Quasilinear Theory of the 2D Euler Equation

We develop a quasilinear theory of the 2D Euler equation and derive an integrodifferential equation for the evolution of the coarse-grained vorticity $\overline{\omega }(\mathbf{r},t)$. This equation respects all of the invariance properties of the Euler equation and conserves angular momentum in a circular domain and linear impulse in a channel. We show under which hypothesis we can derive an $H$ theorem for the Fermi-Dirac entropy and make the connection with statistical theories of 2D turbulence.