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

Authors:
Rabi N. Bhattacharya, Larry Chen, Scott Dobson, Ronald B. Guenther, Chris Orum, Mina Ossiander, Enrique Thomann and Edward C. Waymire

Journal:
Trans. Amer. Math. Soc. **355** (2003), 5003-5040

MSC (2000):
Primary 35Q30, 76D05; Secondary 60J80, 76M35

DOI:
https://doi.org/10.1090/S0002-9947-03-03246-X

Published electronically:
July 24, 2003

MathSciNet review:
1997593

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Abstract | References | Similar Articles | Additional Information

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:
https://doi.org/10.1090/S0002-9947-03-03246-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

Published electronically:
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

Article copyright:
© Copyright 2003
American Mathematical Society