# American Institute of Mathematical Sciences

October  2005, 13(5): 1153-1186. doi: 10.3934/dcds.2005.13.1153

## Multiscale analysis in Lagrangian formulation for the 2-D incompressible Euler equation

 1 Department of Applied and Computational Mathematics, California Institute of Technology, Pasadena, CA 91125, United States 2 School of Mathematics and System Science, Shandong University, Jinan, 250100, China 3 Applied and Computational Mathematics, California Institute of Technology, Pasadena, CA 91125, United States

Received  November 2004 Revised  March 2005 Published  September 2005

We perform a systematic multiscale analysis for the 2-D incompressible Euler equation with rapidly oscillating initial data using a Lagrangian approach. The Lagrangian formulation enables us to capture the propagation of the multiscale solution in a natural way. By making an appropriate multiscale expansion in the vorticity-stream function formulation, we derive a well-posed homogenized equation for the Euler equation. Based on the multiscale analysis in the Lagrangian formulation, we also derive the corresponding multiscale analysis in the Eulerian formulation. Moreover, our multiscale analysis reveals some interesting structure for the Reynolds stress term, which provides a theoretical base for establishing systematic multiscale modeling of 2-D incompressible flow.
Citation: Thomas Y. Hou, Danping Yang, Hongyu Ran. Multiscale analysis in Lagrangian formulation for the 2-D incompressible Euler equation. Discrete and Continuous Dynamical Systems, 2005, 13 (5) : 1153-1186. doi: 10.3934/dcds.2005.13.1153
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