Convexity and the Hele-Shaw equation
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.