The Eulerian Limit for 2D Statistical Hydrodynamics

We consider the 2D Navier–Stokes system, perturbed by a white in time random force, proportional to the square root of the viscosity. We prove that under the limit “time to infinity, viscosity to zero” each of its (random) solution converges in distribution to a non-trivial stationary process, formed by solutions of the (free) Euler equation, while the Reynolds number grows to infinity. We study the convergence and the limiting solutions.

[1]  L. Rogers.  Stochastic differential equations and diffusion processes: Nobuyuki Ikeda and Shinzo Watanabe North-Holland, Amsterdam, 1981, xiv + 464 pages, Dfl.175.00 , 1982 .

[2]  P. Hartman.  Ordinary Differential Equations , 1965 .

[3]  Armen Shirikyan,et al.  A Coupling Approach¶to Randomly Forced Nonlinear PDE's. I , 2001 .

[4]  L. Young,et al.  Ergodic Theory of Infinite Dimensional Systems¶with Applications to Dissipative Parabolic PDEs , 2002 .

[5]  R. M. Dudley,et al.  Real Analysis and Probability , 1989 .

[6]  Uniqueness of the Invariant Measure¶for a Stochastic PDE Driven by Degenerate Noise , 2000, nlin/0009028.

[7]  A. Shirikyan,et al.  Coupling approach to white-forced nonlinear PDEs , 2002 .

[8]  Jonathan C. Mattingly.  Exponential Convergence for the Stochastically Forced Navier-Stokes Equations and Other Partially Dissipative Dynamics , 2002 .

[9]  R. Robert,et al.  Statistical equilibrium states for two-dimensional flows , 1991, Journal of Fluid Mechanics.

[10]  David Montgomery,et al.  Two-Dimensional Turbulence , 2012 .

[11]  Jonathan C. Mattingly,et al.  Ergodicity for the Navier‐Stokes equation with degenerate random forcing: Finite‐dimensional approximation , 2001 .

[12]  Weinan E,et al.  Gibbsian Dynamics and Ergodicity¶for the Stochastically Forced Navier–Stokes Equation , 2001 .

[13]  A. Shirikyan.  Analyticity of solutions for randomly forced two-dimensional Navier-Stokes equations , 2002 .

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

[15]  J. Bricmont,et al.  Exponential Mixing of the 2D Stochastic Navier-Stokes Dynamics , 2000 .

[16]  A. Shirikyan,et al.  Stochastic Dissipative PDE's and Gibbs Measures , 2000 .

[17]  S. Kuksin.  Ergodic Theorems for 2D Statistical Hydrodynamics , 2002 .

[18]  R. Kraichnan.  Inertial Ranges in Two‐Dimensional Turbulence , 1967 .

[19]  R. Temam.  Navier-Stokes Equations , 1977 .

[20]  Jean-Yves Chemin,et al.  Perfect Incompressible Fluids , 1998 .

[21]  G. Gallavotti,et al.  Foundations of Fluid Dynamics , 2002 .

[22]  M. Vishik,et al.  Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type , 1999 .