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Majorizing kernels and stochastic cascades with applications to incompressible Navier-Stokes equations

Author(s): Rabi N. Bhattacharya; Larry Chen; Scott Dobson; Ronald B. Guenther; Chris Orum; Mina Ossiander; Enrique Thomann; Edward C. Waymire
Journal: Trans. Amer. Math. Soc. 355 (2003), 5003-5040.
MSC (2000): Primary 35Q30, 76D05; Secondary 60J80, 76M35
Posted: July 24, 2003
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Abstract: A general method is developed to obtain conditions on initial data and forcing terms for the global existence of unique regular solutions to incompressible 3d Navier-Stokes equations. The basic idea generalizes a probabilistic approach introduced by LeJan and Sznitman (1997) to obtain weak solutions whose Fourier transform may be represented by an expected value of a stochastic cascade. A functional analytic framework is also developed which partially connects stochastic iterations and certain Picard iterates. Some local existence and uniqueness results are also obtained by contractive mapping conditions on the Picard iteration.

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Additional Information:

Rabi N. Bhattacharya
Affiliation: Department of Mathematics, University of Arizona, Tucson, Arizona 85721
Email: rabi@math.arizona.edu

Larry Chen
Affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331-4605
Email: chen@math.orst.edu

Scott Dobson
Affiliation: Department of Mathematics, Linn-Benton Community College, Albany, Oregon 97321
Address at time of publication: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331-4605
Email: dobsons@attbi.com

Ronald B. Guenther
Affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331-4605

Chris Orum
Affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331-4605
Email: orum@math.orst.edu

Mina Ossiander
Affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331-4605
Email: ossiand@math.orst.edu

Enrique Thomann
Affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331-4605
Email: thomann@math.orst.edu

Edward C. Waymire
Affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331-4605
Email: waymire@math.orst.edu

DOI: 10.1090/S0002-9947-03-03246-X
PII: S 0002-9947(03)03246-X
Keywords: Multiplicative cascade, branching random walk, incompressible Navier-Stokes, Feynman-Kac, reaction-diffusion
Received by editor(s): June 15, 2002
Received by editor(s) in revised form: October 16, 2002
Posted: July 24, 2003
Additional Notes: This research was partially supported by Focussed Research Group collaborative awards DMS-0073958, DMS-0073865 to Oregon State University and Indiana University by the National Science Foundation. The first author was also supported by a fellowship from the Guggenheim Foundation
Copyright of article: Copyright 2003, American Mathematical Society

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