Integrable Structures for 2D Euler Equations of Incompressible Inviscid Fluids

The governing equation of turbulence, that we are interested in, is the incompressible 2D Navier– Stokes equation under periodic boundary conditions. We are particularly interested in investigating the dynamics of 2D Navier–Stokes equation in the infinite Reynolds number limit and of 2D Euler equation. Our approach is different from many other studies on 2D Navier–Stokes equation in which one starts with Stokes equation to prove results on 2D Navier–Stokes equation for small Reynolds number. In our studies, we start with 2D Euler equation and view 2D Navier–Stokes equation for large Reynolds number as a (singular) perturbation of 2D Euler equation. 2D Euler equation is a Hamiltonian system with infinitely many Casimirs. To understand the nature of turbulence, we start with investigating the hyperbolic structure of 2D Euler equation. We are especially interested in investigating the possible homoclinic structures. In [1], we studied a linearized 2D Euler equation at a fixed point. The linear system decouples into infinitely many one-dimensional invariant subsystems. The essential spectrum of each invariant subsystem is a band of continuous spectrum on the imaginary axis. Only finitely many of these invariant subsystems have point spectra. The point spectra can be computed through continued fractions. Examples show that there are indeed eigenvalues with positive and negative real parts. Thus, there is linear hyperbolicity. In [2] and [3], a Lax pair and a Bäcklund–Darboux transformation were found for the 2D Euler equation. Typically, Bäcklund–Darboux transformation can be used to generate homoclinic orbits [4]. The 2D Euler equation can be written in the vorticity form,