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

Full-text PDF Free Access

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.

**[1]**Albeverio, S. and R. Høegh-Krohn (1976): Mathematical theory of Feynman path integrals, Lecture Notes in Mathematics 523, Springer-Verlag, NY. MR**58:14535****[2]**Aronszajn, N. and K.T. Smith (1961): Theory of Bessel potentials I,*Ann. Inst. Fourier (Grenoble)***11**385-475. MR**26:1485****[3]**Bhattacharya, R.N. and R.R. Rao (1976): Normal approximation and asymptotic expansions, Wiley, NY. MR**55:9219****[4]**Cannone, M. and F. Planchon (2000): On the regularity of the bilinear term for solutions to the incompressible Navier-Stokes equations,*Revista Matemática Iberoamericana***16**1-16. MR**2001d:35158****[5]**Cannone, M and Y. Meyer (1995): Littlewood-Paley decomposition and the Navier-Stokes equations in ,*Math. Appl. Anal.***2**307-319. MR**96h:35151****[6]**Chen, Zhi Min and Zhouping Xin (2001): Homogeneity criterion for the Navier-Stokes equations in the whole space,*Journal of Mathematical Fluid Mechanics***3**152-182. MR**2002d:76033****[7]**Chen, L., S. Dobson, R. Guenther, C. Orum, M. Ossiander, E. Waymire (2003): On Itô's complex measure condition, IMS Lecture-Notes Monographs Series, Papers in Honor of Rabi Bhattacharya, eds. K. Athreya, M. Majumdar, M. Puri, E. Waymire**41**, 65-80.**[8]**Feller, W. (1971): An Introduction to Probability Theory and its Applications, vol II, Wiley, NY. MR**42:5292****[9]**Folland, Gerald B. (1992): Fourier Analysis and its Applications, Brooks/Cole Publishing Company, Pacific Grove, California. MR**93f:42001****[10]**Foias, C. and R. Temam (1989): Grevey class regularity for the solutions of the Navier-Stokes equations,*J. Functional Analysis***87**359-369. MR**91a:35135****[11]**Fujita, H. and T. Kato (1964): On the Navier-Stokes initial value problem I,*Arch. Rational Mech. Anal.***16**, 269-315. MR**29:3774****[12]**Galdi, G.P. (1994):*An Introduction to the Mathematical Theory of the Navier-Stokes Equations,*Springer-Verlag, NY. MR**91a:35135****[13]**Harris, T. (1989):*The Theory of Branching Processes,*Dover Publ. Inc., NY. MR**29:664****[14]**Itô, K.(1965): Generalized uniform complex measures in the Hilbertian metric space with the application to the Feynman integral,*Proc. Fifth Berkeley Symp. Math. Stat. Probab. II,*145-161. MR**35:7359****[15]**Kato, T. (1984): Strong solutions of the Navier-Stokes equations in with applications to weak solutions,*Math. Z.***187**471-480. MR**86b:35171****[16]**Lemarié-Rieusset, P.G. (2000): Une remarque sur l'analyticité des soutions milds des équations de Navier-Stokes dans ,*C.R. Acad. Sci. Paris,*t.330, Série 1, 183-186. MR**2001c:35190****[17]**LeJan, Y. and A.S. Sznitman (1997): Stochastic cascades and 3-dimensional Navier-Stokes equations,*Prob. Theory and Rel. Fields***109**343-366. MR**98j:35144****[18]**McKean, H.P. (1975): Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov,*Comm.Pure. Appl.Math.***28**323-331. MR**53:4262****[19]**Montgomery-Smith, S. (2001): Finite time blow up for a Navier-Stokes like equation,*Proc. A.M.S.***129**, 3025-3029. MR**2002d:35164****[20]**Orum, Chris (2003): Ph.D. Thesis, Oregon State University.**[21]**Temam, R. (1995):*Navier Stokes Equations and Nonlinear Functional Analysis,*SIAM, Philadelphia, PA. MR**96e:35136****[22]**Woyczynski,W., P. Biler, and T. Funaki (1998): Fractal Burgers equations,*J. Diff. Equations***148**, 9-46. MR**99g:35111****[23]**Yamazato, M. (1978): Unimodality of infinitely divisible distribution functions of class L, Ann. Probab. 6, no. 4, 523-531. MR**58:2976**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
35Q30,
76D05,
60J80,
76M35

Retrieve articles in all journals with MSC (2000): 35Q30, 76D05, 60J80, 76M35

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