Majorizing kernels and stochastic cascades with applications to incompressible NavierStokes 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), 50035040
MSC (2000):
Primary 35Q30, 76D05; Secondary 60J80, 76M35
Published electronically:
July 24, 2003
MathSciNet review:
1997593
Fulltext PDF Free Access
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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 NavierStokes 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 973314605
Email:
chen@math.orst.edu
Scott Dobson
Affiliation:
Department of Mathematics, LinnBenton Community College, Albany, Oregon 97321
Address at time of publication:
Department of Mathematics, Oregon State University, Corvallis, Oregon 973314605
Email:
dobsons@attbi.com
Ronald B. Guenther
Affiliation:
Department of Mathematics, Oregon State University, Corvallis, Oregon 973314605
Chris Orum
Affiliation:
Department of Mathematics, Oregon State University, Corvallis, Oregon 973314605
Email:
orum@math.orst.edu
Mina Ossiander
Affiliation:
Department of Mathematics, Oregon State University, Corvallis, Oregon 973314605
Email:
ossiand@math.orst.edu
Enrique Thomann
Affiliation:
Department of Mathematics, Oregon State University, Corvallis, Oregon 973314605
Email:
thomann@math.orst.edu
Edward C. Waymire
Affiliation:
Department of Mathematics, Oregon State University, Corvallis, Oregon 973314605
Email:
waymire@math.orst.edu
DOI:
http://dx.doi.org/10.1090/S000299470303246X
PII:
S 00029947(03)03246X
Keywords:
Multiplicative cascade,
branching random walk,
incompressible NavierStokes,
FeynmanKac,
reactiondiffusion
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 DMS0073958, DMS0073865 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
