On 2D Euler Equations: III. A Line Model

To understand the nature of turbulence, we select 2D Euler equation under periodic boundary condition as our primary example to study. 2D Navier-Stokes equation at high Reynolds number is regarded as a singularly perturbed 2D Euler equation. That is, we are interested in studying the zero viscosity limit problem. To begin an infinite dimensional dynamical system study, we consider a simple fixed point and study the spectrum of the linear 2D Euler operator in [1]. The spectral theorem in l2 space is proved. As a corollary of the spectral theorem in l2 space, we will present the spectral theorem in Sobolev spaces in this article. Sobolev spaces are of more interest to us, since we are interested in understanding the invariant manifolds of 2D Euler equation at the fixed point. The main obstacle toward proving the invariant manifold theorem is that the nonlinear term is nonLipschitzian. In [2], a (dashed) line model is introduced to understand the invariant manifold structure of 2D Euler equation. At a special parameter value, the explicit expression of the invariant manifolds of the dashed line model can be calculated. The stable and unstable manifolds are two dimensional ellipsoidal surfaces, and together they form a lip-shape hyperbolic structure. Such structure appears to be robust with repsect to the parameter. In this article, we will prove the existence of invariant manifolds for the line model. Another more exciting development is the discovery of a Lax pair for 2D Euler equation [3]. From the Lax pair, we have obtained a Darboux transformation for the 2D Euler equation [4]. In principle, explicit expressions of the hyperbolic structures can be obtained from Darboux transformations [5].