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

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]**Sergio A. Albeverio and Raphael J. Høegh-Krohn,*Mathematical theory of Feynman path integrals*, Lecture Notes in Mathematics, Vol. 523, Springer-Verlag, Berlin-New York, 1976. MR**0495901****[2]**N. Aronszajn and K. T. Smith,*Theory of Bessel potentials. I*, Ann. Inst. Fourier (Grenoble)**11**(1961), 385–475 (English, with French summary). MR**0143935****[3]**R. N. Bhattacharya and R. Ranga Rao,*Normal approximation and asymptotic expansions*, John Wiley & Sons, New York-London-Sydney, 1976. Wiley Series in Probability and Mathematical Statistics. MR**0436272****[4]**Marco Cannone and Fabrice Planchon,*On the regularity of the bilinear term for solutions to the incompressible Navier-Stokes equations*, Rev. Mat. Iberoamericana**16**(2000), no. 1, 1–16. MR**1768531**, 10.4171/RMI/268**[5]**Marco Cannone and Yves Meyer,*Littlewood-Paley decomposition and Navier-Stokes equations*, Methods Appl. Anal.**2**(1995), no. 3, 307–319. MR**1362019**, 10.4310/MAA.1995.v2.n3.a4**[6]**Zhi Min Chen and Zhouping Xin,*Homogeneity criterion for the Navier-Stokes equations in the whole spaces*, J. Math. Fluid Mech.**3**(2001), no. 2, 152–182. MR**1838955**, 10.1007/PL00000967**[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]**William Feller,*An introduction to probability theory and its applications. Vol. II.*, Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR**0270403****[9]**Gerald B. Folland,*Fourier analysis and its applications*, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992. MR**1145236****[10]**C. Foias and R. Temam,*Gevrey class regularity for the solutions of the Navier-Stokes equations*, J. Funct. Anal.**87**(1989), no. 2, 359–369. MR**1026858**, 10.1016/0022-1236(89)90015-3**[11]**Hiroshi Fujita and Tosio Kato,*On the Navier-Stokes initial value problem. I*, Arch. Rational Mech. Anal.**16**(1964), 269–315. MR**0166499****[12]**C. Foias and R. Temam,*Gevrey class regularity for the solutions of the Navier-Stokes equations*, J. Funct. Anal.**87**(1989), no. 2, 359–369. MR**1026858**, 10.1016/0022-1236(89)90015-3**[13]**Theodore E. Harris,*The theory of branching processes*, Die Grundlehren der Mathematischen Wissenschaften, Bd. 119, Springer-Verlag, Berlin; Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963. MR**0163361****[14]**Kiyoshi Itô,*Generalized uniform complex measures in the Hilbertian metric space with their application to the Feynman integral*, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Univ. California Press, Berkeley, Calif., 1967, pp. 145–161. MR**0216528****[15]**Tosio Kato,*Strong 𝐿^{𝑝}-solutions of the Navier-Stokes equation in 𝑅^{𝑚}, with applications to weak solutions*, Math. Z.**187**(1984), no. 4, 471–480. MR**760047**, 10.1007/BF01174182**[16]**Pierre Gilles Lemarié-Rieusset,*Une remarque sur l’analyticité des solutions milds des équations de Navier-Stokes dans 𝑅³*, C. R. Acad. Sci. Paris Sér. I Math.**330**(2000), no. 3, 183–186 (French, with English and French summaries). MR**1748305**, 10.1016/S0764-4442(00)00103-8**[17]**Y. Le Jan and A. S. Sznitman,*Stochastic cascades and 3-dimensional Navier-Stokes equations*, Probab. Theory Related Fields**109**(1997), no. 3, 343–366. MR**1481125**, 10.1007/s004400050135**[18]**H. P. McKean,*Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov*, Comm. Pure Appl. Math.**28**(1975), no. 3, 323–331. MR**0400428****[19]**Stephen Montgomery-Smith,*Finite time blow up for a Navier-Stokes like equation*, Proc. Amer. Math. Soc.**129**(2001), no. 10, 3025–3029. MR**1840108**, 10.1090/S0002-9939-01-06062-2**[20]**Orum, Chris (2003): Ph.D. Thesis, Oregon State University.**[21]**Roger Temam,*Navier-Stokes equations and nonlinear functional analysis*, 2nd ed., CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 66, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. MR**1318914****[22]**Piotr Biler, Tadahisa Funaki, and Wojbor A. Woyczynski,*Fractal Burgers equations*, J. Differential Equations**148**(1998), no. 1, 9–46. MR**1637513**, 10.1006/jdeq.1998.3458**[23]**Makoto Yamazato,*Unimodality of infinitely divisible distribution functions of class 𝐿*, Ann. Probab.**6**(1978), no. 4, 523–531. MR**0482941**

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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