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 -