Eulerian limit for 2D Navier-Stokes equation and damped/driven KdV equation as its model

We discuss the inviscid limits for the randomly forced 2D Navier-Stokes equation (NSE) and the damped/driven KdV equation. The former describes the space-periodic 2D turbulence in terms of a special class of solutions for the free Euler equation, and we view the latter as its model. We review and revise recent results on the inviscid limit for the perturbed KdV and use them to suggest a setup which could be used to make a next step in the study of the inviscid limit of 2D NSE. The proposed approach is based on an ergodic hypothesis for the flow of the 2D Euler equation on iso-integral surfaces. It invokes a Whitham equation for the 2D Navier-Stokes equation, written in terms of the ergodic measures. Dedicated to Vladimir Igorevich Arnold on his 70th birthday     

Eulerian limit for 2D Navier-Stokes equation and damped/driven KdV equation as its model

We discuss the inviscid limits for the randomly forced 2D Navier-Stokes equation (NSE) and the damped/driven KdV equation. The former describes the space-periodic 2D turbulence in terms of a special class of solutions for the free Euler equation, and we view the latter as its model. We review and revise recent results on the inviscid limit for the perturbed KdV and use them to suggest a setup which could be used to make a next step in the study of the inviscid limit of 2D NSE. The proposed approach is based on an ergodic hypothesis for the flow of the 2D Euler equation on iso-integral surfaces. It invokes a Whitham equation for the 2D Navier-Stokes equation, written in terms of the ergodic measures.     

Infinite-dimensional symplectic capacities and a squeezing theorem for Hamiltonian PDE's

We study partial differential equations of hamiltonian form and treat them as infinite-dimensional hamiltonian systems in a functional phase-space ofx-dependent functions. In this phase space we construct an invariant symplectic capacity and prove a version of Gromov's (non)squeezing theorem. We give an interpretation of the theorem in terms of the energy transition to high frequencies problem.     

KAM theory and the 3D Euler equation

We prove that the dynamical system defined by the hydrodynamical Euler equation on any closed Riemannian 3-manifold M   is not mixing in the C^k$C^k$ topology (k>4$k>4$ and non-integer) for any prescribed value of helicity and sufficiently large values of energy. This can be regarded as a 3D version of Nadirashvili's and Shnirelman's theorems showing the existence of wandering solutions for the 2D Euler equation. Moreover, we obtain an obstruction for the mixing under the Euler flow of C^k$C^k$-neighborhoods of divergence-free vectorfields on M  . On the way we construct a family of functionals on the space of divergence-free C¹$C^1$ vectorfields on the manifold, which are integrals of motion of the 3D Euler equation. Given a vectorfield these functionals measure the part of the manifold foliated by ergodic invariant tori of fixed isotopy types. We use the KAM theory to establish some continuity properties of these functionals in the C^k$C^k$-topology. This allows one to get a lower bound for the C^k$C^k$-distance between a divergence-free vectorfield (in particular, a steady solution) and a trajectory of the Euler flow.     

A Coupling Approach to Randomly Forced Nonlinear PDE's. II

 We consider a class of discrete time random dynamical systems and establish the exponential convergence of its trajectories to a unique stationary measure. The result obtained applies, in particular, to the 2D Navier–Stokes system and multidimensional complex Ginzburg–Landau equation with random kick-force. Received: 7 February 2002 / Accepted: 29 April 2002 Published online: 12 August 2002     

On the Zakharov–L’vov Stochastic Model for Wave Turbulence

Doklady Mathematics - In this paper we discuss a number of rigorous results in the stochastic model for wave turbulence due to Zakharov–L’vov. Namely, we consider the damped/driven...     

Remarks on the Balance Relations for the Two-Dimensional Navier–Stokes Equation with Random Forcing

We use the balance relations for the stationary in time solutions of the randomly forced 2D Navier-Stokes equations, found in [10], to study these solutions further. We show that the vorticity ξ(t,x) of a stationary solution has a finite exponential moment, and that for any the expectation of the integral of over the level-set , up to a constant factor equals the expectation of the integral of over the same set.     

On the parametrization of finite-gap solutions by frequency and wavenumber vectors and a theorem of I. krichever

We discuss the parametrization of real finite-gap solutions of an integrable equation by frequency and wavenumber vectors. This parametrization underlies perturbation and averaging theories for the finite-gap solutions. Out of the framework of integrable equations, the parametrization gives a convenient coordinate system on the corresponding manifold of Riemann curves.     

Quasi-linear elliptic differential equations for mappings of manifolds,II

We study questions related to the orientability of the infinite-dimensional moduli spaces formed by solutions of elliptic equations for mappings of manifolds. The principal result states that the first Stiefel–Whitney class of such a moduli space is given by the ℤ₂-spectral flow of the families of linearised operators. Under an additional compactness hypotheses, we develop elements of Morse–Bott theory and express the algebraic number of solutions of a non-homogeneous equation with a generic right-hand side in terms of the Euler characteristic of the space of solutions corresponding to the homogeneous equation. The applications of this include estimates for the number of homotopic maps with prescribed tension field and for the number of the perturbed pseudoholomorphic tori, sharpening some known results. Mathematics Subject Classifications (2000): 35J05, 58B15, 58E05, 58E20, 53D45