  This article is devoted to the study of the Hele-Shaw equation. We introduce
an approach inspired by the water-wave theory. Starting from a reduction to the
boundary, introducing the Dirichlet to Neumann operator and exploiting various
cancellations, we exhibit parabolic evolution equations for the horizontal and
vertical traces of the velocity on the free surface. This allows to
quasi-linearize the equations in a very simple way. By combining these exact
identities with convexity inequalities, we prove the existence of hidden
Lyapunov functions of different natures. We also deduce from these identities
and previous works on the water wave problem a simple proof of the
well-posedness of the Cauchy problem. The analysis contains two side results of
independent interest. Firstly, we give a principle to derive estimates for the
modulus of continuity of a PDE under general assumptions on the flow. Secondly
we prove a convexity inequality for the Dirichlet to Neumann operator.
