Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation

被引:142
作者
Weinan E. [1 ,2 ]
Mattingly J.C. [3 ]
Sinai Y. [4 ,5 ]
机构
[1] Dept. of Math./Prog. in Appl./C., Princeton University, Princeton
[2] School of Mathematics, Peking University, Beijing
[3] Department of Mathematics, Stanford University, Stanford
[4] Department of Mathematics, Princeton University, Princeton
[5] Landau Inst. of Theoretical Physics, Moscow
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D O I
10.1007/s002201224083
中图分类号
学科分类号
摘要
We study stationary measures for the two-dimensional Navier-Stokes equation with periodic boundary condition and random forcing. We prove uniqueness of the stationary measure under the condition that all "determining modes" are forced. The main idea behind the proof is to study the Gibbsian dynamics of the low modes obtained by representing the high modes as functionals of the time-history of the low modes.
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页码:83 / 106
页数:23
相关论文
共 20 条
[1]  
Bricmont J., Kupiainen A., Lefevere R.
[2]  
Crauel H., Debussche A., Flandoli F., Random attractors, J. Dynam. Diff. Eqs., 9, 2, pp. 307-341, (1997)
[3]  
Constantin P., Foias C., Navier-Stokes Equations, (1988)
[4]  
Mattingly J.C., Ergodicity for the Navier-Stokes Equation with Degenerate Random Forcing: Finite Dimensional Approximation
[5]  
Eckmann J.P., Hairer M., Uniqueness of the Invariant Measure for a Stochastic PDE Driven by Degenerate Noise
[6]  
Eden A., Foias C., Nicolaenko B., Temam R., Exponential attractors for dissipative evolution equations, Research in Applied Mathematics, (1994)
[7]  
Ferrario B., Ergodic results for stochastic Navier-Stokes equation, Stochastics and Stochastics Rep., 60, 3-4, pp. 271-288, (1997)
[8]  
Flandoli F., Dissipativity and invariant measures for stochastic Navier-Stokes equations, NoDEA, 1, pp. 403-426, (1994)
[9]  
Flandoli F., Maslowski B., Ergodicity of the 2-D Navier-Stokes equation under random perturbations, Commun. Math. Phys., 171, pp. 119-141, (1995)
[10]  
Foias C., Manley O., Rosa R., Temam R., Navier-Stokes Equations and Turbulence