By Sergei Kuksin, Armen Shirikyan
This ebook is devoted to the mathematical examine of two-dimensional statistical hydrodynamics and turbulence, defined via the second Navier–Stokes procedure with a random strength. The authors' major aim is to justify the statistical homes of a fluid's pace box u(t,x) that physicists suppose of their paintings. They conscientiously end up that u(t,x) converges, as time grows, to a statistical equilibrium, self reliant of preliminary facts. They use this to review ergodic homes of u(t,x) – proving, specifically, that observables f(u(t,.)) fulfill the powerful legislation of enormous numbers and significant restrict theorem. in addition they speak about the inviscid restrict while viscosity is going to 0, normalising the strength in order that the power of suggestions remains consistent, whereas their Reynolds numbers develop to infinity. They exhibit that then the statistical equilibria converge to invariant measures of the second Euler equation and research those measures. The tools practice to different nonlinear PDEs perturbed by means of random forces.
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Extra resources for Mathematics of Two-Dimensional Turbulence (Cambridge Tracts in Mathematics)
The idea of coupling goes back to Doeblin [Doe38; Doe40], but its systematic application in the theory of stochastic processes started much later; see the papers [Har55; Pit74] by Harris and Pitman. 24 was often used by Dobrushin [Dob68; Dob74] in his celebrated work on Gibbs systems. , see the books of Revuz [Rev84], Kifer [Kif86], Karatzas and Shreve [KS91], and L. Arnold [Arn98]. Our presentation of this subject essentially follows one of these references. 2 Two-dimensional Navier–Stokes equations In this chapter we present some well-known results on the 2D Navier–Stokes equations.
We shall denote by D the space of R2 -valued distributions on (0, T ) × T2 . 10 A pair of functions (u, p) is called a solution of Eq. 1) holds in D . 3). 1) is not a system of evolution equations in the sense that it does not contain the time derivative of the unknown function p. However, it is possible to exclude the pressure from the problem in question and to obtain a nonlocal nonlinear PDE which can be regarded as an evolution equation in a Hilbert space. 1). 19) where B(u) = B(u, u). 19).
7) holds for uj ∈ Vj , then F extends to an r-linear continuous map from X1 × · · · × Xr to Y . The proof of this lemma is straightforward. To formulate the second result, we introduce some notation. For any sufficiently regular vector fields u and v on the torus T2 , we set Lu = − u, B(u, v) = u, ∇ v . 8) For the periodic boundary conditions, Lu equals − u, but this is not the case for bounded domains and Dirichlet boundary conditions. It is easy to check with the help of a Fourier expansion that L is a self-adjoint operator in L2σ with domain D(L) = Hσ2 .