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Cubo (Temuco)

versión On-line ISSN 0719-0646

Cubo vol.12 no.2 Temuco  2010

http://dx.doi.org/10.4067/S0719-06462010000200006 

CUBO A Mathematical Journal Vol.12, N°02, (77–96). June 2010

 

Generalized solutions of the Cauchy problem for the Navier-Stokes system and diffusion processes

 

S. Albeverio and Ya. Belopolskaya

Institut für Angewandte Mathematik, Universität Bonn, Wegelerstr. 6, D-53115 Bonn, Germany SFB 611,HCM, Bonn, BiBoS, Bielefeld - Bonn, CERFIM, Locarno and USI (Switzerland) email: albeverio@uni-bonn.de

St.Petersburg State University for Architecture and Civil Engineering, 2-ja Krasnoarmejskaja 4, 190005, St.Petersburg, Russia email: yana@yb1569.spb.edu


ABSTRACT

We reduce the construction of a weak solution of the Cauchy problem for the Navier-Stokes system to the construction of a stochastic problem solution. Under suitable conditions we solve the stochastic problem and prove that simultaneously we obtain a weak (generalized) solution to the Cauchy problem for the Navier-Stokes system.

Key words and phrases: Stochastic flows, diffusion processes, nonlinear parabolic equations, Cauchy problem.


RESUMEN

Nosotros reducimos la construcción de una solución débil de un problema de Cauchy para el sistema de Navier-Stokes para la construcción de la resolución de un problema estocástico. Bajo condiciones convenientes resolvimos el problema estocástico y probamos que simultáneamente obtenemos una solución débil (generalizada) para el problema de Cauchy del sistema de Navier-Stokes.

AMS Subj. Class.: 60H10, 60J60 , 35G05, 35K45


References

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Received: February 2009.

Revised: March 2009.