# Constructing Continuous Solutions

# Constructing Continuous Solutions

This chapter demonstrates how the preceding construction, combined with a few estimates from Part V, can be used to prove the Main Lemma for continuous solutions. The first step is to mollify the velocity, followed by mollification of the stress. The lifespan is then chosen, preferring a small parameter to ensure that the first term in the parametrix for the High–High term is controlled. The chapter proceeds by discussing the bounds for the new stress and solving the divergence equation, along with the bounds for the corrections and finally, control of the energy increment. The equation for the energy increment includes a smooth vector field and involves bounding the error term.

*Keywords:*
mollification, Main Lemma, continuous solution, velocity, stress, divergence equation, correction, energy increment, smooth vector field

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