Abstract
In this chapter we shall construct several explicit solutions to the Hele-Shaw problem, more precisely, to the Polubarinova–Galin equation, starting with the classical ones of Polubarinova-Kochina [438], [439], Galin [199] and Saffman, Taylor [488], [489]. Some properties of polynomial and rational solutions will be discussed, and it will be proved that the property of the conformal map to the fluid domain of being a polynomial or a rational function is preserved under the time evolution. The same is true also when logarithmic singularities are allowed. From these properties one easily deduces local existence and uniqueness of solutions within such classes.
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