×

A probabilistic representation for the vorticity of a three-dimensional viscous fluid and for general systems of parabolic equations. (English) Zbl 1075.76019

Summary: We give a probabilistic representation formula for general systems of linear parabolic equations, coupled only through the zero-order term. On this basis, an implicit probabilistic representation for the vorticity in a three-dimensional viscous fluid (described by the Navier-Stokes equations) is 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 [J. T. Beale, T. Kato and A. Majda, Commun. Math. Phys. 94, 61–66 (1984; Zbl 0573.76029)], is proved.

MSC:

76D06 Statistical solutions of Navier-Stokes and related equations
35Q30 Navier-Stokes equations
35R60 PDEs with randomness, stochastic partial differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H30 Applications of stochastic analysis (to PDEs, etc.)
60J45 Probabilistic potential theory
76M35 Stochastic analysis applied to problems in fluid mechanics

Citations:

Zbl 0573.76029
PDFBibTeX XMLCite
Full Text: DOI