On Discrete Models of the Euler Equation

Abstract

We consider two discrete models for the Euler equation describing incompressible fluid dynamics. These models are infinite coupled systems of ODEs for the functions $u_j$ which can be thought of as wavelet coefficients of the fluid velocity. The first model has been proposed and studied by Katz and Pavlovic. The second has been recently discussed by Waleffe and goes back to Obukhov studies of the energy cascade in developed turbulence. These are the only basic models of this type satisfying some natural scaling and conservation conditions. We prove that the Katz-Pavlovic model leads to finite time blowup for any initial datum, while the Obukhov model has a global solution for any sufficiently smooth initial datum.