Homogenization of Incompressible Euler Equations

Abstract

In this paper, we perform a nonlinear multiscale analysis for incompressible Euler equations with rapidly oscillating initial data. The initial condition for velocity field is assumed to have two scales. The fast scale velocity component is periodic and is of order one. One of the important questions is how the two-scale velocity structure propagates in time and whether nonlinear interaction will generate more scales dynamically. By using a Lagrangian framework to describe the propagation of small scale solution, we show that the two-scale structure is preserved dynamically. Moreover, we derive a well-posed homogenized equation for the incompressible Euler equations. Preliminary numerical experiments are presented to demonstrate that the homogenized equation captures the correct averaged solution of the incompressible Euler equation.