  Walter Craig's seminal works on the water-waves problem established the
importance of several exact identities: Zakharov's hamiltonian formulation,
shape derivative formula for the Dirichlet-to-Neumann operator, normal forms
transformations. In this paper, we introduce several identities for the
Hele-Shaw equation which are inspired by his nonlinear approach. Firstly, we
study convex changes of unknowns and obtain a large class of strong Lyapounov
functions; in addition to be non-increasing, these Lyapounov functions are
convex functions of time. The analysis relies on a simple elliptic formulation
of the Hele-Shaw equation, which is of independent interest. Then we study the
role of convexity to control the spatial derivatives of the solutions. We
consider the evolution equation for the Rayleigh-Taylor coefficient $a$ (this
is a positive function proportional to the opposite of the normal derivative of
the pressure at the free surface). Inspired by the study of entropies for
elliptic or parabolic equations, we consider the special function
$\varphi(x)=x\log x$ and find that $\varphi(1/\sqrt{a})$ is a sub-solution of a
well-posed equation. As an application, we give another proof of the maximum
principe for $1/a$ obtained with Nicolas Meunier and Didier Smets.
