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Lyapunov Functions, Identities and the Cauchy Problem for the Hele–Shaw Equation

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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 natural 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 and give applications of a convexity inequality for the Dirichlet to Neumann operator.

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Notes

  1. We do not need to (and in fact we cannot) apply directly the conclusion of Theorem 2.12 to infer that \(1-B(t,x)\ge {\underline{a}}\) for all time t. This is because we do not have a similar result for an iterative scheme converging to the solution.

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Acknowledgements

We thank the reviewers for their careful readings of the manuscript. T.A. and D.S. acknowledge the support of the SingFlows project, grant ANR-18-CE40-0027 of the French National Research Agency (ANR).

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Authors and Affiliations

  1. ENS Paris-Saclay, CNRS, Centre Borelli UMR9010, Université Paris-Saclay, avenue des Sciences, 91190, Gif-sur-Yvette, France

    Thomas Alazard

  2. Laboratoire de Mathématiques et Modélisation d’Evry (LaMME), UMR 8071, Université d’Evry Val d’Essonne, 23 Boulevard de France, 91037, Evry, France

    Nicolas Meunier

  3. Laboratoire Jacques-Louis Lions, Sorbonne Université, 4 Place Jussieu, 75005, Paris, France

    Didier Smets

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  1. Thomas Alazard
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Correspondence to Thomas Alazard.

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Communicated by A. Ionescu

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Alazard, T., Meunier, N. & Smets, D. Lyapunov Functions, Identities and the Cauchy Problem for the Hele–Shaw Equation. Commun. Math. Phys. 377, 1421–1459 (2020). https://doi.org/10.1007/s00220-020-03761-w

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