# Transport Estimates

# Transport Estimates

This chapter derives estimates for quantities which are transported by the coarse scale flow and for their derivatives. It first considers the phase functions which satisfy the Transport equation, with the goal of choosing the lifespan parameter τ sufficiently small so that all the phase functions which appear in the analysis can be guaranteed to remain nonstationary in the time interval, and so that the Stress equation can be solved. In order for these requirements to be met, τ small enough is chosen so that the gradients of the phase functions do not depart significantly from their initial configurations. The chapter presents a proposition that bounds the separation of the phase gradients from their initial values in terms of *b* (*b* is less than or equal to 1, a form related to τ). Finally, it gathers estimates for relative velocity and relative acceleration.

*Keywords:*
coarse scale flow, phase function, Transport equation, lifespan parameter, Stress equation, phase gradient, relative velocity, relative acceleration

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