This chapter solves the underdetermined, elliptic equation ∂ⱼQsuperscript jl = Usuperscript l and Qsuperscript jl = Qsuperscript lj (Equation 1069) in order to eliminate the error term in the parametrix. For the proof of the Main Lemma, estimates for Q and the material derivative as well as its spatial derivatives are derived. The chapter finds a solution to Equation (1069) with good transport properties by solving it via a Transport equation obtained by commuting the divergence operator with the material derivative. It concludes by showing the solutions, spatial derivative estimates, and material derivative estimates for the Transport-Elliptic equation, as well as cutting off the solution to the Transport-Elliptic equation.
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