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A PROBABILISTIC REPRESENTATION FOR THE VORTICITY OF A THREE-DIMENSIONAL VISCOUS FLUID AND FOR GENERAL SYSTEMS OF PARABOLIC EQUATIONS

Published online by Cambridge University Press:  23 May 2005

Barbara Busnello
Affiliation:
Dipartimento di Matematica, Università di Pisa, Via Buonarroti 2, 56127 Pisa, Italy
Franco Flandoli
Affiliation:
Dipartimento di Matematica Applicata, Università di Pisa, Via Bonanno 25/b, 56126 Pisa, Italy ([email protected])
Marco Romito
Affiliation:
Dipartimento di Matematica, Università di Firenze, Viale Morgagni 67/a, 50134 Firenze, Italy ([email protected])
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Abstract

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A probabilistic representation formula for general systems of linear parabolic equations, coupled only through the zero-order term, is given. On this basis, an implicit probabilistic representation for the vorticity in a three-dimensional viscous fluid (described by the Navier–Stokes equations) is carefully analysed, and a theorem of local existence and uniqueness is proved. The aim of the probabilistic representation is to provide an extension of the Lagrangian formalism from the non-viscous (Euler equations) to the viscous case. As an application, a continuation principle, similar to the Beale–Kato–Majda blow-up criterion, is proved.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2005