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. (preprint available at http://arxiv.org/pdf/math.PR/0306075.pdf)
A probabilistic representation for the vorticity of a 3D viscous fluid and for general systems of parabolic equations / B. BUSNELLO; F. FLANDOLI; M. ROMITO. - In: PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY. - ISSN 0013-0915. - STAMPA. - 48 (no. 2):(2005), pp. 295-336. [10.1017/S0013091503000506]
A probabilistic representation for the vorticity of a 3D viscous fluid and for general systems of parabolic equations
ROMITO, MARCO
2005
Abstract
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. (preprint available at http://arxiv.org/pdf/math.PR/0306075.pdf)I documenti in FLORE sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.