# Bounds for the Corrections

# Bounds for the Corrections

This chapter derives the bounds for the correction terms, starting with bounds for the velocity correction. Based on *V* of the form *V* = Δ x *W*, it introduces a proposition for estimating the spatial derivatives of *W*. Since the number of *W*subscript *I* supported at any given region of ℝ x ³ is bounded by a universal constant, it suffices to estimate *W*subscript *I* uniformly in *I*. For an individual wave, it is easy to see that the estimate will hold. During repeated differentiation, the derivative hits either the oscillatory factor, the phase direction, or the amplitude *w*subscript *I* or one of its derivatives. In any case, the largest cost happens when differentiating the phase function. The chapter also gives estimates for derivatives of the coarse scale material derivative of *W* and concludes with bounds for the pressure correction.

*Keywords:*
correction term, velocity correction, spatial derivative, oscillatory factor, phase direction, amplitude, phase function, pressure correction

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