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We investigate into the local well-posedness of incompressible fluid equations. The method adopted in this lecture is different from mild solution, since the dissipation does not occur in the Euler flow. We will establish the claim by the regular approximation and Picard contraction.

The C-K-N theory, originated from the paper in 1982, study the singular behavior of weak solutions of the incompressible Navie-Stokes equations. In this lecture, we adapt the compactness method by Lin's paper to prove the whole theory.

Mild solution is an important object to investigate the strong solution of hydrodynamic flows with dissipation. The investigation will apply semigroup theory and some other functional tools.