A Quasilinear Approximation for the Three-Dimensional Navier—Stokes System

Abstract

In this paper a modification of the 3-dimensional Navier-Stokes system which defines some system of quasilinear equations in Fourier space is considered. Properties of the obtained system and its finite-dimensional approximations are studied.

On some approximation of the 3D Euler system

“…Property (1) is obvious, properties (2) and (3) are proven in [1]. To prove property (4), we note that det…”

“…Here x = (x 1 , x 2 , x 3 ) ∈ R 3 , k = (k 1 , k 2 , k 3 ) ∈ R 3 andũ is pure imaginary and odd. Puttingũ(k, t) = iv(k, t) with v(−k, t) = −v(−k, t) we can write our approximation as a system of quasi-linear equations (see [1]):…”

“…In what follows we consider systems (1) and (2) for ν ≥ 0 satisfying, at t = 0, the incompressibility condition (2). As was shown in [1], any solution v(k, t) satisfies the incompressibility condition (2) for all t for which a solution exists.…”

An inviscid dyadic model of turbulence: The global attractor

“…1 discussed in [3,16]. Analysis of more general "shell" models and motivation in terms of turbulence modeling can be found in [2,8,9,14,20,21].…”

Untitled

“…Recently, a number of simpler models have been proposed and studied by several authors as a way to gain insight into the possible behavior of solutions to Euler and Navier-Stokes equations. Different models have been suggested by Katz and Pavlović [9], Friedlander and Pavlović [5], Dinaburg and Sinaǐ [3], and Waleffe [13]. Although these models are fairly drastic simplifications of the original problem, they do keep a few of the most important characteristic features of the Euler equation.…”