Semi-Markov Cascade Representations of Local Solutions to 3-D Incompressible Navier-Stokes

Abstract

AbstractA probabilistic approach int roduced by LeJan and Sznitrnan (1997) perm its derivation of weak solutions to 3-d incompressible Navier-Stokes equations whose Fourier transform may be rep resented by an expected value of a stochastic cascade. This approach was extended in Bhattacharya et al. (2003) by met hods which would yield unique global solutions by a stochastic representation under “small initial data conditions”. A connection to iterative contraction maps on appropriate function space was also provided which would also yield local existence and uniqueness under “short time” constraint s, but with out stochastic cascade representations. In t he present paper the authors (i.) Provide a stochastic cascade representation for local solutions, and (ii.) Provide time-asymptotics for global solutions from t he stochastic representation.Key wordssemi-Markov multiplicative stochastic cascadebranching random walkincompressible Navier -Stokeslocal solutionstime-asymptoticsAMS(MOS) subject classificationsPrimary 35Q3076D05Secondary 60J8076M3S