Revista Matemática Iberoamericana


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Volume 3, Issue 3, 1987, pp. 275–310
DOI: 10.4171/RMI/51

Published online: 1987-12-31

The Relation Between the Porous Medium and the Eikonal Equations in Several Space Dimensions

Pierre-Louis Lions[1], Panagiotis E. Souganidis[2] and Juan Luis Vázquez[3]

(1) Université de Paris-Dauphine, Paris, France
(2) University of Chicago, USA
(3) Universidad Autónoma de Madrid, Spain

We study the relation between the porous medium equation $u_t = \Delta(u^m), \; m>1$, and the eikonal equation $\nu_t = |D\nu|^2$. Under quite general assumptions, we prove that the pressure and the interface of the solution of the Cauchy problem for the porous medium equation converge as $m \downarrow 1$ to the viscosity solution and the interface of the Cauchy problem for the eikonal equation. We also address the same questions for the case of the Dirichlet boundary value problem.

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Lions Pierre-Louis, Souganidis Panagiotis, Vázquez Juan Luis: The Relation Between the Porous Medium and the Eikonal Equations in Several Space Dimensions. Rev. Mat. Iberoamericana 3 (1987), 275-310. doi: 10.4171/RMI/51