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This article is devoted to the study of the Hele-Shaw equation. We introducean approach inspired by the water-wave theory. Starting from a reduction to theboundary, introducing the Dirichlet to Neumann operator and exploiting variouscancellations, we exhibit parabolic evolution equations for the horizontal andvertical traces of the velocity on the free surface. This allows toquasi-linearize the equations in a very simple way. By combining these exactidentities with convexity inequalities, we prove the existence of hiddenLyapunov functions of different natures. We also deduce from these identitiesand previous works on the water wave problem a simple proof of thewell-posedness of the Cauchy problem. The analysis contains two side results ofindependent interest. Firstly, we give a principle to derive estimates for themodulus of continuity of a PDE under general assumptions on the flow. Secondlywe prove a convexity inequality for the Dirichlet to Neumann operator.

Lyapunov functions, Identities and the Cauchy problem for the Hele-Shaw equation