# The Coarse Scale Flow and Commutator Estimates

# The Coarse Scale Flow and Commutator Estimates

This chapter derives estimates for the coarse scale flow and commutator. Instead of mollifying the velocity field in the time variable, it derives a Transport equation for *v*subscript Element and some estimates that will be necessary for the proof. Here the quadratic term arises from the failure of the nonlinearity to commute with the averaging. Commutator estimates are then derived. To observe cancellation in the quadratic term, the control over the higher-frequency part of *v* is used, and cancellation is obtained from the lower-frequency parts. It becomes clear that the commutator terms can be estimated using the control of only the derivatives of *v*. The chapter concludes by presenting the theorem for coarse scale flow estimates.

*Keywords:*
coarse scale flow, commutator estimate, Transport equation, commutator, commutator term

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