0 where u = W[A] is given by the Weber formula W[A] = P{(∇A)*v} in terms of the gradient of A and the passive field v = u0 (A). (P is the projector on the divergence-free part.) The initial data is A(x,0)= x, so for short times this is a distortion of the identity map. After a short time one obtains a new u and starts again from the identity map, using the new u instead of uo in the Weber formula. The viscous Navier-Stokes equations admit the same representation, with a diffusive back-to-labels map A and a v that is no longer passive.

AB - The incompressible Euler equations can be written as the active vector system (∂tgt; + u. ∇) A = 0 where u = W[A] is given by the Weber formula W[A] = P{(∇A)*v} in terms of the gradient of A and the passive field v = u0 (A). (P is the projector on the divergence-free part.) The initial data is A(x,0)= x, so for short times this is a distortion of the identity map. After a short time one obtains a new u and starts again from the identity map, using the new u instead of uo in the Weber formula. The viscous Navier-Stokes equations admit the same representation, with a diffusive back-to-labels map A and a v that is no longer passive.

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U2 - 10.1007/0-306-48420-x_35

DO - 10.1007/0-306-48420-x_35

M3 - Chapter

AN - SCOPUS:84859853004

SN - 1402009801

SN - 9781402009808

T3 - Fluid Mechanics and its Applications

SP - 285

EP - 294

BT - Tubes, Sheets and Singularities in Fluid Dynamics

PB - Kluwer Academic Publishers

ER -