# The Euler-Reynolds System

# The Euler-Reynolds System

This chapter provides a background on the Euler-Reynolds system, starting with some of the underlying philosophy behind the argument. It describes low frequency parts and ensemble averages of Euler flows and shows that the average of any family of solutions to Euler will be a solution of the Euler-Reynolds equations. It explains how the most relevant type of averaging to convex integration arises during the operation of taking weak limits, which can be regarded as an averaging process. The chapter proceeds by focusing on weak limits of Euler flows and the hierarchy of frequencies, concluding with a discussion of the method of convex integration and the *h*-principle for weak limits. The method inherently proves that weak solutions to Euler may fail to be solutions.

*Keywords:*
weak solution, Euler flow, Euler-Reynolds equations, convex integration, weak limit, frequencies, h-principle

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