A STOCHASTIC NAVIER-STOKES EQUATION FOR THE VORTICITY OF A TWO-DIMENSIONAL FLUID

被引:22
作者
Kotelenez, Peter [1 ]
机构
[1] Case Western Reserve Univ, Dept Math & Stat, Cleveland, OH 44106 USA
关键词
Stochastic partial differential equation; Navier-Stokes equation; random vortices; macroscopic limit; viscous diffusion; eddy diffusion; stochastic temperature field;
D O I
10.1214/aoap/1177004609
中图分类号
O21 [概率论与数理统计]; C8 [统计学];
学科分类号
020208 ; 070103 ; 0714 ;
摘要
The Navier-Stokes equation for the vorticity of a viscous and incompressible fluid in R-2 is analyzed as a macroscopic equation for an underlying microscopic model of randomly moving vortices. We consider N point vortices whose positions satisfy a stochastic ordinary differential equation on R-2N, where the fluctuation forces are state dependent and driven by Brownian sheets. The state dependence is modeled to yield a short correlation length e between the fluctuation forces of different vortices. The associated signed point measure-valued empirical process turns out to be a weak solution to a stochastic Navier-Stokes equation (SNSE) whose stochastic term is state dependent and small if e is small. Thereby we generalize the well known approach to the Euler equation to the viscous case. The solution is extended to a large class of signed measures conserving the total positive and negative vorticities, and it is shown to be a weak solution of the SNSE. For initial conditions in L-2(R-2, dr) the solutions are shown to live on the same space with continuous sample paths and an equation for the square of the L-2-norm is derived. Finally we obtain the macroscopic NSE as the correlation length epsilon --> 0 and N --> infinity (macroscopic limit), where we assume that the initial conditions are sums of N point measures. As a corollary to the above results we obtain the solution to a bilinear stochastic partial differential equation which can be interpreted as the temperature field in a stochastic flow.
引用
收藏
页码:1126 / 1160
页数:35
相关论文
共 35 条
[1]  
ALBEVERIO S., 1990, COMMUN MATH PHYS, V129, P432
[2]  
Bensoussan A., 1973, J FUNCT ANAL, V13, P195, DOI [10.1016/0022-1236(73)90045-1, DOI 10.1016/0022-1236(73)90045-1]
[3]   EQUILIBRIUM STATISTICS OF A VORTEX FILAMENT WITH APPLICATIONS [J].
CHORIN, AJ .
COMMUNICATIONS IN MATHEMATICAL PHYSICS, 1991, 141 (03) :619-631
[4]   Numerical study of slightly viscous flow [J].
Chorin, Alexandre Joel .
JOURNAL OF FLUID MECHANICS, 1973, 57 :785-796
[5]  
Davies E. B., 1980, ONE PARAMETER SEMIGR
[6]  
Dawson D. A., 1975, J MULTIVARIATE ANAL, V5, P1, DOI 10.1016/0047-259X(75)90054-8
[8]  
DUDLEY RM, 1989, REAL ANAL PROBABILIT
[9]  
Dynkin E.B., 1965, MARKOV PROCESSES, VI
[10]  
Ethier S.N., 2005, MARKOV PROCESSES CHA, Vsecond