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

## TL;DR

This paper develops a novel approach inspired by water-wave theory to analyze the Hele-Shaw equation, introducing Lyapunov functions and identities that simplify the proof of well-posedness and provide new estimates.

## Contribution

It introduces a new method using identities and convexity inequalities to study the Hele-Shaw equation, leading to simplified proofs and new estimates.

## Key findings

- Existence of hidden Lyapunov functions for the Hele-Shaw equation.
- A simple proof of the well-posedness of the Cauchy problem.
- New principles for estimating the modulus of continuity of PDEs.

## Abstract

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.