Long time behaviour of a stochastic-Lagrangian particle system for the Navier-Stokes equations

Abstract. This paper is based on a formulation of the Navier-Stokes equations developed in arxiv:math.PR/0511067 (to appear in Commun. Pure Appl. Math), where the velocity field of a viscous incompressible fluid is written as the expected value of a stochastic process. In this paper, we take N copies of the above process (each based on independent Wiener processes), and replace the expected value with 1 N times the sum over these N copies. We prove that in two dimensions, this system has (time) global solutions with C initial data. Further, we show that as N → ∞ the system converges to the solution of Navier-Stokes equations on any finite interval [0, T ]. However for fixed N , we prove that this system retains roughly O( 1 N ) times it’s original energy t → ∞. For general flows, we only provide a lower bound to this effect. In the special case of shear flows, we compute the behaviour as t → ∞ explicitly.

[1]  Gautam Iyer.  A Stochastic Perturbation of Inviscid Flows , 2005, math/0505066.

[2]  P. Constantin,et al.  A stochastic Lagrangian representation of the three‐dimensional incompressible Navier‐Stokes equations , 2005, math/0511067.

[3]  A. Rollett,et al.  The Monte Carlo Method , 2004 .

[4]  Peter Constantin,et al.  Some Open Problems and Research Directions in the Mathematical Study of Fluid Dynamics , 2001 .

[5]  Hoon Kim,et al.  Monte Carlo Statistical Methods , 2000, Technometrics.

[6]  John Odentrantz,et al.  Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues , 2000, Technometrics.

[7]  P. Constantin.  An Eulerian-Lagrangian approach for incompressible fluids: Local theory , 2000, math/0004059.

[8]  Hantaek Bae.  Navier-Stokes equations , 1992 .

[9]  A. Sznitman.  Topics in propagation of chaos , 1991 .

[10]  École d'été de probabilités de Saint-Flour,et al.  Ecole d'été de probabilités de Saint-Flour XIX, 1989 , 1991 .

[11]  L. E. Fraenkel,et al.  NAVIER-STOKES EQUATIONS (Chicago Lectures in Mathematics) , 1990 .

[12]  Mtw,et al.  Stochastic flows and stochastic differential equations , 1990 .

[13]  国田 寛.  Stochastic flows and stochastic differential equations , 1990 .

[14]  Tosio Kato,et al.  Remarks on the breakdown of smooth solutions for the 3-D Euler equations , 1984 .

[15]  J. Holton.  Geophysical fluid dynamics. , 1983, Science.

[16]  Boris Rozovskii,et al.  Stochastic partial differential equations and diffusion processes , 1982 .

[17]  J. Marsden,et al.  A mathematical introduction to fluid mechanics , 1979 .

[18]  D. Vere-Jones.  Markov Chains , 1972, Nature.

[19]  H. J.,et al.  Hydrodynamics , 1924, Nature.