# The Main Iteration Lemma

# The Main Iteration Lemma

This chapter properly formalizes the Main Lemma, first by discussing the frequency energy levels for the Euler-Reynolds equations. Here the bounds are all consistent with the symmetries of the Euler equations, and the scaling symmetry is reflected by dimensional analysis. The chapter proceeds by making assumptions that are consistent with the Galilean invariance of the Euler equations and the Euler-Reynolds equations. If (*v*, *p*, *R*) solve the Euler-Reynolds equations, then a new solution to Euler-Reynolds with the same frequency energy levels can be obtained. The chapter also states the Main Lemma, taking into account dimensional analysis, energy regularity, and Onsager's conjecture. Finally, it introduces the main theorem (Theorem 10.1), which states that there exists a nonzero solution to the Euler equations with compact support in time.

*Keywords:*
frequency energy levels, Euler-Reynolds equations, scaling symmetry, dimensional analysis, Euler equations, Main Lemma, energy regularity, Onsager's conjecture, nonzero solution

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